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| Boilerpipe Text | Sine and cosine
General information
General definition
Fields of application
Trigonometry
,
Fourier series
,
Mathematical analysis
.Domain, codomain and image
Domain
real number
Image
In
mathematics
,
sine
and
cosine
are
trigonometric functions
of an
angle
. The sine and cosine of an
acute angle
are defined in the context of a
right triangle
: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the
triangle
(the
hypotenuse
), and the cosine is the
ratio
of the length of the adjacent leg to that of the
hypotenuse
. For an angle
, the sine and cosine functions are denoted as
and
.
The definitions of sine and cosine have been extended to any
real
value in terms of the lengths of certain line segments in a
unit circle
. More modern definitions express the sine and cosine as
infinite series
, or as the solutions of certain
differential equations
, allowing their extension to arbitrary positive and negative values and even to
complex numbers
.
The sine and cosine functions are commonly used to model
periodic
phenomena such as
sound
and
light waves
, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the
jyā
and
koṭi-jyā
functions used in
Indian astronomy
during the
Gupta period
.
Elementary descriptions
[
edit
]
Right-angled triangle definition
[
edit
]
For the angle
α
, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.
To define the sine and cosine of an acute angle
, start with a
right triangle
that contains an angle of measure
; in the accompanying figure, angle
in a right triangle
is the angle of interest. The three sides of the triangle are named as follows:
[
1
]
Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, and the cosine of the angle is equal to the length of the adjacent side divided by the length of the hypotenuse:
[
1
]
The other trigonometric functions of the angle can be defined similarly; for example, the
tangent
is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and cosine functions. The
reciprocal
of sine is cosecant, which gives the ratio of the hypotenuse length to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the hypotenuse length to that of the adjacent side. The cotangent function is the ratio between the adjacent and opposite sides, a reciprocal of a tangent function. These functions can be formulated as:
[
1
]
Special angle measures
[
edit
]
As stated, the values
and
appear to depend on the choice of a right triangle containing an angle of measure
. However, this is not the case as all such triangles are
similar
, and so the ratios are the same for each of them. For example, each
leg
of the 45-45-90 right triangle is 1 unit, and its hypotenuse is
; therefore,
.
[
2
]
The following table shows the special value of each input for both sine and cosine with the domain between
. The input in this table provides various unit systems such as degree, radian, and so on. The angles other than those five can be obtained by using a calculator.
[
3
]
[
4
]
Angle,
x
sin(
x
)
cos(
x
)
Degrees
Radians
Gradians
Turns
Exact
Decimal
Exact
Decimal
0°
0
0
0
0
1
1
30°
0.5
0.866
45°
0.707
0.707
60°
0.866
0.5
90°
1
1
0
0
Law of sines and cosines' illustration
The
law of sines
is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known.
[
5
]
Given a triangle
with sides
,
, and
, and angles opposite those sides
,
, and
, the law states,
This is equivalent to the equality of the first three expressions below:
where
is the triangle's
circumradius
.
The
law of cosines
is useful for computing the length of an unknown side if two other sides and an angle are known.
[
5
]
The law states,
In the case where
from which
, the resulting equation becomes the
Pythagorean theorem
.
[
6
]
The
cross product
and
dot product
are operations on two
vectors
in
Euclidean vector space
. The sine and cosine functions can be defined in terms of the cross product and dot product. If
and
are vectors, and
is the angle between
and
, then sine and cosine can be defined as:
[
7
]
[
8
]
Analytic descriptions
[
edit
]
Unit circle definition
[
edit
]
The sine and cosine functions may also be defined in a more general way by using
unit circle
, a circle of radius one centered at the origin
, formulated as the equation of
in the
Cartesian coordinate system
. Let a line through the origin intersect the unit circle, making an angle of
with the positive half of the
-
axis. The
-
and
-
coordinates of this point of intersection are equal to
and
, respectively; that is,
[
9
]
This definition is consistent with the right-angled triangle definition of sine and cosine when
because the length of the hypotenuse of the unit circle is always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the
-
coordinate. A similar argument can be made for the cosine function to show that the cosine of an angle when
, even under the new definition using the unit circle.
[
10
]
[
11
]
Graph of a function and its elementary properties
[
edit
]
Animation demonstrating how the sine function (in red) is graphed from the
y
-
coordinate (red dot) of a point on the
unit circle
(in green), at an angle of
θ
. The cosine (in blue) is the
x
-
coordinate.
Using the unit circle definition has the advantage of drawing the graph of sine and cosine functions. This can be done by rotating counterclockwise a point along the circumference of a circle, depending on the input
. In a sine function, if the input is
, the point is rotated counterclockwise and stopped exactly on the
-
axis. If
, the point is at the circle's halfway point. If
, the point returns to its origin. This results in both sine and cosine functions having the
range
between
.
[
12
]
Extending the angle to any real domain, the point rotated counterclockwise continuously. This can be done similarly for the cosine function as well, although the point is rotated initially from the
-
coordinate. In other words, both sine and cosine functions are
periodic
, meaning any angle added by the circle's circumference is the angle itself. Mathematically,
[
13
]
A function
is said to be
odd
if
, and is said to be
even
if
. The sine function is odd, whereas the cosine function is even.
[
14
]
Both sine and cosine functions are similar, with their difference being
shifted
by
. This phase shift can be expressed as
or
. This is distinct from the cofunction identities that follow below, which arise from right-triangle geometry and are not phase shifts:
[
15
]
The fixed point iteration
x
n
+1
= cos(
x
n
)
with initial value
x
0
= −1
converges to the Dottie number.
Zero is the only real
fixed point
of the sine function; in other words the only intersection of the sine function and the
identity function
is
. The only real fixed point of the cosine function is called the
Dottie number
. The Dottie number is the unique real root of the equation
. The decimal expansion of the Dottie number is approximately 0.739085.
[
16
]
Continuity and differentiation
[
edit
]
The quadrants of the unit circle and of sin(
x
), using the
Cartesian coordinate system
The sine and cosine functions are infinitely differentiable.
[
17
]
The derivative of sine is cosine, and the derivative of cosine is negative sine:
[
18
]
Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself.
[
17
]
These derivatives can be applied to the
first derivative test
, according to which the
monotonicity
of a function can be defined as the inequality of function's first derivative greater or less than equal to zero.
[
19
]
It can also be applied to
second derivative test
, according to which the
concavity
of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero.
[
20
]
The following table shows that both sine and cosine functions have concavity and monotonicity—the positive sign (
) denotes a graph is increasing (going upward) and the negative sign (
) is decreasing (going downward)—in certain intervals.
[
3
]
This information can be represented as a Cartesian coordinates system divided into four quadrants.
Quadrant
Angle
Sine
Cosine
Degrees
Radians
Sign
Monotony
Convexity
Sign
Monotony
Convexity
1st quadrant, I
Increasing
Concave
Decreasing
Concave
2nd quadrant, II
Decreasing
Concave
Decreasing
Convex
3rd quadrant, III
Decreasing
Convex
Increasing
Convex
4th quadrant, IV
Increasing
Convex
Increasing
Concave
Both sine and cosine functions can be defined by using differential equations. The pair of
is the solution
to the two-dimensional system of
differential equations
and
with the
initial conditions
and
. One could interpret the unit circle in the above definitions as defining the
phase space trajectory
of the differential equation with the given initial conditions. It can be interpreted as a phase space trajectory of the system of differential equations
and
starting from the initial conditions
and
.
[
citation needed
]
Integral and the usage in mensuration
[
edit
]
Their area under a curve can be obtained by using the
integral
with a certain bounded interval. Their antiderivatives are:
where
denotes the
constant of integration
.
[
21
]
These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval. For example, the
arc length
of the sine curve between
and
is
where
is the
incomplete elliptic integral of the second kind
with modulus
. It cannot be expressed using
elementary functions
.
[
22
]
In the case of a full period, its arc length is
where
is the
gamma function
and
is the
lemniscate constant
.
[
23
]
[
24
]
The usual principal values of the
arcsin(
x
)
and
arccos(
x
)
functions graphed on the Cartesian plane
The functions
and
(as well as those functions with the same function rule and domain whose codomain is a subset of
containing the interval
) are not bijective and therefore do not have inverse functions. For example,
, but also
,
. Sine's "inverse", called arcsine, can then be described not as a function but a relation (for example, all integer multiples of
would have an arcsine of zero). To define the inverse functions of sine and cosine, they must be restricted to their
principal branches
by restricting their domain and codomain; the standard functions used to define arcsine and arccosine are then
and
.
[
25
]
These are bijective and have inverses:
and
. Alternative notation is
for arcsine and
for arccosine. Using these definitions, one obtains the identity maps:
and
An acute angle
is given by:
where for some integer
,
By definition, both functions satisfy the equations:
and
According to
Pythagorean theorem
, the squared hypotenuse is the sum of two squared legs of a right triangle. Dividing the formula on both sides with squared hypotenuse resulting in the
Pythagorean trigonometric identity
, the sum of a squared sine and a squared cosine equals 1:
[
26
]
[
a
]
Sine and cosine satisfy the following double-angle formulas:
[
27
]
Sine function in blue and sine squared function in red. The
x
-
axis is in radians.
The cosine double angle formula implies that sin
2
and cos
2
are, themselves, shifted and scaled sine waves. Specifically,
[
28
]
The graph shows both sine and sine squared functions, with the sine in blue and the sine squared in red. Both graphs have the same shape but with different ranges of values and different periods. Sine squared has only positive values, but twice the number of periods.
[
citation needed
]
Series and polynomials
[
edit
]
This animation shows how including more and more terms in the partial sum of its Taylor series approaches a sine curve.
Both sine and cosine functions can be defined by using a
Taylor series
, a
power series
involving the higher-order derivatives. As mentioned in
§ Continuity and differentiation
, the
derivative
of sine is cosine and the derivative of cosine is the negative of sine. This means the successive derivatives of
are
,
,
,
, continuing to repeat those four functions. The
-
th derivative, evaluated at the point 0:
where the superscript represents repeated differentiation. This implies the following Taylor series expansion at
. One can then use the theory of
Taylor series
to show that the following identities hold for all
real numbers
—where
is the angle in radians.
[
29
]
More generally, for all
complex numbers
:
[
30
]
Taking the derivative of each term gives the Taylor series for cosine:
[
29
]
[
30
]
Both sine and cosine functions with multiple angles may appear as their
linear combination
, resulting in a polynomial. Such a polynomial is known as the
trigonometric polynomial
. The trigonometric polynomial's ample applications may be acquired in
its interpolation
, and its extension of a periodic function known as the
Fourier series
. Let
and
be any coefficients, then the trigonometric polynomial of a degree
—denoted as
—is defined as:
[
31
]
[
32
]
The
trigonometric series
can be defined similarly analogous to the trigonometric polynomial, its infinite inversion. Let
and
be any coefficients, then the trigonometric series can be defined as:
[
33
]
In the case of a Fourier series with a given integrable function
, the coefficients of a trigonometric series are:
[
34
]
Complex numbers relationship
[
edit
]
Complex exponential function definitions
[
edit
]
Both sine and cosine can be extended further via
complex number
, a set of numbers composed of both
real
and
imaginary numbers
. For real number
, the definition of both sine and cosine functions can be extended in a
complex plane
in terms of an
exponential function
as follows:
[
35
]
Alternatively, both functions can be defined in terms of
Euler's formula
:
[
35
]
When plotted on the
complex plane
, the function
for real values of
traces out the
unit circle
in the complex plane. Both sine and cosine functions may be simplified to the imaginary and real parts of
as:
[
36
]
When
for real values
and
, where
, both sine and cosine functions can be expressed in terms of real sines, cosines, and
hyperbolic functions
as:
[
37
]
Both functions
and
are the real and imaginary parts of
.
Sine and cosine are used to connect the real and imaginary parts of a
complex number
with its
polar coordinates
:
and the real and imaginary parts are
where
and
represent the magnitude and angle of the complex number
.
[
38
]
For any real number
, Euler's formula in terms of polar coordinates is stated as
.
[
35
]
Domain coloring
of sin(
z
) in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.
Vector field rendering of sin(
z
)
Applying the series definition of the sine and cosine to a complex argument,
z
, gives:
where sinh and cosh are the
hyperbolic sine and cosine
. These are
entire functions
.
It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument:
Partial fraction and product expansions of complex sine
[
edit
]
Using the partial fraction expansion technique in
complex analysis
, one can find that the infinite series
both converge and are equal to
. Similarly, one can show that
Using product expansion technique, one can derive
Usage of complex sine
[
edit
]
sin(
z
) is found in the
functional equation
for the
Gamma function
,
which in turn is found in the
functional equation
for the
Riemann zeta-function
,
As a
holomorphic function
, sin
z
is a 2D solution of
Laplace's equation
:
The complex sine function is also related to the level curves of
pendulums
.
[
how?
]
[
39
]
[
better source needed
]
Sine function in the complex plane
Real component
Imaginary component
Magnitude
Arcsine function in the complex plane
Real component
Imaginary component
Magnitude
The word
sine
is derived, indirectly, from the
Sanskrit
word
jyā
'bow-string' or more specifically its synonym
jīvá
(both adopted from
Ancient Greek
χορδή
'string; chord'), due to visual similarity between the arc of a circle with its corresponding chord and a bow with its string (see
jyā, koti-jyā and utkrama-jyā
;
sine
and
chord
are closely related in a circle of unit diameter, see
Ptolemy's Theorem
). This was
transliterated
in
Arabic
as
jība
, which is meaningless in that language and written as
jb
(
جب
). Since Arabic is written without short vowels,
jb
was interpreted as the
homograph
jayb
(
جيب
), which means 'bosom', 'pocket', or 'fold'.
[
40
]
[
41
]
When the Arabic texts of
Al-Battani
and
al-Khwārizmī
were translated into
Medieval Latin
in the 12th century by
Gerard of Cremona
, he used the Latin equivalent
sinus
(which also means 'bay' or 'fold', and more specifically 'the hanging fold of a
toga
over the breast').
[
42
]
[
43
]
[
44
]
Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage.
[
45
]
[
46
]
The English form
sine
was introduced in
Thomas Fale
's 1593
Horologiographia
.
[
47
]
The word
cosine
derives from an abbreviation of the Latin
complementi sinus
'sine of the
complementary angle
' as
cosinus
in
Edmund Gunter
's
Canon triangulorum
(1620), which also includes a similar definition of
cotangens
.
[
48
]
Quadrant from 1840s
Ottoman Turkey
with axes for looking up the sine and
versine
of angles
While the early study of trigonometry can be traced to antiquity, the
trigonometric functions
as they are in use today were developed in the medieval period. The
chord
function was discovered by
Hipparchus
of
Nicaea
(180–125 BCE) and
Ptolemy
of
Roman Egypt
(90–165 CE).
[
49
]
The sine and cosine functions are closely related to the
jyā
and
koṭi-jyā
functions used in
Indian astronomy
during the
Gupta period
(
Aryabhatiya
and
Surya Siddhanta
), via translation from Sanskrit to Arabic and then from Arabic to Latin.
[
42
]
[
50
]
All six trigonometric functions in current use were known in
Islamic mathematics
by the 9th century, as was the
law of sines
, used in
solving triangles
.
[
51
]
Al-Khwārizmī
(c. 780–850) produced tables of sines, cosines and tangents.
[
52
]
[
53
]
Muhammad ibn Jābir al-Harrānī al-Battānī
(853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.
[
53
]
In the early 17th-century, the French mathematician
Albert Girard
published the first use of the abbreviations
sin
,
cos
, and
tan
; these were further promulgated by Euler (see below). The
Opus palatinum de triangulis
of
Georg Joachim Rheticus
, a student of
Copernicus
, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.
In a paper published in 1682,
Leibniz
proved that sin
x
is not an
algebraic function
of
x
.
[
54
]
Roger Cotes
computed the derivative of sine in his
Harmonia Mensurarum
(1722).
[
55
]
Leonhard Euler
's
Introductio in analysin infinitorum
(1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "
Euler's formula
", as well as the near-modern abbreviations
sin.
,
cos.
,
tang.
,
cot.
,
sec.
, and
cosec.
[
42
]
Software implementations
[
edit
]
There is no standard algorithm for calculating sine and cosine.
IEEE 754
, the most widely used standard for the specification of reliable floating-point computation, does not address calculating trigonometric functions such as sine. The reason is that no efficient algorithm is known for computing sine and cosine with a specified accuracy, especially for large inputs.
[
56
]
Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g.
sin(10
22
)
.
A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest pre-calculated value, or
linearly interpolate
between the 2 closest values to approximate it. This allows results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage.
[
citation needed
]
The
CORDIC
algorithm is commonly used in scientific calculators.
The sine and cosine functions, along with other trigonometric functions, are widely available across programming languages and platforms. In computing, they are typically abbreviated to
sin
and
cos
.
Some CPU architectures have a built-in instruction for sine, including the Intel
x87
FPUs since the 80387.
In programming languages,
sin
and
cos
are typically either a built-in function or found within the language's standard math library. For example, the
C standard library
defines sine functions within
math.h
:
sin(
double
)
,
sinf(
float
)
, and
sinl(
long double
)
. The parameter of each is a
floating point
value, specifying the angle in radians. Each function returns the same
data type
as it accepts. Many other trigonometric functions are also defined in
math.h
, such as for cosine, arc sine, and hyperbolic sine (sinh). Similarly,
Python
defines
math.sin(x)
and
math.cos(x)
within the built-in
math
module. Complex sine and cosine functions are also available within the
cmath
module, e.g.
cmath.sin(z)
.
CPython
's math functions call the
C
math
library, and use a
double-precision floating-point format
.
Turns based implementations
[
edit
]
"sinpi" redirects here. For the township in Pingtung County, Taiwan, see
Xinpi
.
"cospi" redirects here. For the 17th-century Bolognese nobleman, see
Ferdinando Cospi
.
Some software libraries provide implementations of sine and cosine using the input angle in half-
turns
, a half-turn being an angle of 180 degrees or
radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.
[
57
]
[
58
]
These functions are called
sinpi
and
cospi
in MATLAB,
[
57
]
OpenCL
,
[
59
]
R,
[
58
]
Julia,
[
60
]
CUDA
,
[
61
]
and ARM.
[
62
]
For example,
sinpi(x)
would evaluate to
where
x
is expressed in half-turns, and consequently the final input to the function,
πx
can be interpreted in radians by
sin
.
SciPy
provides similar functions
sindg
and
cosdg
with input in degrees.
[
63
]
The accuracy advantage stems from the ability to perfectly represent key angles like full-turn, half-turn, and quarter-turn losslessly in binary floating-point or fixed-point. In contrast, representing
,
, and
in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits.
Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing modulo
involves inaccuracies in representing
.
For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to 4096 over one complete revolution.
[
64
]
If half-turns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to
would be incurred.
Āryabhaṭa's sine table
Bhaskara I's sine approximation formula
Discrete sine transform
Dixon elliptic functions
Euler's formula
Generalized trigonometry
Hyperbolic function
Lemniscate elliptic functions
Law of sines
List of periodic functions
List of trigonometric identities
Madhava series
Madhava's sine table
Optical sine theorem
Polar sine
—a generalization to vertex angles
Proofs of trigonometric identities
Sinc function
Sine and cosine transforms
Sine integral
Sine quadrant
Sine wave
Sine–Gordon equation
Sinusoidal model
SOH-CAH-TOA
Trigonometric functions
Trigonometric integral
^
a
b
c
Young (2017)
, p.
27
.
^
Young (2017)
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^
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68
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^
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^
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^
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165
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^
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, p. 199.
^
Vince (2023)
, p.
162
.
^
Adlaj (2012)
.
^
OEIS
sequence A105419 (Decimal expansion of the arc length of the sine or cosine curve for one full period.)
^
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, p. 365.
^
Young (2017)
, p.
99
.
^
Dennis G. Zill (2013).
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. Jones & Bartlett Publishers. p. 238.
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^
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^
Various sources credit the first use of
sinus
to either
Plato Tiburtinus
's 1116 translation of the
Astronomy
of
Al-Battani
Gerard of Cremona
's translation of the
Algebra
of
al-Khwārizmī
Robert of Chester
's 1145 translation of the tables of al-Khwārizmī
See
Merlet (2004)
. See
Maor (1998)
, Chapter 3, for an earlier etymology crediting Gerard. See
Katz (2008)
, p. 210.
^
Fale's book alternately uses the spellings "sine", "signe", or "sign".
Fale, Thomas (1593).
Horologiographia. The Art of Dialling: Teaching, an Easie and Perfect Way to make all Kindes of Dials ...
London: F. Kingston. p. 11, for example.
^
Gunter (1620)
.
^
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The Mathematics Teacher
.
58
(2):
141–
149.
doi
:
10.5951/MT.58.2.0141
.
JSTOR
27967990
.
^
Van Brummelen, Glen
(2009). "India".
The Mathematics of the Heavens and the Earth
. Princeton University Press. Ch. 3, pp. 94–134.
ISBN
978-0-691-12973-0
.
^
Gingerich, Owen (1986).
"Islamic Astronomy"
.
Scientific American
. Vol. 254. p. 74. Archived from
the original
on 2013-10-19
. Retrieved
2010-07-13
.
^
Jacques Sesiano, "Islamic mathematics", p. 157, in
Selin, Helaine
;
D'Ambrosio, Ubiratan
, eds. (2000).
Mathematics Across Cultures: The History of Non-western Mathematics
.
Springer Science+Business Media
.
ISBN
978-1-4020-0260-1
.
^
a
b
"trigonometry"
. Encyclopedia Britannica. 17 June 2024.
^
Nicolás Bourbaki (1994).
Elements of the History of Mathematics
. Springer.
ISBN
9783540647676
.
^
"
Why the sine has a simple derivative
Archived
2011-07-20 at the
Wayback Machine
", in
Historical Notes for Calculus Teachers
Archived
2011-07-20 at the
Wayback Machine
by
V. Frederick Rickey
Archived
2011-07-20 at the
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^
Zimmermann (2006)
.
^
a
b
"sinpi - Compute sin(X*pi) accurately"
.
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Archived
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. Retrieved
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.
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a
b
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.
www.rdocumentation.org
. Retrieved
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.
^
"sin, sincos, sinh, sinpi"
.
registry.khronos.org
. Retrieved
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.
^
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.
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. Retrieved
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.
^
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.
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. Archived from
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. Retrieved
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.
^
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. Retrieved
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.
^
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.
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. Retrieved
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.
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. Archived from
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on 2019-04-17
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.
Abramowitz, Milton
;
Stegun, Irene A.
(1970),
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
, New York:
Dover Publications
, Ninth printing
Adlaj, Semjon (2012),
"An Eloquent Formula for the Perimeter of an Ellipse"
(PDF)
,
American Mathematical Society
,
59
(8): 1097
Axler, Sheldon (2012),
Algebra and Trigonometry
,
John Wiley & Sons
,
ISBN
978-0470-58579-5
Bourchtein, Ludmila
; Bourchtein, Andrei (2022),
Theory of Infinite Sequences and Series
, Springer,
doi
:
10.1007/978-3-030-79431-6
,
ISBN
978-3-030-79431-6
Gunter, Edmund
(1620),
Canon triangulorum
Howie, John M. (2003),
Complex Analysis
, Springer Undergraduate Mathematics Series, Springer,
doi
:
10.1007/978-1-4471-0027-0
,
ISBN
978-1-4471-0027-0
Traupman, Ph.D., John C. (1966),
The New College Latin & English Dictionary
, Toronto: Bantam,
ISBN
0-553-27619-0
Katz, Victor J. (2008),
A History of Mathematics
(PDF)
(3rd ed.), Boston: Addison-Wesley,
The English word "sine" comes from a series of mistranslations of the Sanskrit
jyā-ardha
(chord-half). Āryabhaṭa frequently abbreviated this term to
jyā
or its synonym
jīvá
. When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an otherwise meaningless Arabic word
jiba
. But since Arabic is written without vowels, later writers interpreted the consonants
jb
as
jaib
, which means bosom or breast. In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word
sinus
, which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf.
Maor, Eli (1998),
Trigonometric Delights
,
Princeton University Press
,
ISBN
1-4008-4282-4
Merlet, Jean-Pierre (2004), "A Note on the History of the Trigonometric Functions", in Ceccarelli, Marco (ed.),
International Symposium on History of Machines and Mechanisms
, Springer,
doi
:
10.1007/1-4020-2204-2
,
ISBN
978-1-4020-2203-6
Merzbach, Uta C.
;
Boyer, Carl B.
(2011),
A History of Mathematics
(3rd ed.),
John Wiley & Sons
,
It was Robert of Chester's translation from Arabic that resulted in our word "sine". The Hindus had given the name jiva to the half-chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language, there is also the word jaib meaning "bay" or "inlet". When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence, he used the word sinus, the Latin word for "bay" or "inlet".
Plofker (2009),
Mathematics in India
,
Princeton University Press
Powell, Michael J. D.
(1981),
Approximation Theory and Methods
,
Cambridge University Press
,
ISBN
978-0-521-29514-7
Rudin, Walter
(1987),
Real and complex analysis
(3rd ed.), New York:
McGraw-Hill
,
ISBN
978-0-07-054234-1
,
MR
0924157
Smith, D. E. (1958) [1925],
History of Mathematics
, vol. I,
Dover Publications
,
ISBN
0-486-20429-4
Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007),
Calculus
(9th ed.),
Pearson Prentice Hall
,
ISBN
978-0131469686
Vince, John (2023),
Calculus for Computer Graphics
, Springer,
doi
:
10.1007/978-3-031-28117-4
,
ISBN
978-3-031-28117-4
Young, Cynthia
(2012),
Trigonometry
(3rd ed.), John Wiley & Sons,
ISBN
978-1-119-32113-2
——— (2017),
Trigonometry
(4th ed.), John Wiley & Sons,
ISBN
978-1-119-32113-2
Zimmermann, Paul (2006), "Can we trust floating-point numbers?",
Grand Challenges of Informatics
(PDF)
, p. 14/31
Zygmund, Antoni
(1968),
Trigonometric Series
(2nd, reprinted ed.),
Cambridge University Press
,
MR
0236587
Look up
sine
in Wiktionary, the free dictionary.
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- [1 Elementary descriptions](https://en.wikipedia.org/wiki/Sine_and_cosine#Elementary_descriptions)
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- [2\.1.3 Integral and the usage in mensuration](https://en.wikipedia.org/wiki/Sine_and_cosine#Integral_and_the_usage_in_mensuration)
- [2\.1.4 Inverse functions](https://en.wikipedia.org/wiki/Sine_and_cosine#Inverse_functions)
- [2\.1.5 Other identities](https://en.wikipedia.org/wiki/Sine_and_cosine#Other_identities)
- [2\.2 Series and polynomials](https://en.wikipedia.org/wiki/Sine_and_cosine#Series_and_polynomials)
- [3 Complex numbers relationship](https://en.wikipedia.org/wiki/Sine_and_cosine#Complex_numbers_relationship)
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- [3\.3 Complex arguments](https://en.wikipedia.org/wiki/Sine_and_cosine#Complex_arguments)
- [3\.3.1 Partial fraction and product expansions of complex sine](https://en.wikipedia.org/wiki/Sine_and_cosine#Partial_fraction_and_product_expansions_of_complex_sine)
- [3\.3.2 Usage of complex sine](https://en.wikipedia.org/wiki/Sine_and_cosine#Usage_of_complex_sine)
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# Sine and cosine
10 languages
- [বাংলা](https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%BE%E0%A6%87%E0%A6%A8_%E0%A6%93_%E0%A6%95%E0%A7%8B%E0%A6%B8%E0%A6%BE%E0%A6%87%E0%A6%A8 "সাইন ও কোসাইন – Bangla")
- [Deutsch](https://de.wikipedia.org/wiki/Sinus_und_Kosinus "Sinus und Kosinus – German")
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From Wikipedia, the free encyclopedia
Fundamental trigonometric functions
"Sine" and "Cosine" redirect here. For other uses, see [Sine (disambiguation)](https://en.wikipedia.org/wiki/Sine_\(disambiguation\) "Sine (disambiguation)") and [Cosine (disambiguation)](https://en.wikipedia.org/wiki/Cosine_\(disambiguation\) "Cosine (disambiguation)"). "Sine" is not to be confused with [Sign](https://en.wikipedia.org/wiki/Sign "Sign"), [Sign (mathematics)](https://en.wikipedia.org/wiki/Sign_\(mathematics\) "Sign (mathematics)") or the [sign function](https://en.wikipedia.org/wiki/Sign_function "Sign function").
| Sine and cosine | |
|---|---|
| [](https://en.wikipedia.org/wiki/File:Sine_cosine_one_period.svg) | |
| General information | |
| General definition | sin ( θ ) \= opposite hypotenuse cos ( θ ) \= adjacent hypotenuse {\\displaystyle {\\begin{aligned}&\\sin(\\theta )={\\frac {\\textrm {opposite}}{\\textrm {hypotenuse}}}\\\\\[8pt\]&\\cos(\\theta )={\\frac {\\textrm {adjacent}}{\\textrm {hypotenuse}}}\\\\\[8pt\]\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}&\\sin(\\theta )={\\frac {\\textrm {opposite}}{\\textrm {hypotenuse}}}\\\\\[8pt\]&\\cos(\\theta )={\\frac {\\textrm {adjacent}}{\\textrm {hypotenuse}}}\\\\\[8pt\]\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52374fd3474dfab1331993d6c170e9cac82f4a4a) |
In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), **sine** and **cosine** are [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions") of an [angle](https://en.wikipedia.org/wiki/Angle "Angle"). The sine and cosine of an [acute angle](https://en.wikipedia.org/wiki/Acute_angle "Acute angle") are defined in the context of a [right triangle](https://en.wikipedia.org/wiki/Right_triangle "Right triangle"): for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the [triangle](https://en.wikipedia.org/wiki/Triangle "Triangle") (the [hypotenuse](https://en.wikipedia.org/wiki/Hypotenuse "Hypotenuse")), and the cosine is the [ratio](https://en.wikipedia.org/wiki/Ratio "Ratio") of the length of the adjacent leg to that of the [hypotenuse](https://en.wikipedia.org/wiki/Hypotenuse "Hypotenuse"). For an angle θ {\\displaystyle \\theta } , the sine and cosine functions are denoted as sin ( θ ) {\\displaystyle \\sin(\\theta )}  and cos ( θ ) {\\displaystyle \\cos(\\theta )} .
The definitions of sine and cosine have been extended to any [real](https://en.wikipedia.org/wiki/Real_number "Real number") value in terms of the lengths of certain line segments in a [unit circle](https://en.wikipedia.org/wiki/Unit_circle "Unit circle"). More modern definitions express the sine and cosine as [infinite series](https://en.wikipedia.org/wiki/Series_\(mathematics\) "Series (mathematics)"), or as the solutions of certain [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation"), allowing their extension to arbitrary positive and negative values and even to [complex numbers](https://en.wikipedia.org/wiki/Complex_number "Complex number").
The sine and cosine functions are commonly used to model [periodic](https://en.wikipedia.org/wiki/Periodic_function "Periodic function") phenomena such as [sound](https://en.wikipedia.org/wiki/Sound "Sound") and [light waves](https://en.wikipedia.org/wiki/Light_waves "Light waves"), the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the [*jyā* and *koṭi-jyā*](https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81 "Jyā, koti-jyā and utkrama-jyā") functions used in [Indian astronomy](https://en.wikipedia.org/wiki/Indian_astronomy "Indian astronomy") during the [Gupta period](https://en.wikipedia.org/wiki/Gupta_period "Gupta period").
## Elementary descriptions
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=1 "Edit section: Elementary descriptions")\]
### Right-angled triangle definition
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=2 "Edit section: Right-angled triangle definition")\]
[](https://en.wikipedia.org/wiki/File:Trigono_sine_en2.svg)
For the angle *α*, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.
To define the sine and cosine of an acute angle α {\\displaystyle \\alpha } , start with a [right triangle](https://en.wikipedia.org/wiki/Right_triangle "Right triangle") that contains an angle of measure α {\\displaystyle \\alpha } ; in the accompanying figure, angle α {\\displaystyle \\alpha }  in a right triangle A B C {\\displaystyle ABC}  is the angle of interest. The three sides of the triangle are named as follows:[\[1\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]-1)
- The *opposite side* is the side opposite to the angle of interest; in this case, it is
a
{\\displaystyle a}
.
- The *hypotenuse* is the side opposite the right angle; in this case, it is
h
{\\displaystyle h}
. The hypotenuse is always the longest side of a right-angled triangle.
- The *adjacent side* is the remaining side; in this case, it is
b
{\\displaystyle b}
. It forms a side of (and is adjacent to) both the angle of interest and the right angle.
Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, and the cosine of the angle is equal to the length of the adjacent side divided by the length of the hypotenuse:[\[1\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]-1) sin ( α ) \= opposite hypotenuse , cos ( α ) \= adjacent hypotenuse . {\\displaystyle \\sin(\\alpha )={\\frac {\\text{opposite}}{\\text{hypotenuse}}},\\qquad \\cos(\\alpha )={\\frac {\\text{adjacent}}{\\text{hypotenuse}}}.} 
The other trigonometric functions of the angle can be defined similarly; for example, the [tangent](https://en.wikipedia.org/wiki/Trigonometric_functions#Right-angled_triangle_definitions "Trigonometric functions") is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and cosine functions. The [reciprocal](https://en.wikipedia.org/wiki/Multiplicative_inverse "Multiplicative inverse") of sine is cosecant, which gives the ratio of the hypotenuse length to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the hypotenuse length to that of the adjacent side. The cotangent function is the ratio between the adjacent and opposite sides, a reciprocal of a tangent function. These functions can be formulated as:[\[1\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]-1) tan ( θ ) \= sin ( θ ) cos ( θ ) \= opposite adjacent , cot ( θ ) \= 1 tan ( θ ) \= adjacent opposite , csc ( θ ) \= 1 sin ( θ ) \= hypotenuse opposite , sec ( θ ) \= 1 cos ( θ ) \= hypotenuse adjacent . {\\displaystyle {\\begin{aligned}\\tan(\\theta )&={\\frac {\\sin(\\theta )}{\\cos(\\theta )}}={\\frac {\\text{opposite}}{\\text{adjacent}}},\\\\\\cot(\\theta )&={\\frac {1}{\\tan(\\theta )}}={\\frac {\\text{adjacent}}{\\text{opposite}}},\\\\\\csc(\\theta )&={\\frac {1}{\\sin(\\theta )}}={\\frac {\\text{hypotenuse}}{\\text{opposite}}},\\\\\\sec(\\theta )&={\\frac {1}{\\cos(\\theta )}}={\\frac {\\textrm {hypotenuse}}{\\textrm {adjacent}}}.\\end{aligned}}} 
### Special angle measures
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=3 "Edit section: Special angle measures")\]
As stated, the values sin ( α ) {\\displaystyle \\sin(\\alpha )}  and cos ( α ) {\\displaystyle \\cos(\\alpha )}  appear to depend on the choice of a right triangle containing an angle of measure α {\\displaystyle \\alpha } . However, this is not the case as all such triangles are [similar](https://en.wikipedia.org/wiki/Similarity_\(geometry\) "Similarity (geometry)"), and so the ratios are the same for each of them. For example, each [leg](https://en.wikipedia.org/wiki/Catheti "Catheti") of the 45-45-90 right triangle is 1 unit, and its hypotenuse is 2 {\\displaystyle {\\sqrt {2}}} ; therefore, sin 45 ∘ \= cos 45 ∘ \= 2 2 {\\textstyle \\sin 45^{\\circ }=\\cos 45^{\\circ }={\\frac {\\sqrt {2}}{2}}} .[\[2\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA36_36]-2) The following table shows the special value of each input for both sine and cosine with the domain between 0 \< α \< π 2 {\\textstyle 0\<\\alpha \<{\\frac {\\pi }{2}}} . The input in this table provides various unit systems such as degree, radian, and so on. The angles other than those five can be obtained by using a calculator.[\[3\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200742-3)[\[4\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA37_37],_[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA78_78]-4)
| Angle, x | sin(*x*) | cos(*x*) | | | | | |
|---|---|---|---|---|---|---|---|
| [Degrees](https://en.wikipedia.org/wiki/Degree_\(angle\) "Degree (angle)") | [Radians](https://en.wikipedia.org/wiki/Radian "Radian") | [Gradians](https://en.wikipedia.org/wiki/Gradian "Gradian") | [Turns](https://en.wikipedia.org/wiki/Turn_\(geometry\) "Turn (geometry)") | Exact | Decimal | Exact | Decimal |
| 0° | 0 | 0 g {\\displaystyle 0^{g}}  | | | | | |
### Laws
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=4 "Edit section: Laws")\]
Main articles: [Law of sines](https://en.wikipedia.org/wiki/Law_of_sines "Law of sines") and [Law of cosines](https://en.wikipedia.org/wiki/Law_of_cosines "Law of cosines")
[](https://en.wikipedia.org/wiki/File:Law_of_sines_\(simple\).svg)
Law of sines and cosines' illustration
The [law of sines](https://en.wikipedia.org/wiki/Law_of_sines "Law of sines") is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known.[\[5\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]-5) Given a triangle A B C {\\displaystyle ABC}  with sides a {\\displaystyle a} , b {\\displaystyle b} , and c {\\displaystyle c} , and angles opposite those sides α {\\displaystyle \\alpha } , β {\\displaystyle \\beta } , and γ {\\displaystyle \\gamma } , the law states, sin α a \= sin β b \= sin γ c . {\\displaystyle {\\frac {\\sin \\alpha }{a}}={\\frac {\\sin \\beta }{b}}={\\frac {\\sin \\gamma }{c}}.}  This is equivalent to the equality of the first three expressions below: a sin α \= b sin β \= c sin γ \= 2 R , {\\displaystyle {\\frac {a}{\\sin \\alpha }}={\\frac {b}{\\sin \\beta }}={\\frac {c}{\\sin \\gamma }}=2R,}  where R {\\displaystyle R}  is the triangle's [circumradius](https://en.wikipedia.org/wiki/Circumcircle "Circumcircle").
The [law of cosines](https://en.wikipedia.org/wiki/Law_of_cosines "Law of cosines") is useful for computing the length of an unknown side if two other sides and an angle are known.[\[5\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]-5) The law states, a 2 \+ b 2 − 2 a b cos ( γ ) \= c 2 {\\displaystyle a^{2}+b^{2}-2ab\\cos(\\gamma )=c^{2}}  In the case where γ \= π / 2 {\\displaystyle \\gamma =\\pi /2}  from which cos ( γ ) \= 0 {\\displaystyle \\cos(\\gamma )=0} , the resulting equation becomes the [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem").[\[6\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA632_632]-6)
### Vector definition
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=5 "Edit section: Vector definition")\]
The [cross product](https://en.wikipedia.org/wiki/Cross_product "Cross product") and [dot product](https://en.wikipedia.org/wiki/Dot_product "Dot product") are operations on two [vectors](https://en.wikipedia.org/wiki/Vector_\(mathematics_and_physics\) "Vector (mathematics and physics)") in [Euclidean vector space](https://en.wikipedia.org/wiki/Euclidean_vector_space "Euclidean vector space"). The sine and cosine functions can be defined in terms of the cross product and dot product. If a {\\displaystyle \\mathbf {a} }  and b {\\displaystyle \\mathbf {b} }  are vectors, and θ {\\displaystyle \\theta }  is the angle between a {\\displaystyle \\mathbf {a} }  and b {\\displaystyle \\mathbf {b} } , then sine and cosine can be defined as:[\[7\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-7)[\[8\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-8) sin ( θ ) \= \| a × b \| \| a \| \| b \| , cos ( θ ) \= a ⋅ b \| a \| \| b \| . {\\displaystyle {\\begin{aligned}\\sin(\\theta )&={\\frac {\|\\mathbf {a} \\times \\mathbf {b} \|}{\|\\mathbf {a} \|\|\\mathbf {b} \|}},\\\\\\cos(\\theta )&={\\frac {\\mathbf {a} \\cdot \\mathbf {b} }{\|\\mathbf {a} \|\|\\mathbf {b} \|}}.\\end{aligned}}} 
## Analytic descriptions
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=6 "Edit section: Analytic descriptions")\]
### Unit circle definition
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=7 "Edit section: Unit circle definition")\]
The sine and cosine functions may also be defined in a more general way by using [unit circle](https://en.wikipedia.org/wiki/Unit_circle#Trigonometric_functions_on_the_unit_circle "Unit circle"), a circle of radius one centered at the origin ( 0 , 0 ) {\\displaystyle (0,0)} , formulated as the equation of x 2 \+ y 2 \= 1 {\\displaystyle x^{2}+y^{2}=1}  in the [Cartesian coordinate system](https://en.wikipedia.org/wiki/Cartesian_coordinate_system "Cartesian coordinate system"). Let a line through the origin intersect the unit circle, making an angle of θ {\\displaystyle \\theta }  with the positive half of the x {\\displaystyle x}  \-axis. The x {\\displaystyle x}  \- and y {\\displaystyle y}  \-coordinates of this point of intersection are equal to cos ( θ ) {\\displaystyle \\cos(\\theta )}  and sin ( θ ) {\\displaystyle \\sin(\\theta )} , respectively; that is,[\[9\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200741-9) sin ( θ ) \= y , cos ( θ ) \= x . {\\displaystyle \\sin(\\theta )=y,\\qquad \\cos(\\theta )=x.} 
This definition is consistent with the right-angled triangle definition of sine and cosine when 0 \< θ \< π 2 {\\textstyle 0\<\\theta \<{\\frac {\\pi }{2}}}  because the length of the hypotenuse of the unit circle is always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the y {\\displaystyle y}  \-coordinate. A similar argument can be made for the cosine function to show that the cosine of an angle when 0 \< θ \< π 2 {\\textstyle 0\<\\theta \<{\\frac {\\pi }{2}}} , even under the new definition using the unit circle.[\[10\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA68_68]-10)[\[11\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200747-11)
#### Graph of a function and its elementary properties
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=8 "Edit section: Graph of a function and its elementary properties")\]
[](https://en.wikipedia.org/wiki/File:Circle_cos_sin.gif)
Animation demonstrating how the sine function (in red) is graphed from the *y*\-coordinate (red dot) of a point on the [unit circle](https://en.wikipedia.org/wiki/Unit_circle "Unit circle") (in green), at an angle of *θ*. The cosine (in blue) is the *x*\-coordinate.
Using the unit circle definition has the advantage of drawing the graph of sine and cosine functions. This can be done by rotating counterclockwise a point along the circumference of a circle, depending on the input θ \> 0 {\\displaystyle \\theta \>0} . In a sine function, if the input is θ \= π 2 {\\textstyle \\theta ={\\frac {\\pi }{2}}} , the point is rotated counterclockwise and stopped exactly on the y {\\displaystyle y}  \-axis. If θ \= π {\\displaystyle \\theta =\\pi } , the point is at the circle's halfway point. If θ \= 2 π {\\displaystyle \\theta =2\\pi } , the point returns to its origin. This results in both sine and cosine functions having the [range](https://en.wikipedia.org/wiki/Range_of_a_function "Range of a function") between − 1 ≤ y ≤ 1 {\\displaystyle -1\\leq y\\leq 1} .[\[12\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200741–42-12)
Extending the angle to any real domain, the point rotated counterclockwise continuously. This can be done similarly for the cosine function as well, although the point is rotated initially from the y {\\displaystyle y}  \-coordinate. In other words, both sine and cosine functions are [periodic](https://en.wikipedia.org/wiki/Periodic_function "Periodic function"), meaning any angle added by the circle's circumference is the angle itself. Mathematically,[\[13\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200741,_43-13) sin ( θ \+ 2 π ) \= sin ( θ ) , cos ( θ \+ 2 π ) \= cos ( θ ) . {\\displaystyle \\sin(\\theta +2\\pi )=\\sin(\\theta ),\\qquad \\cos(\\theta +2\\pi )=\\cos(\\theta ).} 
A function f {\\displaystyle f}  is said to be [odd](https://en.wikipedia.org/wiki/Odd_function "Odd function") if f ( − x ) \= − f ( x ) {\\displaystyle f(-x)=-f(x)} , and is said to be [even](https://en.wikipedia.org/wiki/Even_function "Even function") if f ( − x ) \= f ( x ) {\\displaystyle f(-x)=f(x)} . The sine function is odd, whereas the cosine function is even.[\[14\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2012[httpsbooksgooglecombooksidOMrcN0a3LxICpgRA1-PA165_165]-14) Both sine and cosine functions are similar, with their difference being [shifted](https://en.wikipedia.org/wiki/Translation_\(geometry\) "Translation (geometry)") by π 2 {\\textstyle {\\frac {\\pi }{2}}} . This phase shift can be expressed as cos ( θ ) \= sin ( θ \+ π 2 ) {\\textstyle \\cos(\\theta )=\\sin \\left(\\theta +{\\frac {\\pi }{2}}\\right)}  or sin ( θ ) \= cos ( θ − π 2 ) {\\textstyle \\sin(\\theta )=\\cos \\left(\\theta -{\\frac {\\pi }{2}}\\right)} . This is distinct from the cofunction identities that follow below, which arise from right-triangle geometry and are not phase shifts: [\[15\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200742,_47-15) sin ( θ ) \= cos ( π 2 − θ ) , cos ( θ ) \= sin ( π 2 − θ ) . {\\displaystyle {\\begin{aligned}\\sin(\\theta )&=\\cos \\left({\\frac {\\pi }{2}}-\\theta \\right),\\\\\\cos(\\theta )&=\\sin \\left({\\frac {\\pi }{2}}-\\theta \\right).\\end{aligned}}} 
[](https://en.wikipedia.org/wiki/File:Cosine_fixed_point.svg)
The fixed point iteration *x**n*\+1 = cos(*xn*) with initial value *x*0 = −1 converges to the Dottie number.
Zero is the only real [fixed point](https://en.wikipedia.org/wiki/Fixed_point_\(mathematics\) "Fixed point (mathematics)") of the sine function; in other words the only intersection of the sine function and the [identity function](https://en.wikipedia.org/wiki/Identity_function "Identity function") is sin ( 0 ) \= 0 {\\displaystyle \\sin(0)=0} . The only real fixed point of the cosine function is called the [Dottie number](https://en.wikipedia.org/wiki/Dottie_number "Dottie number"). The Dottie number is the unique real root of the equation cos ( x ) \= x {\\displaystyle \\cos(x)=x} . The decimal expansion of the Dottie number is approximately 0.739085.[\[16\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-16)
#### Continuity and differentiation
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=9 "Edit section: Continuity and differentiation")\]
Main article: [Differentiation of trigonometric functions](https://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions "Differentiation of trigonometric functions")
[](https://en.wikipedia.org/wiki/File:Sine_quads_01_Pengo.svg)
The quadrants of the unit circle and of sin(*x*), using the [Cartesian coordinate system](https://en.wikipedia.org/wiki/Cartesian_coordinate_system "Cartesian coordinate system")
The sine and cosine functions are infinitely differentiable.[\[17\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]-17) The derivative of sine is cosine, and the derivative of cosine is negative sine:[\[18\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007115-18) d d x sin ( x ) \= cos ( x ) , d d x cos ( x ) \= − sin ( x ) . {\\displaystyle {\\frac {d}{dx}}\\sin(x)=\\cos(x),\\qquad {\\frac {d}{dx}}\\cos(x)=-\\sin(x).}  Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself.[\[17\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]-17) These derivatives can be applied to the [first derivative test](https://en.wikipedia.org/wiki/First_derivative_test "First derivative test"), according to which the [monotonicity](https://en.wikipedia.org/wiki/Monotone_function "Monotone function") of a function can be defined as the inequality of function's first derivative greater or less than equal to zero.[\[19\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007155-19) It can also be applied to [second derivative test](https://en.wikipedia.org/wiki/Second_derivative_test "Second derivative test"), according to which the [concavity](https://en.wikipedia.org/wiki/Concave_function "Concave function") of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero.[\[20\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007157-20) The following table shows that both sine and cosine functions have concavity and monotonicity—the positive sign (\+ {\\displaystyle +} ) denotes a graph is increasing (going upward) and the negative sign (− {\\displaystyle -} ) is decreasing (going downward)—in certain intervals.[\[3\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200742-3) This information can be represented as a Cartesian coordinates system divided into four quadrants.
| [Quadrant](https://en.wikipedia.org/wiki/Cartesian_coordinate_system#Quadrants_and_octants "Cartesian coordinate system") | Angle | Sine | Cosine | | | | |
|---|---|---|---|---|---|---|---|
| [Degrees](https://en.wikipedia.org/wiki/Degree_\(angle\) "Degree (angle)") | [Radians](https://en.wikipedia.org/wiki/Radian "Radian") | [Sign](https://en.wikipedia.org/wiki/Sign_\(mathematics\) "Sign (mathematics)") | [Monotony](https://en.wikipedia.org/wiki/Monotonic_function "Monotonic function") | [Convexity](https://en.wikipedia.org/wiki/Convex_function "Convex function") | [Sign](https://en.wikipedia.org/wiki/Sign_\(mathematics\) "Sign (mathematics)") | [Monotony](https://en.wikipedia.org/wiki/Monotonic_function "Monotonic function") | [Convexity](https://en.wikipedia.org/wiki/Convex_function "Convex function") |
| 1st quadrant, I | 0 ∘ \< x \< 90 ∘ {\\displaystyle 0^{\\circ }\<x\<90^{\\circ }}  | | | | | | |
Both sine and cosine functions can be defined by using differential equations. The pair of ( cos θ , sin θ ) {\\displaystyle (\\cos \\theta ,\\sin \\theta )}  is the solution ( x ( θ ) , y ( θ ) ) {\\displaystyle (x(\\theta ),y(\\theta ))}  to the two-dimensional system of [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") y ′ ( θ ) \= x ( θ ) {\\displaystyle y'(\\theta )=x(\\theta )}  and x ′ ( θ ) \= − y ( θ ) {\\displaystyle x'(\\theta )=-y(\\theta )}  with the [initial conditions](https://en.wikipedia.org/wiki/Initial_conditions "Initial conditions") y ( 0 ) \= 0 {\\displaystyle y(0)=0}  and x ( 0 ) \= 1 {\\displaystyle x(0)=1} . One could interpret the unit circle in the above definitions as defining the [phase space trajectory](https://en.wikipedia.org/wiki/Phase_space_trajectory "Phase space trajectory") of the differential equation with the given initial conditions. It can be interpreted as a phase space trajectory of the system of differential equations y ′ ( θ ) \= x ( θ ) {\\displaystyle y'(\\theta )=x(\\theta )}  and x ′ ( θ ) \= − y ( θ ) {\\displaystyle x'(\\theta )=-y(\\theta )}  starting from the initial conditions y ( 0 ) \= 0 {\\displaystyle y(0)=0}  and x ( 0 ) \= 1 {\\displaystyle x(0)=1} .\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
#### Integral and the usage in mensuration
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=10 "Edit section: Integral and the usage in mensuration")\]
Main article: [List of integrals of trigonometric functions](https://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions "List of integrals of trigonometric functions")
Their area under a curve can be obtained by using the [integral](https://en.wikipedia.org/wiki/Integral "Integral") with a certain bounded interval. Their antiderivatives are: ∫ sin ( x ) d x \= − cos ( x ) \+ C ∫ cos ( x ) d x \= sin ( x ) \+ C , {\\displaystyle \\int \\sin(x)\\,dx=-\\cos(x)+C\\qquad \\int \\cos(x)\\,dx=\\sin(x)+C,}  where C {\\displaystyle C}  denotes the [constant of integration](https://en.wikipedia.org/wiki/Constant_of_integration "Constant of integration").[\[21\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007199-21) These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval. For example, the [arc length](https://en.wikipedia.org/wiki/Arc_length "Arc length") of the sine curve between 0 {\\displaystyle 0}  and t {\\displaystyle t}  is ∫ 0 t 1 \+ cos 2 ( x ) d x \= 2 E ( t , 1 2 ) , {\\displaystyle \\int \_{0}^{t}\\!{\\sqrt {1+\\cos ^{2}(x)}}\\,dx={\\sqrt {2}}\\operatorname {E} \\left(t,{\\frac {1}{\\sqrt {2}}}\\right),}  where E ( φ , k ) {\\displaystyle \\operatorname {E} (\\varphi ,k)}  is the [incomplete elliptic integral of the second kind](https://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind "Elliptic integral") with modulus k {\\displaystyle k} . It cannot be expressed using [elementary functions](https://en.wikipedia.org/wiki/Elementary_function "Elementary function").[\[22\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVince2023[httpsbooksgooglecombooksidGnW6EAAAQBAJpgPA162_162]-22) In the case of a full period, its arc length is L \= 4 2 π 3 Γ ( 1 / 4 ) 2 \+ Γ ( 1 / 4 ) 2 2 π \= 2 π ϖ \+ 2 ϖ ≈ 7\.6404 {\\displaystyle L={\\frac {4{\\sqrt {2\\pi ^{3}}}}{\\Gamma (1/4)^{2}}}+{\\frac {\\Gamma (1/4)^{2}}{\\sqrt {2\\pi }}}={\\frac {2\\pi }{\\varpi }}+2\\varpi \\approx 7.6404}  where Γ {\\displaystyle \\Gamma }  is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function "Gamma function") and ϖ {\\displaystyle \\varpi }  is the [lemniscate constant](https://en.wikipedia.org/wiki/Lemniscate_constant "Lemniscate constant").[\[23\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAdlaj2012-23)[\[24\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-24)
#### Inverse functions
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=11 "Edit section: Inverse functions")\]
[](https://en.wikipedia.org/wiki/File:Arcsine_Arccosine.svg)
The usual principal values of the arcsin(*x*) and arccos(*x*) functions graphed on the Cartesian plane
The functions sin : R → R {\\textstyle \\sin :\\mathbb {R} \\to \\mathbb {R} }  and cos : R → R {\\displaystyle \\cos :\\mathbb {R} \\to \\mathbb {R} }  (as well as those functions with the same function rule and domain whose codomain is a subset of R {\\displaystyle \\mathbb {R} }  containing the interval \[ − 1 , 1 \] {\\displaystyle \\left\[-1,1\\right\]} ![{\\displaystyle \\left\[-1,1\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79566f857ac1fcd0ef0f62226298a4ed15b796ad)) are not bijective and therefore do not have inverse functions. For example, sin ( 0 ) \= 0 {\\displaystyle \\sin(0)=0} , but also sin ( π ) \= 0 {\\displaystyle \\sin(\\pi )=0} , sin ( 2 π ) \= 0 {\\displaystyle \\sin(2\\pi )=0} . Sine's "inverse", called arcsine, can then be described not as a function but a relation (for example, all integer multiples of π {\\displaystyle \\pi }  would have an arcsine of zero). To define the inverse functions of sine and cosine, they must be restricted to their [principal branches](https://en.wikipedia.org/wiki/Principal_branch "Principal branch") by restricting their domain and codomain; the standard functions used to define arcsine and arccosine are then sin : \[ − π / 2 , π / 2 \] → \[ − 1 , 1 \] {\\displaystyle \\sin :\\left\[-\\pi /2,\\pi /2\\right\]\\to \\left\[-1,1\\right\]} ![{\\displaystyle \\sin :\\left\[-\\pi /2,\\pi /2\\right\]\\to \\left\[-1,1\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34676d3cc30ba3520fb506118910a65ae45fe25f) and cos : \[ 0 , π \] → \[ − 1 , 1 \] {\\displaystyle \\cos :\\left\[0,\\pi \\right\]\\to \\left\[-1,1\\right\]} ![{\\displaystyle \\cos :\\left\[0,\\pi \\right\]\\to \\left\[-1,1\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2864b5c77dc4b8af5111e9d4e78e9ea9a876b7).[\[25\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007365-25) These are bijective and have inverses: arcsin : \[ − 1 , 1 \] → \[ − π / 2 , π / 2 \] {\\displaystyle \\arcsin :\\left\[-1,1\\right\]\\to \\left\[-\\pi /2,\\pi /2\\right\]} ![{\\displaystyle \\arcsin :\\left\[-1,1\\right\]\\to \\left\[-\\pi /2,\\pi /2\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/024a4d91e022705b40fb40e554d294bdd9e2eea4) and arccos : \[ − 1 , 1 \] → \[ 0 , π \] {\\displaystyle \\arccos :\\left\[-1,1\\right\]\\to \\left\[0,\\pi \\right\]} ![{\\displaystyle \\arccos :\\left\[-1,1\\right\]\\to \\left\[0,\\pi \\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1126b78ee63338e44c6a25dae46d50fdad0b5c1). Alternative notation is sin − 1 {\\displaystyle \\sin ^{-1}}  for arcsine and cos − 1 {\\displaystyle \\cos ^{-1}}  for arccosine. Using these definitions, one obtains the identity maps:
sin ∘ arcsin ( x ) \= x x ∈ \[ − 1 , 1 \] arcsin ∘ sin ( x ) \= x x ∈ \[ − π / 2 , π / 2 \] {\\displaystyle {\\begin{aligned}\\sin \\circ \\arcsin \\,(x)&=x\\qquad x\\in \\left\[-1,1\\right\]\\\\\\arcsin \\circ \\sin \\,(x)&=x\\qquad x\\in \\left\[-\\pi /2,\\pi /2\\right\]\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\sin \\circ \\arcsin \\,(x)&=x\\qquad x\\in \\left\[-1,1\\right\]\\\\\\arcsin \\circ \\sin \\,(x)&=x\\qquad x\\in \\left\[-\\pi /2,\\pi /2\\right\]\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/631b526ef79de3155b8c652483e136009c87f6ef)and
cos ∘ arccos ( x ) \= x x ∈ \[ − 1 , 1 \] arccos ∘ cos ( x ) \= x x ∈ \[ 0 , π \] {\\displaystyle {\\begin{aligned}\\cos \\circ \\arccos \\,(x)&=x\\qquad x\\in \\left\[-1,1\\right\]\\\\\\arccos \\circ \\cos \\,(x)&=x\\qquad x\\in \\left\[0,\\pi \\right\]\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\cos \\circ \\arccos \\,(x)&=x\\qquad x\\in \\left\[-1,1\\right\]\\\\\\arccos \\circ \\cos \\,(x)&=x\\qquad x\\in \\left\[0,\\pi \\right\]\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81aa8f5001ec127005efd5c2508631440ff6ab8d)
An acute angle θ {\\displaystyle \\theta }  is given by: θ \= arcsin ( opposite hypotenuse ) \= arccos ( adjacent hypotenuse ) , {\\displaystyle \\theta =\\arcsin \\left({\\frac {\\text{opposite}}{\\text{hypotenuse}}}\\right)=\\arccos \\left({\\frac {\\text{adjacent}}{\\text{hypotenuse}}}\\right),}  where for some integer k {\\displaystyle k} , sin ( y ) \= x ⟺ y \= arcsin ( x ) \+ 2 π k , or y \= π − arcsin ( x ) \+ 2 π k cos ( y ) \= x ⟺ y \= arccos ( x ) \+ 2 π k , or y \= − arccos ( x ) \+ 2 π k {\\displaystyle {\\begin{aligned}\\sin(y)=x\\iff \&y=\\arcsin(x)+2\\pi k,{\\text{ or }}\\\\\&y=\\pi -\\arcsin(x)+2\\pi k\\\\\\cos(y)=x\\iff \&y=\\arccos(x)+2\\pi k,{\\text{ or }}\\\\\&y=-\\arccos(x)+2\\pi k\\end{aligned}}}  By definition, both functions satisfy the equations: sin ( arcsin ( x ) ) \= x cos ( arccos ( x ) ) \= x {\\displaystyle \\sin(\\arcsin(x))=x\\qquad \\cos(\\arccos(x))=x}  and arcsin ( sin ( θ ) ) \= θ for − π 2 ≤ θ ≤ π 2 arccos ( cos ( θ ) ) \= θ for 0 ≤ θ ≤ π {\\displaystyle {\\begin{aligned}\\arcsin(\\sin(\\theta ))=\\theta \\quad &{\\text{for}}\\quad -{\\frac {\\pi }{2}}\\leq \\theta \\leq {\\frac {\\pi }{2}}\\\\\\arccos(\\cos(\\theta ))=\\theta \\quad &{\\text{for}}\\quad 0\\leq \\theta \\leq \\pi \\end{aligned}}} 
#### Other identities
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=12 "Edit section: Other identities")\]
Main article: [List of trigonometric identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities "List of trigonometric identities")
According to [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem"), the squared hypotenuse is the sum of two squared legs of a right triangle. Dividing the formula on both sides with squared hypotenuse resulting in the [Pythagorean trigonometric identity](https://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity "Pythagorean trigonometric identity"), the sum of a squared sine and a squared cosine equals 1:[\[26\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA99_99]-26)[\[a\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-27) sin 2 ( θ ) \+ cos 2 ( θ ) \= 1\. {\\displaystyle \\sin ^{2}(\\theta )+\\cos ^{2}(\\theta )=1.} 
Sine and cosine satisfy the following double-angle formulas:[\[27\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-28) sin ( 2 θ ) \= 2 sin ( θ ) cos ( θ ) , cos ( 2 θ ) \= cos 2 ( θ ) − sin 2 ( θ ) \= 2 cos 2 ( θ ) − 1 \= 1 − 2 sin 2 ( θ ) {\\displaystyle {\\begin{aligned}\\sin(2\\theta )&=2\\sin(\\theta )\\cos(\\theta ),\\\\\\cos(2\\theta )&=\\cos ^{2}(\\theta )-\\sin ^{2}(\\theta )\\\\&=2\\cos ^{2}(\\theta )-1\\\\&=1-2\\sin ^{2}(\\theta )\\end{aligned}}} 
[](https://en.wikipedia.org/wiki/File:SinSquared.png)
Sine function in blue and sine squared function in red. The *x*\-axis is in radians.
The cosine double angle formula implies that sin2 and cos2 are, themselves, shifted and scaled sine waves. Specifically,[\[28\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-29) sin 2 ( θ ) \= 1 − cos ( 2 θ ) 2 cos 2 ( θ ) \= 1 \+ cos ( 2 θ ) 2 {\\displaystyle \\sin ^{2}(\\theta )={\\frac {1-\\cos(2\\theta )}{2}}\\qquad \\cos ^{2}(\\theta )={\\frac {1+\\cos(2\\theta )}{2}}}  The graph shows both sine and sine squared functions, with the sine in blue and the sine squared in red. Both graphs have the same shape but with different ranges of values and different periods. Sine squared has only positive values, but twice the number of periods.\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
### Series and polynomials
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=13 "Edit section: Series and polynomials")\]
[](https://en.wikipedia.org/wiki/File:Sine.gif)
This animation shows how including more and more terms in the partial sum of its Taylor series approaches a sine curve.
Both sine and cosine functions can be defined by using a [Taylor series](https://en.wikipedia.org/wiki/Taylor_series "Taylor series"), a [power series](https://en.wikipedia.org/wiki/Power_series "Power series") involving the higher-order derivatives. As mentioned in [§ Continuity and differentiation](https://en.wikipedia.org/wiki/Sine_and_cosine#Continuity_and_differentiation), the [derivative](https://en.wikipedia.org/wiki/Derivative "Derivative") of sine is cosine and the derivative of cosine is the negative of sine. This means the successive derivatives of sin ( x ) {\\displaystyle \\sin(x)}  are cos ( x ) {\\displaystyle \\cos(x)} , − sin ( x ) {\\displaystyle -\\sin(x)} , − cos ( x ) {\\displaystyle -\\cos(x)} , sin ( x ) {\\displaystyle \\sin(x)} , continuing to repeat those four functions. The ( 4 n \+ k ) {\\displaystyle (4n+k)}  \-th derivative, evaluated at the point 0: sin ( 4 n \+ k ) ( 0 ) \= { 0 when k \= 0 1 when k \= 1 0 when k \= 2 − 1 when k \= 3 {\\displaystyle \\sin ^{(4n+k)}(0)={\\begin{cases}0&{\\text{when }}k=0\\\\1&{\\text{when }}k=1\\\\0&{\\text{when }}k=2\\\\-1&{\\text{when }}k=3\\end{cases}}}  where the superscript represents repeated differentiation. This implies the following Taylor series expansion at x \= 0 {\\displaystyle x=0} . One can then use the theory of [Taylor series](https://en.wikipedia.org/wiki/Taylor_series "Taylor series") to show that the following identities hold for all [real numbers](https://en.wikipedia.org/wiki/Real_number "Real number") x {\\displaystyle x} —where x {\\displaystyle x}  is the angle in radians.[\[29\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007491–492-30) More generally, for all [complex numbers](https://en.wikipedia.org/wiki/Complex_number "Complex number"):[\[30\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]-31) sin ( x ) \= x − x 3 3 \! \+ x 5 5 \! − x 7 7 \! \+ ⋯ \= ∑ n \= 0 ∞ ( − 1 ) n ( 2 n \+ 1 ) \! x 2 n \+ 1 {\\displaystyle {\\begin{aligned}\\sin(x)&=x-{\\frac {x^{3}}{3!}}+{\\frac {x^{5}}{5!}}-{\\frac {x^{7}}{7!}}+\\cdots \\\\&=\\sum \_{n=0}^{\\infty }{\\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\end{aligned}}}  Taking the derivative of each term gives the Taylor series for cosine:[\[29\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007491–492-30)[\[30\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]-31) cos ( x ) \= 1 − x 2 2 \! \+ x 4 4 \! − x 6 6 \! \+ ⋯ \= ∑ n \= 0 ∞ ( − 1 ) n ( 2 n ) \! x 2 n {\\displaystyle {\\begin{aligned}\\cos(x)&=1-{\\frac {x^{2}}{2!}}+{\\frac {x^{4}}{4!}}-{\\frac {x^{6}}{6!}}+\\cdots \\\\&=\\sum \_{n=0}^{\\infty }{\\frac {(-1)^{n}}{(2n)!}}x^{2n}\\end{aligned}}} 
Both sine and cosine functions with multiple angles may appear as their [linear combination](https://en.wikipedia.org/wiki/Linear_combination "Linear combination"), resulting in a polynomial. Such a polynomial is known as the [trigonometric polynomial](https://en.wikipedia.org/wiki/Trigonometric_polynomial "Trigonometric polynomial"). The trigonometric polynomial's ample applications may be acquired in [its interpolation](https://en.wikipedia.org/wiki/Trigonometric_interpolation "Trigonometric interpolation"), and its extension of a periodic function known as the [Fourier series](https://en.wikipedia.org/wiki/Fourier_series "Fourier series"). Let a n {\\displaystyle a\_{n}}  and b n {\\displaystyle b\_{n}}  be any coefficients, then the trigonometric polynomial of a degree N {\\displaystyle N} —denoted as T ( x ) {\\displaystyle T(x)} —is defined as:[\[31\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEPowell1981150-32)[\[32\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTERudin198788-33) T ( x ) \= a 0 \+ ∑ n \= 1 N a n cos ( n x ) \+ ∑ n \= 1 N b n sin ( n x ) . {\\displaystyle T(x)=a\_{0}+\\sum \_{n=1}^{N}a\_{n}\\cos(nx)+\\sum \_{n=1}^{N}b\_{n}\\sin(nx).} 
The [trigonometric series](https://en.wikipedia.org/wiki/Trigonometric_series "Trigonometric series") can be defined similarly analogous to the trigonometric polynomial, its infinite inversion. Let A n {\\displaystyle A\_{n}}  and B n {\\displaystyle B\_{n}}  be any coefficients, then the trigonometric series can be defined as:[\[33\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEZygmund19681-34) 1 2 A 0 \+ ∑ n \= 1 ∞ A n cos ( n x ) \+ B n sin ( n x ) . {\\displaystyle {\\frac {1}{2}}A\_{0}+\\sum \_{n=1}^{\\infty }A\_{n}\\cos(nx)+B\_{n}\\sin(nx).}  In the case of a Fourier series with a given integrable function f {\\displaystyle f} , the coefficients of a trigonometric series are:[\[34\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEZygmund196811-35) A n \= 1 π ∫ 0 2 π f ( x ) cos ( n x ) d x , B n \= 1 π ∫ 0 2 π f ( x ) sin ( n x ) d x . {\\displaystyle {\\begin{aligned}A\_{n}&={\\frac {1}{\\pi }}\\int \_{0}^{2\\pi }f(x)\\cos(nx)\\,dx,\\\\B\_{n}&={\\frac {1}{\\pi }}\\int \_{0}^{2\\pi }f(x)\\sin(nx)\\,dx.\\end{aligned}}} 
## Complex numbers relationship
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=14 "Edit section: Complex numbers relationship")\]
| | |
|---|---|
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### Complex exponential function definitions
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=15 "Edit section: Complex exponential function definitions")\]
Both sine and cosine can be extended further via [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number"), a set of numbers composed of both [real](https://en.wikipedia.org/wiki/Real_number "Real number") and [imaginary numbers](https://en.wikipedia.org/wiki/Imaginary_number "Imaginary number"). For real number θ {\\displaystyle \\theta } , the definition of both sine and cosine functions can be extended in a [complex plane](https://en.wikipedia.org/wiki/Complex_plane "Complex plane") in terms of an [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") as follows:[\[35\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]-36) sin ( θ ) \= e i θ − e − i θ 2 i , cos ( θ ) \= e i θ \+ e − i θ 2 , {\\displaystyle {\\begin{aligned}\\sin(\\theta )&={\\frac {e^{i\\theta }-e^{-i\\theta }}{2i}},\\\\\\cos(\\theta )&={\\frac {e^{i\\theta }+e^{-i\\theta }}{2}},\\end{aligned}}} 
Alternatively, both functions can be defined in terms of [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"):[\[35\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]-36) e i θ \= cos ( θ ) \+ i sin ( θ ) , e − i θ \= cos ( θ ) − i sin ( θ ) . {\\displaystyle {\\begin{aligned}e^{i\\theta }&=\\cos(\\theta )+i\\sin(\\theta ),\\\\e^{-i\\theta }&=\\cos(\\theta )-i\\sin(\\theta ).\\end{aligned}}} 
When plotted on the [complex plane](https://en.wikipedia.org/wiki/Complex_plane "Complex plane"), the function e i x {\\displaystyle e^{ix}}  for real values of x {\\displaystyle x}  traces out the [unit circle](https://en.wikipedia.org/wiki/Unit_circle "Unit circle") in the complex plane. Both sine and cosine functions may be simplified to the imaginary and real parts of e i θ {\\displaystyle e^{i\\theta }}  as:[\[36\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTERudin19872-37) sin θ \= Im ( e i θ ) , cos θ \= Re ( e i θ ) . {\\displaystyle {\\begin{aligned}\\sin \\theta &=\\operatorname {Im} (e^{i\\theta }),\\\\\\cos \\theta &=\\operatorname {Re} (e^{i\\theta }).\\end{aligned}}} 
When z \= x \+ i y {\\displaystyle z=x+iy}  for real values x {\\displaystyle x}  and y {\\displaystyle y} , where i \= − 1 {\\displaystyle i={\\sqrt {-1}}} , both sine and cosine functions can be expressed in terms of real sines, cosines, and [hyperbolic functions](https://en.wikipedia.org/wiki/Hyperbolic_function "Hyperbolic function") as:[\[37\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-38) sin z \= sin x cosh y \+ i cos x sinh y , cos z \= cos x cosh y − i sin x sinh y . {\\displaystyle {\\begin{aligned}\\sin z&=\\sin x\\cosh y+i\\cos x\\sinh y,\\\\\\cos z&=\\cos x\\cosh y-i\\sin x\\sinh y.\\end{aligned}}} 
### Polar coordinates
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=16 "Edit section: Polar coordinates")\]
[](https://en.wikipedia.org/wiki/File:Sinus_und_Kosinus_am_Einheitskreis_3.svg)
Both functions
cos
(
θ
)
{\\displaystyle \\cos(\\theta )}
and
sin
(
θ
)
{\\displaystyle \\sin(\\theta )}
are the real and imaginary parts of
e
i
θ
{\\displaystyle e^{i\\theta }}
.
Sine and cosine are used to connect the real and imaginary parts of a [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number") with its [polar coordinates](https://en.wikipedia.org/wiki/Polar_coordinates "Polar coordinates") ( r , θ ) {\\displaystyle (r,\\theta )} : z \= r ( cos ( θ ) \+ i sin ( θ ) ) \= r cos ( θ ) \+ i r sin ( θ ) , {\\displaystyle z=r(\\cos(\\theta )+i\\sin(\\theta ))=r\\cos(\\theta )+ir\\sin(\\theta ),}  and the real and imaginary parts are Re ( z ) \= r cos ( θ ) , Im ( z ) \= r sin ( θ ) , {\\displaystyle {\\begin{aligned}\\operatorname {Re} (z)&=r\\cos(\\theta ),\\\\\\operatorname {Im} (z)&=r\\sin(\\theta ),\\end{aligned}}}  where r {\\displaystyle r}  and θ {\\displaystyle \\theta }  represent the magnitude and angle of the complex number z {\\displaystyle z} .[\[38\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_23–24]-39)
For any real number θ {\\displaystyle \\theta } , Euler's formula in terms of polar coordinates is stated as z \= r e i θ {\\textstyle z=re^{i\\theta }} .[\[35\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]-36)
### Complex arguments
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=17 "Edit section: Complex arguments")\]
[](https://en.wikipedia.org/wiki/File:Complex_sin.jpg)
[Domain coloring](https://en.wikipedia.org/wiki/Domain_coloring "Domain coloring") of sin(*z*) in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.
[](https://en.wikipedia.org/wiki/File:Sin_z_vector_field_02_Pengo.svg)
Vector field rendering of sin(*z*)
Applying the series definition of the sine and cosine to a complex argument, *z*, gives:
sin
(
z
)
\=
∑
n
\=
0
∞
(
−
1
)
n
(
2
n
\+
1
)
\!
z
2
n
\+
1
\=
e
i
z
−
e
−
i
z
2
i
\=
sinh
(
i
z
)
i
\=
−
i
sinh
(
i
z
)
cos
(
z
)
\=
∑
n
\=
0
∞
(
−
1
)
n
(
2
n
)
\!
z
2
n
\=
e
i
z
\+
e
−
i
z
2
\=
cosh
(
i
z
)
{\\displaystyle {\\begin{aligned}\\sin(z)&=\\sum \_{n=0}^{\\infty }{\\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\\\\&={\\frac {e^{iz}-e^{-iz}}{2i}}\\\\&={\\frac {\\sinh \\left(iz\\right)}{i}}\\\\&=-i\\sinh \\left(iz\\right)\\\\\\cos(z)&=\\sum \_{n=0}^{\\infty }{\\frac {(-1)^{n}}{(2n)!}}z^{2n}\\\\&={\\frac {e^{iz}+e^{-iz}}{2}}\\\\&=\\cosh(iz)\\\\\\end{aligned}}}
where sinh and cosh are the [hyperbolic sine and cosine](https://en.wikipedia.org/wiki/Hyperbolic_function "Hyperbolic function"). These are [entire functions](https://en.wikipedia.org/wiki/Entire_function "Entire function").
It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument:
sin
(
x
\+
i
y
)
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sin
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cos
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{\\displaystyle {\\begin{aligned}\\sin(x+iy)&=\\sin(x)\\cos(iy)+\\cos(x)\\sin(iy)\\\\&=\\sin(x)\\cosh(y)+i\\cos(x)\\sinh(y)\\\\\\cos(x+iy)&=\\cos(x)\\cos(iy)-\\sin(x)\\sin(iy)\\\\&=\\cos(x)\\cosh(y)-i\\sin(x)\\sinh(y)\\\\\\end{aligned}}}
#### Partial fraction and product expansions of complex sine
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=18 "Edit section: Partial fraction and product expansions of complex sine")\]
Using the partial fraction expansion technique in [complex analysis](https://en.wikipedia.org/wiki/Complex_analysis "Complex analysis"), one can find that the infinite series ∑ n \= − ∞ ∞ ( − 1 ) n z − n \= 1 z − 2 z ∑ n \= 1 ∞ ( − 1 ) n n 2 − z 2 {\\displaystyle \\sum \_{n=-\\infty }^{\\infty }{\\frac {(-1)^{n}}{z-n}}={\\frac {1}{z}}-2z\\sum \_{n=1}^{\\infty }{\\frac {(-1)^{n}}{n^{2}-z^{2}}}}  both converge and are equal to π sin ( π z ) {\\textstyle {\\frac {\\pi }{\\sin(\\pi z)}}} . Similarly, one can show that π 2 sin 2 ( π z ) \= ∑ n \= − ∞ ∞ 1 ( z − n ) 2 . {\\displaystyle {\\frac {\\pi ^{2}}{\\sin ^{2}(\\pi z)}}=\\sum \_{n=-\\infty }^{\\infty }{\\frac {1}{(z-n)^{2}}}.} 
Using product expansion technique, one can derive sin ( π z ) \= π z ∏ n \= 1 ∞ ( 1 − z 2 n 2 ) . {\\displaystyle \\sin(\\pi z)=\\pi z\\prod \_{n=1}^{\\infty }\\left(1-{\\frac {z^{2}}{n^{2}}}\\right).} 
#### Usage of complex sine
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=19 "Edit section: Usage of complex sine")\]
sin(*z*) is found in the [functional equation](https://en.wikipedia.org/wiki/Functional_equation "Functional equation") for the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function "Gamma function"),
Γ
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{\\displaystyle \\Gamma (s)\\Gamma (1-s)={\\pi \\over \\sin(\\pi s)},}
which in turn is found in the [functional equation](https://en.wikipedia.org/wiki/Functional_equation "Functional equation") for the [Riemann zeta-function](https://en.wikipedia.org/wiki/Riemann_zeta-function "Riemann zeta-function"),
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{\\displaystyle \\zeta (s)=2(2\\pi )^{s-1}\\Gamma (1-s)\\sin \\left({\\frac {\\pi }{2}}s\\right)\\zeta (1-s).}
As a [holomorphic function](https://en.wikipedia.org/wiki/Holomorphic_function "Holomorphic function"), sin *z* is a 2D solution of [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation"):
Δ
u
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1
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\=
0\.
{\\displaystyle \\Delta u(x\_{1},x\_{2})=0.}
The complex sine function is also related to the level curves of [pendulums](https://en.wikipedia.org/wiki/Pendulums "Pendulums").\[*[how?](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify "Wikipedia:Please clarify")*\][\[39\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-40)\[*[better source needed](https://en.wikipedia.org/wiki/Wikipedia:Verifiability#Questionable_sources "Wikipedia:Verifiability")*\]
### Complex graphs
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=20 "Edit section: Complex graphs")\]
| | | |
|---|---|---|
| [](https://en.wikipedia.org/wiki/File:Complex_sin_real_01_Pengo.svg) | [](https://en.wikipedia.org/wiki/File:Complex_sin_imag_01_Pengo.svg) | [](https://en.wikipedia.org/wiki/File:Complex_sin_abs_01_Pengo.svg) |
| Real component | Imaginary component | Magnitude |
| | | |
|---|---|---|
| [](https://en.wikipedia.org/wiki/File:Complex_arcsin_real_01_Pengo.svg) | [](https://en.wikipedia.org/wiki/File:Complex_arcsin_imag_01_Pengo.svg) | [](https://en.wikipedia.org/wiki/File:Complex_arcsin_abs_01_Pengo.svg) |
| Real component | Imaginary component | Magnitude |
## Background
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=21 "Edit section: Background")\]
### Etymology
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=22 "Edit section: Etymology")\]
Main article: [History of trigonometry § Etymology](https://en.wikipedia.org/wiki/History_of_trigonometry#Etymology "History of trigonometry")
The word *sine* is derived, indirectly, from the [Sanskrit](https://en.wikipedia.org/wiki/Sanskrit "Sanskrit") word *jyā* 'bow-string' or more specifically its synonym *jīvá* (both adopted from [Ancient Greek](https://en.wikipedia.org/wiki/Ancient_Greek_language "Ancient Greek language") χορδή 'string; chord'), due to visual similarity between the arc of a circle with its corresponding chord and a bow with its string (see [jyā, koti-jyā and utkrama-jyā](https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81 "Jyā, koti-jyā and utkrama-jyā"); *sine* and *chord* are closely related in a circle of unit diameter, see [Ptolemy's Theorem](https://en.wikipedia.org/wiki/Ptolemy%27s_theorem#Corollaries "Ptolemy's theorem")). This was [transliterated](https://en.wikipedia.org/wiki/Transliteration "Transliteration") in [Arabic](https://en.wikipedia.org/wiki/Arabic_language "Arabic language") as *jība*, which is meaningless in that language and written as *jb* (جب). Since Arabic is written without short vowels, *jb* was interpreted as the [homograph](https://en.wikipedia.org/wiki/Homograph "Homograph") *jayb* ([جيب](https://en.wiktionary.org/wiki/%D8%AC%D9%8A%D8%A8 "wikt:جيب")), which means 'bosom', 'pocket', or 'fold'.[\[40\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEPlofker2009[httpsbooksgooglecombooksidDHvThPNp9yMCpgPA257_257]-41)[\[41\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEMaor1998[httpsbooksgooglecombooksidr9aMrneWFpUCpgPA35_35]-42) When the Arabic texts of [Al-Battani](https://en.wikipedia.org/wiki/Al-Battani "Al-Battani") and [al-Khwārizmī](https://en.wikipedia.org/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB "Muḥammad ibn Mūsā al-Khwārizmī") were translated into [Medieval Latin](https://en.wikipedia.org/wiki/Medieval_Latin "Medieval Latin") in the 12th century by [Gerard of Cremona](https://en.wikipedia.org/wiki/Gerard_of_Cremona "Gerard of Cremona"), he used the Latin equivalent [*sinus*](https://en.wiktionary.org/wiki/sinus "wikt:sinus") (which also means 'bay' or 'fold', and more specifically 'the hanging fold of a [toga](https://en.wikipedia.org/wiki/Toga "Toga") over the breast').[\[42\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEMerzbachBoyer2011-43)[\[43\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEMaor199835–36-44)[\[44\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEKatz2008253-45) Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage.[\[45\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTESmith1958202-46)[\[46\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-47) The English form *sine* was introduced in [Thomas Fale](https://en.wikipedia.org/wiki/Thomas_Fale "Thomas Fale")'s 1593 *Horologiographia*.[\[47\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-48)
The word *cosine* derives from an abbreviation of the Latin *complementi sinus* 'sine of the [complementary angle](https://en.wikipedia.org/wiki/Complementary_angle "Complementary angle")' as *cosinus* in [Edmund Gunter](https://en.wikipedia.org/wiki/Edmund_Gunter "Edmund Gunter")'s *Canon triangulorum* (1620), which also includes a similar definition of *cotangens*.[\[48\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEGunter1620-49)
### History
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=23 "Edit section: History")\]
Main article: [History of trigonometry](https://en.wikipedia.org/wiki/History_of_trigonometry "History of trigonometry")
[](https://en.wikipedia.org/wiki/File:Khalili_Collection_Islamic_Art_sci_0040.1_CROP.jpg)
Quadrant from 1840s [Ottoman Turkey](https://en.wikipedia.org/wiki/Ottoman_Empire "Ottoman Empire") with axes for looking up the sine and [versine](https://en.wikipedia.org/wiki/Versine "Versine") of angles
While the early study of trigonometry can be traced to antiquity, the [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions") as they are in use today were developed in the medieval period. The [chord](https://en.wikipedia.org/wiki/Chord_\(geometry\) "Chord (geometry)") function was discovered by [Hipparchus](https://en.wikipedia.org/wiki/Hipparchus "Hipparchus") of [Nicaea](https://en.wikipedia.org/wiki/%C4%B0znik "İznik") (180–125 BCE) and [Ptolemy](https://en.wikipedia.org/wiki/Ptolemy "Ptolemy") of [Roman Egypt](https://en.wikipedia.org/wiki/Egypt_\(Roman_province\) "Egypt (Roman province)") (90–165 CE).[\[49\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-50)
The sine and cosine functions are closely related to the [*jyā* and *koṭi-jyā*](https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81 "Jyā, koti-jyā and utkrama-jyā") functions used in [Indian astronomy](https://en.wikipedia.org/wiki/Indian_astronomy "Indian astronomy") during the [Gupta period](https://en.wikipedia.org/wiki/Gupta_period "Gupta period") (*[Aryabhatiya](https://en.wikipedia.org/wiki/Aryabhatiya "Aryabhatiya")* and *[Surya Siddhanta](https://en.wikipedia.org/wiki/Surya_Siddhanta "Surya Siddhanta")*), via translation from Sanskrit to Arabic and then from Arabic to Latin.[\[42\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEMerzbachBoyer2011-43)[\[50\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-51)
All six trigonometric functions in current use were known in [Islamic mathematics](https://en.wikipedia.org/wiki/Islamic_mathematics "Islamic mathematics") by the 9th century, as was the [law of sines](https://en.wikipedia.org/wiki/Law_of_sines "Law of sines"), used in [solving triangles](https://en.wikipedia.org/wiki/Solving_triangles "Solving triangles").[\[51\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-Gingerich_1986-52) [Al-Khwārizmī](https://en.wikipedia.org/wiki/Al-Khw%C4%81rizm%C4%AB "Al-Khwārizmī") (c. 780–850) produced tables of sines, cosines and tangents.[\[52\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-Sesiano-53)[\[53\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-Britannica-54) [Muhammad ibn Jābir al-Harrānī al-Battānī](https://en.wikipedia.org/wiki/Muhammad_ibn_J%C4%81bir_al-Harr%C4%81n%C4%AB_al-Batt%C4%81n%C4%AB "Muhammad ibn Jābir al-Harrānī al-Battānī") (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.[\[53\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-Britannica-54)
In the early 17th-century, the French mathematician [Albert Girard](https://en.wikipedia.org/wiki/Albert_Girard "Albert Girard") published the first use of the abbreviations *sin*, *cos*, and *tan*; these were further promulgated by Euler (see below). The *Opus palatinum de triangulis* of [Georg Joachim Rheticus](https://en.wikipedia.org/wiki/Georg_Joachim_Rheticus "Georg Joachim Rheticus"), a student of [Copernicus](https://en.wikipedia.org/wiki/Copernicus "Copernicus"), was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.
In a paper published in 1682, [Leibniz](https://en.wikipedia.org/wiki/Gottfried_Leibniz "Gottfried Leibniz") proved that sin *x* is not an [algebraic function](https://en.wikipedia.org/wiki/Algebraic_function "Algebraic function") of *x*.[\[54\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-55) [Roger Cotes](https://en.wikipedia.org/wiki/Roger_Cotes "Roger Cotes") computed the derivative of sine in his *Harmonia Mensurarum* (1722).[\[55\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-56) [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler")'s *Introductio in analysin infinitorum* (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "[Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula")", as well as the near-modern abbreviations *sin.*, *cos.*, *tang.*, *cot.*, *sec.*, and *cosec.*[\[42\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEMerzbachBoyer2011-43)
## Software implementations
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=24 "Edit section: Software implementations")\]
| | |
|---|---|
| [](https://en.wikipedia.org/wiki/File:Question_book-new.svg) | This section **needs additional citations for [verification](https://en.wikipedia.org/wiki/Wikipedia:Verifiability "Wikipedia:Verifiability")**. Please help [improve this article](https://en.wikipedia.org/wiki/Special:EditPage/Sine_and_cosine "Special:EditPage/Sine and cosine") by [adding citations to reliable sources](https://en.wikipedia.org/wiki/Help:Referencing_for_beginners "Help:Referencing for beginners") in this section. Unsourced material may be challenged and removed. *(August 2024)* *([Learn how and when to remove this message](https://en.wikipedia.org/wiki/Help:Maintenance_template_removal "Help:Maintenance template removal"))* |
See also: [Lookup table § Computing sines](https://en.wikipedia.org/wiki/Lookup_table#Computing_sines "Lookup table")
There is no standard algorithm for calculating sine and cosine. [IEEE 754](https://en.wikipedia.org/wiki/IEEE_754 "IEEE 754"), the most widely used standard for the specification of reliable floating-point computation, does not address calculating trigonometric functions such as sine. The reason is that no efficient algorithm is known for computing sine and cosine with a specified accuracy, especially for large inputs.[\[56\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEZimmermann2006-57)
Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. `sin(1022)`.
A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest pre-calculated value, or [linearly interpolate](https://en.wikipedia.org/wiki/Linear_interpolation "Linear interpolation") between the 2 closest values to approximate it. This allows results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage.\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
The [CORDIC](https://en.wikipedia.org/wiki/CORDIC "CORDIC") algorithm is commonly used in scientific calculators.
The sine and cosine functions, along with other trigonometric functions, are widely available across programming languages and platforms. In computing, they are typically abbreviated to `sin` and `cos`.
Some CPU architectures have a built-in instruction for sine, including the Intel [x87](https://en.wikipedia.org/wiki/X87 "X87") FPUs since the 80387.
In programming languages, `sin` and `cos` are typically either a built-in function or found within the language's standard math library. For example, the [C standard library](https://en.wikipedia.org/wiki/C_standard_library "C standard library") defines sine functions within [math.h](https://en.wikipedia.org/wiki/C_mathematical_functions "C mathematical functions"): `sin(double)`, `sinf(float)`, and `sinl(long double)`. The parameter of each is a [floating point](https://en.wikipedia.org/wiki/Floating_point "Floating point") value, specifying the angle in radians. Each function returns the same [data type](https://en.wikipedia.org/wiki/Data_type "Data type") as it accepts. Many other trigonometric functions are also defined in [math.h](https://en.wikipedia.org/wiki/C_mathematical_functions "C mathematical functions"), such as for cosine, arc sine, and hyperbolic sine (sinh). Similarly, [Python](https://en.wikipedia.org/wiki/Python_\(programming_language\) "Python (programming language)") defines `math.sin(x)` and `math.cos(x)` within the built-in `math` module. Complex sine and cosine functions are also available within the `cmath` module, e.g. `cmath.sin(z)`. [CPython](https://en.wikipedia.org/wiki/CPython "CPython")'s math functions call the [C](https://en.wikipedia.org/wiki/C_\(programming_language\) "C (programming language)") `math` library, and use a [double-precision floating-point format](https://en.wikipedia.org/wiki/Double-precision_floating-point_format "Double-precision floating-point format").
### Turns based implementations
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=25 "Edit section: Turns based implementations")\]
"sinpi" redirects here. For the township in Pingtung County, Taiwan, see [Xinpi](https://en.wikipedia.org/wiki/Xinpi "Xinpi").
"cospi" redirects here. For the 17th-century Bolognese nobleman, see [Ferdinando Cospi](https://en.wikipedia.org/wiki/Ferdinando_Cospi "Ferdinando Cospi").
Some software libraries provide implementations of sine and cosine using the input angle in half-[turns](https://en.wikipedia.org/wiki/Turn_\(angle\) "Turn (angle)"), a half-turn being an angle of 180 degrees or π {\\displaystyle \\pi }  radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.[\[57\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-matlab-58)[\[58\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-r-59) These functions are called `sinpi` and `cospi` in MATLAB,[\[57\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-matlab-58) [OpenCL](https://en.wikipedia.org/wiki/OpenCL "OpenCL"),[\[59\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-60) R,[\[58\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-r-59) Julia,[\[60\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-61) [CUDA](https://en.wikipedia.org/wiki/CUDA "CUDA"),[\[61\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-62) and ARM.[\[62\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-63) For example, `sinpi(x)` would evaluate to sin ( π x ) , {\\displaystyle \\sin(\\pi x),}  where *x* is expressed in half-turns, and consequently the final input to the function, *πx* can be interpreted in radians by sin. [SciPy](https://en.wikipedia.org/wiki/SciPy "SciPy") provides similar functions `sindg` and `cosdg` with input in degrees.[\[63\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-64)
The accuracy advantage stems from the ability to perfectly represent key angles like full-turn, half-turn, and quarter-turn losslessly in binary floating-point or fixed-point. In contrast, representing 2 π {\\displaystyle 2\\pi } , π {\\displaystyle \\pi } , and π 2 {\\textstyle {\\frac {\\pi }{2}}}  in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits.
Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing modulo π 2 {\\textstyle {\\frac {\\pi }{2}}}  involves inaccuracies in representing π 2 {\\textstyle {\\frac {\\pi }{2}}} .
For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to 4096 over one complete revolution.[\[64\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-65) If half-turns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to π 2048 {\\textstyle {\\frac {\\pi }{2048}}}  would be incurred.
## See also
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=26 "Edit section: See also")\]
- [Āryabhaṭa's sine table](https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa%27s_sine_table "Āryabhaṭa's sine table")
- [Bhaskara I's sine approximation formula](https://en.wikipedia.org/wiki/Bhaskara_I%27s_sine_approximation_formula "Bhaskara I's sine approximation formula")
- [Discrete sine transform](https://en.wikipedia.org/wiki/Discrete_sine_transform "Discrete sine transform")
- [Dixon elliptic functions](https://en.wikipedia.org/wiki/Dixon_elliptic_functions "Dixon elliptic functions")
- [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula")
- [Generalized trigonometry](https://en.wikipedia.org/wiki/Generalized_trigonometry "Generalized trigonometry")
- [Hyperbolic function](https://en.wikipedia.org/wiki/Hyperbolic_function "Hyperbolic function")
- [Lemniscate elliptic functions](https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions "Lemniscate elliptic functions")
- [Law of sines](https://en.wikipedia.org/wiki/Law_of_sines "Law of sines")
- [List of periodic functions](https://en.wikipedia.org/wiki/List_of_periodic_functions "List of periodic functions")
- [List of trigonometric identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities "List of trigonometric identities")
- [Madhava series](https://en.wikipedia.org/wiki/Madhava_series "Madhava series")
- [Madhava's sine table](https://en.wikipedia.org/wiki/Madhava%27s_sine_table "Madhava's sine table")
- [Optical sine theorem](https://en.wikipedia.org/wiki/Optical_sine_theorem "Optical sine theorem")
- [Polar sine](https://en.wikipedia.org/wiki/Polar_sine "Polar sine")—a generalization to vertex angles
- [Proofs of trigonometric identities](https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities "Proofs of trigonometric identities")
- [Sinc function](https://en.wikipedia.org/wiki/Sinc_function "Sinc function")
- [Sine and cosine transforms](https://en.wikipedia.org/wiki/Sine_and_cosine_transforms "Sine and cosine transforms")
- [Sine integral](https://en.wikipedia.org/wiki/Sine_integral "Sine integral")
- [Sine quadrant](https://en.wikipedia.org/wiki/Sine_quadrant "Sine quadrant")
- [Sine wave](https://en.wikipedia.org/wiki/Sine_wave "Sine wave")
- [Sine–Gordon equation](https://en.wikipedia.org/wiki/Sine%E2%80%93Gordon_equation "Sine–Gordon equation")
- [Sinusoidal model](https://en.wikipedia.org/wiki/Sinusoidal_model "Sinusoidal model")
- [SOH-CAH-TOA](https://en.wikipedia.org/wiki/Mnemonics_in_trigonometry#SOH-CAH-TOA "Mnemonics in trigonometry")
- [Trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions")
- [Trigonometric integral](https://en.wikipedia.org/wiki/Trigonometric_integral "Trigonometric integral")
## References
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=27 "Edit section: References")\]
### Footnotes
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=28 "Edit section: Footnotes")\]
1. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-27)**
Here,
sin
2
(
x
)
{\\displaystyle \\sin ^{2}(x)}
means the squared sine function
sin
(
x
)
⋅
sin
(
x
)
{\\displaystyle \\sin(x)\\cdot \\sin(x)}
.
### Citations
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=29 "Edit section: Citations")\]
1. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-1) [***c***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-2) [Young (2017)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2017), p. [27](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27).
2. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA36_36]_2-0)** [Young (2017)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2017), p. [36](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA36).
3. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200742_3-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200742_3-1) [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 42.
4. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA37_37],_[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA78_78]_4-0)** [Young (2017)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2017), p. [37](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA37), [78](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA78).
5. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]_5-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]_5-1) [Axler (2012)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFAxler2012), p. [634](https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA634).
6. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA632_632]_6-0)** [Axler (2012)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFAxler2012), p. [632](https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA632).
7. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-7)**
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Cross Product"](https://mathworld.wolfram.com/CrossProduct.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*. Retrieved 5 June 2025.
8. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-8)**
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Dot Product"](https://mathworld.wolfram.com/DotProduct.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*. Retrieved 5 June 2025.
9. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200741_9-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 41.
10. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA68_68]_10-0)** [Young (2017)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2017), p. [68](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA68).
11. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200747_11-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 47.
12. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200741–42_12-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 41–42.
13. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200741,_43_13-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 41, 43.
14. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2012[httpsbooksgooglecombooksidOMrcN0a3LxICpgRA1-PA165_165]_14-0)** [Young (2012)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2012), p. [165](https://books.google.com/books?id=OMrcN0a3LxIC&pg=RA1-PA165).
15. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200742,_47_15-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 42, 47.
16. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-16)**
["OEIS A003957"](https://oeis.org/A003957). *oeis.org*. Retrieved 2019-05-26.
17. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]_17-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]_17-1) [Bourchtein & Bourchtein (2022)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFBourchteinBourchtein2022), p. [294](https://books.google.com/books?id=nGxOEAAAQBAJ&pg=PA294).
18. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007115_18-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 115.
19. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007155_19-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 155.
20. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007157_20-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 157.
21. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007199_21-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 199.
22. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVince2023[httpsbooksgooglecombooksidGnW6EAAAQBAJpgPA162_162]_22-0)** [Vince (2023)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVince2023), p. [162](https://books.google.com/books?id=GnW6EAAAQBAJ&pg=PA162).
23. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAdlaj2012_23-0)** [Adlaj (2012)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFAdlaj2012).
24. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-24)**
OEIS
[sequence A105419 (Decimal expansion of the arc length of the sine or cosine curve for one full period.)](https://oeis.org/A105419)
25. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007365_25-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 365.
26. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA99_99]_26-0)** [Young (2017)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2017), p. [99](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA99).
27. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-28)**
Dennis G. Zill (2013). *Precalculus with Calculus Previews*. Jones & Bartlett Publishers. p. 238. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-4496-4515-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4496-4515-1 "Special:BookSources/978-1-4496-4515-1")
.
[Extract of page 238](https://books.google.com/books?id=dtS5M4lx7scC&pg=PA238)
28. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-29)**
["Sine-squared function"](https://calculus.subwiki.org/wiki/Sine-squared_function#Identities). Retrieved August 9, 2019.
29. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007491–492_30-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007491–492_30-1) [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 491–492.
30. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]_31-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]_31-1) [Abramowitz & Stegun (1970)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFAbramowitzStegun1970), p. [74](https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA74).
31. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEPowell1981150_32-0)** [Powell (1981)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFPowell1981), p. 150.
32. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTERudin198788_33-0)** [Rudin (1987)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFRudin1987), p. 88.
33. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEZygmund19681_34-0)** [Zygmund (1968)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFZygmund1968), p. 1.
34. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEZygmund196811_35-0)** [Zygmund (1968)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFZygmund1968), p. 11.
35. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]_36-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]_36-1) [***c***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]_36-2) [Howie (2003)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFHowie2003), p. [24](https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24).
36. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTERudin19872_37-0)** [Rudin (1987)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFRudin1987), p. 2.
37. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-38)**
Brown, James Ward; [Churchill, Ruel](https://en.wikipedia.org/wiki/Ruel_Vance_Churchill "Ruel Vance Churchill") (2014). *Complex Variables and Applications* (9th ed.). [McGraw-Hill](https://en.wikipedia.org/wiki/McGraw-Hill "McGraw-Hill"). p. 105. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-338317-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-338317-0 "Special:BookSources/978-0-07-338317-0")
.
38. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_23–24]_39-0)** [Howie (2003)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFHowie2003), p. [23–24](https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24).
39. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-40)**
["Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?"](https://math.stackexchange.com/q/220418). *math.stackexchange.com*. Retrieved 2019-08-12.
40. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEPlofker2009[httpsbooksgooglecombooksidDHvThPNp9yMCpgPA257_257]_41-0)** [Plofker (2009)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFPlofker2009), p. [257](https://books.google.com/books?id=DHvThPNp9yMC&pg=PA257).
41. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEMaor1998[httpsbooksgooglecombooksidr9aMrneWFpUCpgPA35_35]_42-0)** [Maor (1998)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFMaor1998), p. [35](https://books.google.com/books?id=r9aMrneWFpUC&pg=PA35).
42. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEMerzbachBoyer2011_43-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEMerzbachBoyer2011_43-1) [***c***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEMerzbachBoyer2011_43-2) [Merzbach & Boyer (2011)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFMerzbachBoyer2011).
43. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEMaor199835–36_44-0)** [Maor (1998)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFMaor1998), p. 35–36.
44. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEKatz2008253_45-0)** [Katz (2008)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFKatz2008), p. 253.
45. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTESmith1958202_46-0)** [Smith (1958)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFSmith1958), p. 202.
46. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-47)**
Various sources credit the first use of *sinus* to either
- [Plato Tiburtinus](https://en.wikipedia.org/wiki/Plato_Tiburtinus "Plato Tiburtinus")'s 1116 translation of the *Astronomy* of [Al-Battani](https://en.wikipedia.org/wiki/Al-Battani "Al-Battani")
- [Gerard of Cremona](https://en.wikipedia.org/wiki/Gerard_of_Cremona "Gerard of Cremona")'s translation of the *Algebra* of [al-Khwārizmī](https://en.wikipedia.org/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB "Muḥammad ibn Mūsā al-Khwārizmī")
- [Robert of Chester](https://en.wikipedia.org/wiki/Robert_of_Chester "Robert of Chester")'s 1145 translation of the tables of al-Khwārizmī
See [Merlet (2004)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFMerlet2004). See [Maor (1998)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFMaor1998), Chapter 3, for an earlier etymology crediting Gerard. See [Katz (2008)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFKatz2008), p. 210.
47. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-48)**
Fale's book alternately uses the spellings "sine", "signe", or "sign".
Fale, Thomas (1593). [*Horologiographia. The Art of Dialling: Teaching, an Easie and Perfect Way to make all Kindes of Dials ...*](https://archive.org/details/b30333106/page/19/mode/1up) London: F. Kingston. p. 11, for example.
48. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEGunter1620_49-0)** [Gunter (1620)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFGunter1620).
49. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-50)**
Brendan, T. (February 1965). "How Ptolemy constructed trigonometry tables". *The Mathematics Teacher*. **58** (2): 141–149\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.5951/MT.58.2.0141](https://doi.org/10.5951%2FMT.58.2.0141). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [27967990](https://www.jstor.org/stable/27967990).
50. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-51)**
[Van Brummelen, Glen](https://en.wikipedia.org/wiki/Glen_Van_Brummelen "Glen Van Brummelen") (2009). "India". *The Mathematics of the Heavens and the Earth*. Princeton University Press. Ch. 3, pp. 94–134. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-691-12973-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-12973-0 "Special:BookSources/978-0-691-12973-0")
.
51. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-Gingerich_1986_52-0)**
Gingerich, Owen (1986). ["Islamic Astronomy"](https://web.archive.org/web/20131019140821/http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm). *[Scientific American](https://en.wikipedia.org/wiki/Scientific_American "Scientific American")*. Vol. 254. p. 74. Archived from [the original](http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm) on 2013-10-19. Retrieved 2010-07-13.
52. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-Sesiano_53-0)**
Jacques Sesiano, "Islamic mathematics", p. 157, in
[Selin, Helaine](https://en.wikipedia.org/wiki/Helaine_Selin "Helaine Selin"); [D'Ambrosio, Ubiratan](https://en.wikipedia.org/wiki/Ubiratan_D%27Ambrosio "Ubiratan D'Ambrosio"), eds. (2000). *Mathematics Across Cultures: The History of Non-western Mathematics*. [Springer Science+Business Media](https://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Media "Springer Science+Business Media"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-4020-0260-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-0260-1 "Special:BookSources/978-1-4020-0260-1")
.
53. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-Britannica_54-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-Britannica_54-1)
["trigonometry"](http://www.britannica.com/EBchecked/topic/605281/trigonometry). Encyclopedia Britannica. 17 June 2024.
54. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-55)**
Nicolás Bourbaki (1994). [*Elements of the History of Mathematics*](https://archive.org/details/elementsofhistor0000bour). Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9783540647676](https://en.wikipedia.org/wiki/Special:BookSources/9783540647676 "Special:BookSources/9783540647676")
.
55. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-56)** "[Why the sine has a simple derivative](http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf) [Archived](https://web.archive.org/web/20110720102700/http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf) 2011-07-20 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine")", in *[Historical Notes for Calculus Teachers](http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm) [Archived](https://web.archive.org/web/20110720102613/http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm) 2011-07-20 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine")* by [V. Frederick Rickey](http://www.math.usma.edu/people/rickey/) [Archived](https://web.archive.org/web/20110720102654/http://www.math.usma.edu/people/rickey/) 2011-07-20 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine")
56. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEZimmermann2006_57-0)** [Zimmermann (2006)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFZimmermann2006).
57. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-matlab_58-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-matlab_58-1)
["sinpi - Compute sin(X\*pi) accurately"](https://www.mathworks.com/help/matlab/ref/double.sinpi.html). *www.mathworks.com*. [Archived](http://web.archive.org/web/20251123002724/https://www.mathworks.com/help/matlab/ref/double.sinpi.html) from the original on 2025-11-23. Retrieved 2026-01-08.
58. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-r_59-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-r_59-1)
["Trig function - RDocumentation"](https://www.rdocumentation.org/packages/base/versions/3.5.3/topics/Trig). *www.rdocumentation.org*. Retrieved 2026-02-17.
59. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-60)**
["sin, sincos, sinh, sinpi"](https://registry.khronos.org/OpenCL/sdk/1.0/docs/man/xhtml/sin.html). *registry.khronos.org*. Retrieved 2026-02-17.
60. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-61)**
["sinpi » Julia Functions"](http://www.jlhub.com/julia/manual/en/function/sinpi). *www.jlhub.com*. Retrieved 2026-02-17.
61. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-62)**
["Double Precision Mathematical Functions"](http://web.archive.org/web/20240723062728/https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__DOUBLE.html). *docs.nvidia.com*. Archived from [the original](https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__DOUBLE.html) on 2024-07-23. Retrieved 2026-01-08.
62. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-63)**
["Documentation – Arm Developer"](https://developer.arm.com/documentation/100614/latest/b-opencl-built-in-functions/b2-math-functions). *developer.arm.com*. Retrieved 2026-02-17.
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["Special functions (scipy.special) — SciPy v1.17.0 Manual"](https://docs.scipy.org/doc/scipy/reference/special.html#convenience-functions). *docs.scipy.org*. Retrieved 25 February 2026.
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["AAS33051: Precision Angle Sensor IC with Incremental and Motor Commutation Outputs and On-Chip Linearization"](http://web.archive.org/web/20190417143715/https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx). *www.allegromicro.com*. Archived from [the original](https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx) on 2019-04-17. Retrieved 2026-02-17.
### Works cited
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=30 "Edit section: Works cited")\]
- [Abramowitz, Milton](https://en.wikipedia.org/wiki/Milton_Abramowitz "Milton Abramowitz"); [Stegun, Irene A.](https://en.wikipedia.org/wiki/Irene_Stegun "Irene Stegun") (1970), *[Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun "Abramowitz and Stegun")*, New York: [Dover Publications](https://en.wikipedia.org/wiki/Dover_Publications "Dover Publications"), Ninth printing
- Adlaj, Semjon (2012), ["An Eloquent Formula for the Perimeter of an Ellipse"](https://www.ams.org/notices/201208/rtx120801094p.pdf) (PDF), *American Mathematical Society*, **59** (8): 1097
- Axler, Sheldon (2012), *Algebra and Trigonometry*, [John Wiley & Sons](https://en.wikipedia.org/wiki/John_Wiley_%26_Sons "John Wiley & Sons"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0470-58579-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0470-58579-5 "Special:BookSources/978-0470-58579-5")
- [Bourchtein, Ludmila](https://en.wikipedia.org/wiki/Ludmila_Bourchtein "Ludmila Bourchtein"); Bourchtein, Andrei (2022), *Theory of Infinite Sequences and Series*, Springer, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-030-79431-6](https://doi.org/10.1007%2F978-3-030-79431-6), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-030-79431-6](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-79431-6 "Special:BookSources/978-3-030-79431-6")
- [Gunter, Edmund](https://en.wikipedia.org/wiki/Edmund_Gunter "Edmund Gunter") (1620), *Canon triangulorum*
- Howie, John M. (2003), *Complex Analysis*, Springer Undergraduate Mathematics Series, Springer, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-1-4471-0027-0](https://doi.org/10.1007%2F978-1-4471-0027-0), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-4471-0027-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4471-0027-0 "Special:BookSources/978-1-4471-0027-0")
- Traupman, Ph.D., John C. (1966), [*The New College Latin & English Dictionary*](https://archive.org/details/boysgirlsbookabo00gard_0), Toronto: Bantam, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-553-27619-0](https://en.wikipedia.org/wiki/Special:BookSources/0-553-27619-0 "Special:BookSources/0-553-27619-0")
- Katz, Victor J. (2008), [*A History of Mathematics*](http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf) (PDF) (3rd ed.), Boston: Addison-Wesley, "The English word "sine" comes from a series of mistranslations of the Sanskrit *jyā-ardha* (chord-half). Āryabhaṭa frequently abbreviated this term to *jyā* or its synonym *jīvá*. When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an otherwise meaningless Arabic word *jiba*. But since Arabic is written without vowels, later writers interpreted the consonants *jb* as *jaib*, which means bosom or breast. In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word *sinus*, which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf."
- Maor, Eli (1998), *Trigonometric Delights*, [Princeton University Press](https://en.wikipedia.org/wiki/Princeton_University_Press "Princeton University Press"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[1-4008-4282-4](https://en.wikipedia.org/wiki/Special:BookSources/1-4008-4282-4 "Special:BookSources/1-4008-4282-4")
- Merlet, Jean-Pierre (2004), "A Note on the History of the Trigonometric Functions", in Ceccarelli, Marco (ed.), *International Symposium on History of Machines and Mechanisms*, Springer, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/1-4020-2204-2](https://doi.org/10.1007%2F1-4020-2204-2), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-4020-2203-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-2203-6 "Special:BookSources/978-1-4020-2203-6")
- [Merzbach, Uta C.](https://en.wikipedia.org/wiki/Uta_Merzbach "Uta Merzbach"); [Boyer, Carl B.](https://en.wikipedia.org/wiki/Carl_B._Boyer "Carl B. Boyer") (2011), *A History of Mathematics* (3rd ed.), [John Wiley & Sons](https://en.wikipedia.org/wiki/John_Wiley_%26_Sons "John Wiley & Sons"), "It was Robert of Chester's translation from Arabic that resulted in our word "sine". The Hindus had given the name jiva to the half-chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language, there is also the word jaib meaning "bay" or "inlet". When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence, he used the word sinus, the Latin word for "bay" or "inlet"."
- Plofker (2009), [*Mathematics in India*](https://en.wikipedia.org/wiki/Mathematics_in_India_\(book\) "Mathematics in India (book)"), [Princeton University Press](https://en.wikipedia.org/wiki/Princeton_University_Press "Princeton University Press")
- [Powell, Michael J. D.](https://en.wikipedia.org/wiki/Michael_J._D._Powell "Michael J. D. Powell") (1981), *Approximation Theory and Methods*, [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-521-29514-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-29514-7 "Special:BookSources/978-0-521-29514-7")
- [Rudin, Walter](https://en.wikipedia.org/wiki/Walter_Rudin "Walter Rudin") (1987), *Real and complex analysis* (3rd ed.), New York: [McGraw-Hill](https://en.wikipedia.org/wiki/McGraw-Hill "McGraw-Hill"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-054234-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-054234-1 "Special:BookSources/978-0-07-054234-1")
, [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0924157](https://mathscinet.ams.org/mathscinet-getitem?mr=0924157)
- Smith, D. E. (1958) \[1925\], *History of Mathematics*, vol. I, [Dover Publications](https://en.wikipedia.org/wiki/Dover_Publications "Dover Publications"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-486-20429-4](https://en.wikipedia.org/wiki/Special:BookSources/0-486-20429-4 "Special:BookSources/0-486-20429-4")
`{{citation}}`: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date "Help:CS1 errors"))
- Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), [*Calculus*](https://archive.org/details/matematika-a-purcell-calculus-9th-ed/mode/2up) (9th ed.), [Pearson Prentice Hall](https://en.wikipedia.org/wiki/Pearson_Prentice_Hall "Pearson Prentice Hall"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0131469686](https://en.wikipedia.org/wiki/Special:BookSources/978-0131469686 "Special:BookSources/978-0131469686")
- Vince, John (2023), *Calculus for Computer Graphics*, Springer, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-031-28117-4](https://doi.org/10.1007%2F978-3-031-28117-4), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-031-28117-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-031-28117-4 "Special:BookSources/978-3-031-28117-4")
- [Young, Cynthia](https://en.wikipedia.org/wiki/Cynthia_Y._Young "Cynthia Y. Young") (2012), *Trigonometry* (3rd ed.), John Wiley & Sons, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-119-32113-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-119-32113-2 "Special:BookSources/978-1-119-32113-2")
- ——— (2017), *Trigonometry* (4th ed.), John Wiley & Sons, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-119-32113-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-119-32113-2 "Special:BookSources/978-1-119-32113-2")
- Zimmermann, Paul (2006), "Can we trust floating-point numbers?", [*Grand Challenges of Informatics*](http://www.jaist.ac.jp/~bjorner/ae-is-budapest/talks/Sept20pm2_Zimmermann.pdf) (PDF), p. 14/31
- [Zygmund, Antoni](https://en.wikipedia.org/wiki/Antoni_Zygmund "Antoni Zygmund") (1968), [*Trigonometric Series*](https://archive.org/details/trigonometricser0012azyg/) (2nd, reprinted ed.), [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"), [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0236587](https://mathscinet.ams.org/mathscinet-getitem?mr=0236587)
## External links
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Look up ***[sine](https://en.wiktionary.org/wiki/sine "wiktionary:sine")*** in Wiktionary, the free dictionary.
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| [v](https://en.wikipedia.org/wiki/Template:Trigonometric_and_hyperbolic_functions "Template:Trigonometric and hyperbolic functions") [t](https://en.wikipedia.org/wiki/Template_talk:Trigonometric_and_hyperbolic_functions "Template talk:Trigonometric and hyperbolic functions") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Trigonometric_and_hyperbolic_functions "Special:EditPage/Template:Trigonometric and hyperbolic functions")Trigonometric and hyperbolic functions | |
|---|---|
| Groups | [Trigonometric](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions") [Sine and cosine]() [Inverse trigonometric](https://en.wikipedia.org/wiki/Inverse_trigonometric_functions "Inverse trigonometric functions") [Hyperbolic](https://en.wikipedia.org/wiki/Hyperbolic_functions "Hyperbolic functions") [Inverse hyperbolic](https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions "Inverse hyperbolic functions") |
| Other | [Versine](https://en.wikipedia.org/wiki/Versine "Versine") [Exsecant](https://en.wikipedia.org/wiki/Exsecant "Exsecant") [Jyā, koti-jyā and utkrama-jyā](https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81 "Jyā, koti-jyā and utkrama-jyā") [atan2](https://en.wikipedia.org/wiki/Atan2 "Atan2") |
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Sine and cosine
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| Readable Markdown | | Sine and cosine | |
|---|---|
| [](https://en.wikipedia.org/wiki/File:Sine_cosine_one_period.svg) | |
| General information | |
| General definition | ![{\\displaystyle {\\begin{aligned}&\\sin(\\theta )={\\frac {\\textrm {opposite}}{\\textrm {hypotenuse}}}\\\\\[8pt\]&\\cos(\\theta )={\\frac {\\textrm {adjacent}}{\\textrm {hypotenuse}}}\\\\\[8pt\]\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52374fd3474dfab1331993d6c170e9cac82f4a4a) |
In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), **sine** and **cosine** are [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions") of an [angle](https://en.wikipedia.org/wiki/Angle "Angle"). The sine and cosine of an [acute angle](https://en.wikipedia.org/wiki/Acute_angle "Acute angle") are defined in the context of a [right triangle](https://en.wikipedia.org/wiki/Right_triangle "Right triangle"): for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the [triangle](https://en.wikipedia.org/wiki/Triangle "Triangle") (the [hypotenuse](https://en.wikipedia.org/wiki/Hypotenuse "Hypotenuse")), and the cosine is the [ratio](https://en.wikipedia.org/wiki/Ratio "Ratio") of the length of the adjacent leg to that of the [hypotenuse](https://en.wikipedia.org/wiki/Hypotenuse "Hypotenuse"). For an angle , the sine and cosine functions are denoted as  and .
The definitions of sine and cosine have been extended to any [real](https://en.wikipedia.org/wiki/Real_number "Real number") value in terms of the lengths of certain line segments in a [unit circle](https://en.wikipedia.org/wiki/Unit_circle "Unit circle"). More modern definitions express the sine and cosine as [infinite series](https://en.wikipedia.org/wiki/Series_\(mathematics\) "Series (mathematics)"), or as the solutions of certain [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation"), allowing their extension to arbitrary positive and negative values and even to [complex numbers](https://en.wikipedia.org/wiki/Complex_number "Complex number").
The sine and cosine functions are commonly used to model [periodic](https://en.wikipedia.org/wiki/Periodic_function "Periodic function") phenomena such as [sound](https://en.wikipedia.org/wiki/Sound "Sound") and [light waves](https://en.wikipedia.org/wiki/Light_waves "Light waves"), the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the [*jyā* and *koṭi-jyā*](https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81 "Jyā, koti-jyā and utkrama-jyā") functions used in [Indian astronomy](https://en.wikipedia.org/wiki/Indian_astronomy "Indian astronomy") during the [Gupta period](https://en.wikipedia.org/wiki/Gupta_period "Gupta period").
## Elementary descriptions
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=1 "Edit section: Elementary descriptions")\]
### Right-angled triangle definition
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=2 "Edit section: Right-angled triangle definition")\]
[](https://en.wikipedia.org/wiki/File:Trigono_sine_en2.svg)
For the angle *α*, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.
To define the sine and cosine of an acute angle , start with a [right triangle](https://en.wikipedia.org/wiki/Right_triangle "Right triangle") that contains an angle of measure ; in the accompanying figure, angle  in a right triangle  is the angle of interest. The three sides of the triangle are named as follows:[\[1\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]-1)
Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, and the cosine of the angle is equal to the length of the adjacent side divided by the length of the hypotenuse:[\[1\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]-1) 
The other trigonometric functions of the angle can be defined similarly; for example, the [tangent](https://en.wikipedia.org/wiki/Trigonometric_functions#Right-angled_triangle_definitions "Trigonometric functions") is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and cosine functions. The [reciprocal](https://en.wikipedia.org/wiki/Multiplicative_inverse "Multiplicative inverse") of sine is cosecant, which gives the ratio of the hypotenuse length to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the hypotenuse length to that of the adjacent side. The cotangent function is the ratio between the adjacent and opposite sides, a reciprocal of a tangent function. These functions can be formulated as:[\[1\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]-1) 
### Special angle measures
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=3 "Edit section: Special angle measures")\]
As stated, the values  and  appear to depend on the choice of a right triangle containing an angle of measure . However, this is not the case as all such triangles are [similar](https://en.wikipedia.org/wiki/Similarity_\(geometry\) "Similarity (geometry)"), and so the ratios are the same for each of them. For example, each [leg](https://en.wikipedia.org/wiki/Catheti "Catheti") of the 45-45-90 right triangle is 1 unit, and its hypotenuse is ; therefore, .[\[2\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA36_36]-2) The following table shows the special value of each input for both sine and cosine with the domain between . The input in this table provides various unit systems such as degree, radian, and so on. The angles other than those five can be obtained by using a calculator.[\[3\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200742-3)[\[4\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA37_37],_[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA78_78]-4)
| Angle, x | sin(*x*) | cos(*x*) | | | | | |
|---|---|---|---|---|---|---|---|
| [Degrees](https://en.wikipedia.org/wiki/Degree_\(angle\) "Degree (angle)") | [Radians](https://en.wikipedia.org/wiki/Radian "Radian") | [Gradians](https://en.wikipedia.org/wiki/Gradian "Gradian") | [Turns](https://en.wikipedia.org/wiki/Turn_\(geometry\) "Turn (geometry)") | Exact | Decimal | Exact | Decimal |
| 0° | 0 |  | | | | | |
[](https://en.wikipedia.org/wiki/File:Law_of_sines_\(simple\).svg)
Law of sines and cosines' illustration
The [law of sines](https://en.wikipedia.org/wiki/Law_of_sines "Law of sines") is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known.[\[5\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]-5) Given a triangle  with sides , , and , and angles opposite those sides , , and , the law states,  This is equivalent to the equality of the first three expressions below:  where  is the triangle's [circumradius](https://en.wikipedia.org/wiki/Circumcircle "Circumcircle").
The [law of cosines](https://en.wikipedia.org/wiki/Law_of_cosines "Law of cosines") is useful for computing the length of an unknown side if two other sides and an angle are known.[\[5\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]-5) The law states,  In the case where  from which , the resulting equation becomes the [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem").[\[6\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA632_632]-6)
The [cross product](https://en.wikipedia.org/wiki/Cross_product "Cross product") and [dot product](https://en.wikipedia.org/wiki/Dot_product "Dot product") are operations on two [vectors](https://en.wikipedia.org/wiki/Vector_\(mathematics_and_physics\) "Vector (mathematics and physics)") in [Euclidean vector space](https://en.wikipedia.org/wiki/Euclidean_vector_space "Euclidean vector space"). The sine and cosine functions can be defined in terms of the cross product and dot product. If  and  are vectors, and  is the angle between  and , then sine and cosine can be defined as:[\[7\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-7)[\[8\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-8) 
## Analytic descriptions
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=6 "Edit section: Analytic descriptions")\]
### Unit circle definition
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=7 "Edit section: Unit circle definition")\]
The sine and cosine functions may also be defined in a more general way by using [unit circle](https://en.wikipedia.org/wiki/Unit_circle#Trigonometric_functions_on_the_unit_circle "Unit circle"), a circle of radius one centered at the origin , formulated as the equation of  in the [Cartesian coordinate system](https://en.wikipedia.org/wiki/Cartesian_coordinate_system "Cartesian coordinate system"). Let a line through the origin intersect the unit circle, making an angle of  with the positive half of the \-axis. The \- and \-coordinates of this point of intersection are equal to  and , respectively; that is,[\[9\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200741-9) 
This definition is consistent with the right-angled triangle definition of sine and cosine when  because the length of the hypotenuse of the unit circle is always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the \-coordinate. A similar argument can be made for the cosine function to show that the cosine of an angle when , even under the new definition using the unit circle.[\[10\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA68_68]-10)[\[11\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200747-11)
#### Graph of a function and its elementary properties
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=8 "Edit section: Graph of a function and its elementary properties")\]
[](https://en.wikipedia.org/wiki/File:Circle_cos_sin.gif)
Animation demonstrating how the sine function (in red) is graphed from the *y*\-coordinate (red dot) of a point on the [unit circle](https://en.wikipedia.org/wiki/Unit_circle "Unit circle") (in green), at an angle of *θ*. The cosine (in blue) is the *x*\-coordinate.
Using the unit circle definition has the advantage of drawing the graph of sine and cosine functions. This can be done by rotating counterclockwise a point along the circumference of a circle, depending on the input . In a sine function, if the input is , the point is rotated counterclockwise and stopped exactly on the \-axis. If , the point is at the circle's halfway point. If , the point returns to its origin. This results in both sine and cosine functions having the [range](https://en.wikipedia.org/wiki/Range_of_a_function "Range of a function") between .[\[12\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200741–42-12)
Extending the angle to any real domain, the point rotated counterclockwise continuously. This can be done similarly for the cosine function as well, although the point is rotated initially from the \-coordinate. In other words, both sine and cosine functions are [periodic](https://en.wikipedia.org/wiki/Periodic_function "Periodic function"), meaning any angle added by the circle's circumference is the angle itself. Mathematically,[\[13\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200741,_43-13) 
A function  is said to be [odd](https://en.wikipedia.org/wiki/Odd_function "Odd function") if , and is said to be [even](https://en.wikipedia.org/wiki/Even_function "Even function") if . The sine function is odd, whereas the cosine function is even.[\[14\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2012[httpsbooksgooglecombooksidOMrcN0a3LxICpgRA1-PA165_165]-14) Both sine and cosine functions are similar, with their difference being [shifted](https://en.wikipedia.org/wiki/Translation_\(geometry\) "Translation (geometry)") by . This phase shift can be expressed as  or . This is distinct from the cofunction identities that follow below, which arise from right-triangle geometry and are not phase shifts: [\[15\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200742,_47-15) 
[](https://en.wikipedia.org/wiki/File:Cosine_fixed_point.svg)
The fixed point iteration *x**n*\+1 = cos(*xn*) with initial value *x*0 = −1 converges to the Dottie number.
Zero is the only real [fixed point](https://en.wikipedia.org/wiki/Fixed_point_\(mathematics\) "Fixed point (mathematics)") of the sine function; in other words the only intersection of the sine function and the [identity function](https://en.wikipedia.org/wiki/Identity_function "Identity function") is . The only real fixed point of the cosine function is called the [Dottie number](https://en.wikipedia.org/wiki/Dottie_number "Dottie number"). The Dottie number is the unique real root of the equation . The decimal expansion of the Dottie number is approximately 0.739085.[\[16\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-16)
#### Continuity and differentiation
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=9 "Edit section: Continuity and differentiation")\]
[](https://en.wikipedia.org/wiki/File:Sine_quads_01_Pengo.svg)
The quadrants of the unit circle and of sin(*x*), using the [Cartesian coordinate system](https://en.wikipedia.org/wiki/Cartesian_coordinate_system "Cartesian coordinate system")
The sine and cosine functions are infinitely differentiable.[\[17\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]-17) The derivative of sine is cosine, and the derivative of cosine is negative sine:[\[18\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007115-18)  Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself.[\[17\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]-17) These derivatives can be applied to the [first derivative test](https://en.wikipedia.org/wiki/First_derivative_test "First derivative test"), according to which the [monotonicity](https://en.wikipedia.org/wiki/Monotone_function "Monotone function") of a function can be defined as the inequality of function's first derivative greater or less than equal to zero.[\[19\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007155-19) It can also be applied to [second derivative test](https://en.wikipedia.org/wiki/Second_derivative_test "Second derivative test"), according to which the [concavity](https://en.wikipedia.org/wiki/Concave_function "Concave function") of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero.[\[20\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007157-20) The following table shows that both sine and cosine functions have concavity and monotonicity—the positive sign () denotes a graph is increasing (going upward) and the negative sign () is decreasing (going downward)—in certain intervals.[\[3\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon200742-3) This information can be represented as a Cartesian coordinates system divided into four quadrants.
| [Quadrant](https://en.wikipedia.org/wiki/Cartesian_coordinate_system#Quadrants_and_octants "Cartesian coordinate system") | Angle | Sine | Cosine | | | | |
|---|---|---|---|---|---|---|---|
| [Degrees](https://en.wikipedia.org/wiki/Degree_\(angle\) "Degree (angle)") | [Radians](https://en.wikipedia.org/wiki/Radian "Radian") | [Sign](https://en.wikipedia.org/wiki/Sign_\(mathematics\) "Sign (mathematics)") | [Monotony](https://en.wikipedia.org/wiki/Monotonic_function "Monotonic function") | [Convexity](https://en.wikipedia.org/wiki/Convex_function "Convex function") | [Sign](https://en.wikipedia.org/wiki/Sign_\(mathematics\) "Sign (mathematics)") | [Monotony](https://en.wikipedia.org/wiki/Monotonic_function "Monotonic function") | [Convexity](https://en.wikipedia.org/wiki/Convex_function "Convex function") |
| 1st quadrant, I |  | | | | | | |
Both sine and cosine functions can be defined by using differential equations. The pair of  is the solution  to the two-dimensional system of [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation")  and  with the [initial conditions](https://en.wikipedia.org/wiki/Initial_conditions "Initial conditions")  and . One could interpret the unit circle in the above definitions as defining the [phase space trajectory](https://en.wikipedia.org/wiki/Phase_space_trajectory "Phase space trajectory") of the differential equation with the given initial conditions. It can be interpreted as a phase space trajectory of the system of differential equations  and  starting from the initial conditions  and .\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
#### Integral and the usage in mensuration
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=10 "Edit section: Integral and the usage in mensuration")\]
Their area under a curve can be obtained by using the [integral](https://en.wikipedia.org/wiki/Integral "Integral") with a certain bounded interval. Their antiderivatives are:  where  denotes the [constant of integration](https://en.wikipedia.org/wiki/Constant_of_integration "Constant of integration").[\[21\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007199-21) These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval. For example, the [arc length](https://en.wikipedia.org/wiki/Arc_length "Arc length") of the sine curve between  and  is  where  is the [incomplete elliptic integral of the second kind](https://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind "Elliptic integral") with modulus . It cannot be expressed using [elementary functions](https://en.wikipedia.org/wiki/Elementary_function "Elementary function").[\[22\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVince2023[httpsbooksgooglecombooksidGnW6EAAAQBAJpgPA162_162]-22) In the case of a full period, its arc length is  where  is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function "Gamma function") and  is the [lemniscate constant](https://en.wikipedia.org/wiki/Lemniscate_constant "Lemniscate constant").[\[23\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAdlaj2012-23)[\[24\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-24)
[](https://en.wikipedia.org/wiki/File:Arcsine_Arccosine.svg)
The usual principal values of the arcsin(*x*) and arccos(*x*) functions graphed on the Cartesian plane
The functions  and  (as well as those functions with the same function rule and domain whose codomain is a subset of  containing the interval ![{\\displaystyle \\left\[-1,1\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79566f857ac1fcd0ef0f62226298a4ed15b796ad)) are not bijective and therefore do not have inverse functions. For example, , but also , . Sine's "inverse", called arcsine, can then be described not as a function but a relation (for example, all integer multiples of  would have an arcsine of zero). To define the inverse functions of sine and cosine, they must be restricted to their [principal branches](https://en.wikipedia.org/wiki/Principal_branch "Principal branch") by restricting their domain and codomain; the standard functions used to define arcsine and arccosine are then ![{\\displaystyle \\sin :\\left\[-\\pi /2,\\pi /2\\right\]\\to \\left\[-1,1\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34676d3cc30ba3520fb506118910a65ae45fe25f) and ![{\\displaystyle \\cos :\\left\[0,\\pi \\right\]\\to \\left\[-1,1\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2864b5c77dc4b8af5111e9d4e78e9ea9a876b7).[\[25\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007365-25) These are bijective and have inverses: ![{\\displaystyle \\arcsin :\\left\[-1,1\\right\]\\to \\left\[-\\pi /2,\\pi /2\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/024a4d91e022705b40fb40e554d294bdd9e2eea4) and ![{\\displaystyle \\arccos :\\left\[-1,1\\right\]\\to \\left\[0,\\pi \\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1126b78ee63338e44c6a25dae46d50fdad0b5c1). Alternative notation is  for arcsine and  for arccosine. Using these definitions, one obtains the identity maps:
![{\\displaystyle {\\begin{aligned}\\sin \\circ \\arcsin \\,(x)&=x\\qquad x\\in \\left\[-1,1\\right\]\\\\\\arcsin \\circ \\sin \\,(x)&=x\\qquad x\\in \\left\[-\\pi /2,\\pi /2\\right\]\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/631b526ef79de3155b8c652483e136009c87f6ef)and
![{\\displaystyle {\\begin{aligned}\\cos \\circ \\arccos \\,(x)&=x\\qquad x\\in \\left\[-1,1\\right\]\\\\\\arccos \\circ \\cos \\,(x)&=x\\qquad x\\in \\left\[0,\\pi \\right\]\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81aa8f5001ec127005efd5c2508631440ff6ab8d)
An acute angle  is given by:  where for some integer ,  By definition, both functions satisfy the equations:  and 
According to [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem"), the squared hypotenuse is the sum of two squared legs of a right triangle. Dividing the formula on both sides with squared hypotenuse resulting in the [Pythagorean trigonometric identity](https://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity "Pythagorean trigonometric identity"), the sum of a squared sine and a squared cosine equals 1:[\[26\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA99_99]-26)[\[a\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-27) 
Sine and cosine satisfy the following double-angle formulas:[\[27\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-28) 
[](https://en.wikipedia.org/wiki/File:SinSquared.png)
Sine function in blue and sine squared function in red. The *x*\-axis is in radians.
The cosine double angle formula implies that sin2 and cos2 are, themselves, shifted and scaled sine waves. Specifically,[\[28\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-29)  The graph shows both sine and sine squared functions, with the sine in blue and the sine squared in red. Both graphs have the same shape but with different ranges of values and different periods. Sine squared has only positive values, but twice the number of periods.\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
### Series and polynomials
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=13 "Edit section: Series and polynomials")\]
[](https://en.wikipedia.org/wiki/File:Sine.gif)
This animation shows how including more and more terms in the partial sum of its Taylor series approaches a sine curve.
Both sine and cosine functions can be defined by using a [Taylor series](https://en.wikipedia.org/wiki/Taylor_series "Taylor series"), a [power series](https://en.wikipedia.org/wiki/Power_series "Power series") involving the higher-order derivatives. As mentioned in [§ Continuity and differentiation](https://en.wikipedia.org/wiki/Sine_and_cosine#Continuity_and_differentiation), the [derivative](https://en.wikipedia.org/wiki/Derivative "Derivative") of sine is cosine and the derivative of cosine is the negative of sine. This means the successive derivatives of  are , , , , continuing to repeat those four functions. The \-th derivative, evaluated at the point 0:  where the superscript represents repeated differentiation. This implies the following Taylor series expansion at . One can then use the theory of [Taylor series](https://en.wikipedia.org/wiki/Taylor_series "Taylor series") to show that the following identities hold for all [real numbers](https://en.wikipedia.org/wiki/Real_number "Real number") —where  is the angle in radians.[\[29\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007491–492-30) More generally, for all [complex numbers](https://en.wikipedia.org/wiki/Complex_number "Complex number"):[\[30\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]-31)  Taking the derivative of each term gives the Taylor series for cosine:[\[29\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEVarbergPurcellRigdon2007491–492-30)[\[30\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]-31) 
Both sine and cosine functions with multiple angles may appear as their [linear combination](https://en.wikipedia.org/wiki/Linear_combination "Linear combination"), resulting in a polynomial. Such a polynomial is known as the [trigonometric polynomial](https://en.wikipedia.org/wiki/Trigonometric_polynomial "Trigonometric polynomial"). The trigonometric polynomial's ample applications may be acquired in [its interpolation](https://en.wikipedia.org/wiki/Trigonometric_interpolation "Trigonometric interpolation"), and its extension of a periodic function known as the [Fourier series](https://en.wikipedia.org/wiki/Fourier_series "Fourier series"). Let  and  be any coefficients, then the trigonometric polynomial of a degree —denoted as —is defined as:[\[31\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEPowell1981150-32)[\[32\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTERudin198788-33) 
The [trigonometric series](https://en.wikipedia.org/wiki/Trigonometric_series "Trigonometric series") can be defined similarly analogous to the trigonometric polynomial, its infinite inversion. Let  and  be any coefficients, then the trigonometric series can be defined as:[\[33\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEZygmund19681-34)  In the case of a Fourier series with a given integrable function , the coefficients of a trigonometric series are:[\[34\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEZygmund196811-35) 
## Complex numbers relationship
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=14 "Edit section: Complex numbers relationship")\]
### Complex exponential function definitions
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=15 "Edit section: Complex exponential function definitions")\]
Both sine and cosine can be extended further via [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number"), a set of numbers composed of both [real](https://en.wikipedia.org/wiki/Real_number "Real number") and [imaginary numbers](https://en.wikipedia.org/wiki/Imaginary_number "Imaginary number"). For real number , the definition of both sine and cosine functions can be extended in a [complex plane](https://en.wikipedia.org/wiki/Complex_plane "Complex plane") in terms of an [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") as follows:[\[35\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]-36) 
Alternatively, both functions can be defined in terms of [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"):[\[35\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]-36) 
When plotted on the [complex plane](https://en.wikipedia.org/wiki/Complex_plane "Complex plane"), the function  for real values of  traces out the [unit circle](https://en.wikipedia.org/wiki/Unit_circle "Unit circle") in the complex plane. Both sine and cosine functions may be simplified to the imaginary and real parts of  as:[\[36\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTERudin19872-37) 
When  for real values  and , where , both sine and cosine functions can be expressed in terms of real sines, cosines, and [hyperbolic functions](https://en.wikipedia.org/wiki/Hyperbolic_function "Hyperbolic function") as:[\[37\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-38) 
[](https://en.wikipedia.org/wiki/File:Sinus_und_Kosinus_am_Einheitskreis_3.svg)
Both functions  and  are the real and imaginary parts of .
Sine and cosine are used to connect the real and imaginary parts of a [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number") with its [polar coordinates](https://en.wikipedia.org/wiki/Polar_coordinates "Polar coordinates") :  and the real and imaginary parts are  where  and  represent the magnitude and angle of the complex number .[\[38\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_23–24]-39)
For any real number , Euler's formula in terms of polar coordinates is stated as .[\[35\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]-36)
[](https://en.wikipedia.org/wiki/File:Complex_sin.jpg)
[Domain coloring](https://en.wikipedia.org/wiki/Domain_coloring "Domain coloring") of sin(*z*) in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.
[](https://en.wikipedia.org/wiki/File:Sin_z_vector_field_02_Pengo.svg)
Vector field rendering of sin(*z*)
Applying the series definition of the sine and cosine to a complex argument, *z*, gives:
where sinh and cosh are the [hyperbolic sine and cosine](https://en.wikipedia.org/wiki/Hyperbolic_function "Hyperbolic function"). These are [entire functions](https://en.wikipedia.org/wiki/Entire_function "Entire function").
It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument:
#### Partial fraction and product expansions of complex sine
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=18 "Edit section: Partial fraction and product expansions of complex sine")\]
Using the partial fraction expansion technique in [complex analysis](https://en.wikipedia.org/wiki/Complex_analysis "Complex analysis"), one can find that the infinite series  both converge and are equal to . Similarly, one can show that 
Using product expansion technique, one can derive 
#### Usage of complex sine
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=19 "Edit section: Usage of complex sine")\]
sin(*z*) is found in the [functional equation](https://en.wikipedia.org/wiki/Functional_equation "Functional equation") for the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function "Gamma function"),
which in turn is found in the [functional equation](https://en.wikipedia.org/wiki/Functional_equation "Functional equation") for the [Riemann zeta-function](https://en.wikipedia.org/wiki/Riemann_zeta-function "Riemann zeta-function"),
As a [holomorphic function](https://en.wikipedia.org/wiki/Holomorphic_function "Holomorphic function"), sin *z* is a 2D solution of [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation"):
The complex sine function is also related to the level curves of [pendulums](https://en.wikipedia.org/wiki/Pendulums "Pendulums").\[*[how?](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify "Wikipedia:Please clarify")*\][\[39\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-40)\[*[better source needed](https://en.wikipedia.org/wiki/Wikipedia:Verifiability#Questionable_sources "Wikipedia:Verifiability")*\]
| | | |
|---|---|---|
| [](https://en.wikipedia.org/wiki/File:Complex_sin_real_01_Pengo.svg) | [](https://en.wikipedia.org/wiki/File:Complex_sin_imag_01_Pengo.svg) | [](https://en.wikipedia.org/wiki/File:Complex_sin_abs_01_Pengo.svg) |
| Real component | Imaginary component | Magnitude |
| | | |
|---|---|---|
| [](https://en.wikipedia.org/wiki/File:Complex_arcsin_real_01_Pengo.svg) | [](https://en.wikipedia.org/wiki/File:Complex_arcsin_imag_01_Pengo.svg) | [](https://en.wikipedia.org/wiki/File:Complex_arcsin_abs_01_Pengo.svg) |
| Real component | Imaginary component | Magnitude |
The word *sine* is derived, indirectly, from the [Sanskrit](https://en.wikipedia.org/wiki/Sanskrit "Sanskrit") word *jyā* 'bow-string' or more specifically its synonym *jīvá* (both adopted from [Ancient Greek](https://en.wikipedia.org/wiki/Ancient_Greek_language "Ancient Greek language") χορδή 'string; chord'), due to visual similarity between the arc of a circle with its corresponding chord and a bow with its string (see [jyā, koti-jyā and utkrama-jyā](https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81 "Jyā, koti-jyā and utkrama-jyā"); *sine* and *chord* are closely related in a circle of unit diameter, see [Ptolemy's Theorem](https://en.wikipedia.org/wiki/Ptolemy%27s_theorem#Corollaries "Ptolemy's theorem")). This was [transliterated](https://en.wikipedia.org/wiki/Transliteration "Transliteration") in [Arabic](https://en.wikipedia.org/wiki/Arabic_language "Arabic language") as *jība*, which is meaningless in that language and written as *jb* (جب). Since Arabic is written without short vowels, *jb* was interpreted as the [homograph](https://en.wikipedia.org/wiki/Homograph "Homograph") *jayb* ([جيب](https://en.wiktionary.org/wiki/%D8%AC%D9%8A%D8%A8 "wikt:جيب")), which means 'bosom', 'pocket', or 'fold'.[\[40\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEPlofker2009[httpsbooksgooglecombooksidDHvThPNp9yMCpgPA257_257]-41)[\[41\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEMaor1998[httpsbooksgooglecombooksidr9aMrneWFpUCpgPA35_35]-42) When the Arabic texts of [Al-Battani](https://en.wikipedia.org/wiki/Al-Battani "Al-Battani") and [al-Khwārizmī](https://en.wikipedia.org/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB "Muḥammad ibn Mūsā al-Khwārizmī") were translated into [Medieval Latin](https://en.wikipedia.org/wiki/Medieval_Latin "Medieval Latin") in the 12th century by [Gerard of Cremona](https://en.wikipedia.org/wiki/Gerard_of_Cremona "Gerard of Cremona"), he used the Latin equivalent [*sinus*](https://en.wiktionary.org/wiki/sinus "wikt:sinus") (which also means 'bay' or 'fold', and more specifically 'the hanging fold of a [toga](https://en.wikipedia.org/wiki/Toga "Toga") over the breast').[\[42\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEMerzbachBoyer2011-43)[\[43\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEMaor199835–36-44)[\[44\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEKatz2008253-45) Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage.[\[45\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTESmith1958202-46)[\[46\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-47) The English form *sine* was introduced in [Thomas Fale](https://en.wikipedia.org/wiki/Thomas_Fale "Thomas Fale")'s 1593 *Horologiographia*.[\[47\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-48)
The word *cosine* derives from an abbreviation of the Latin *complementi sinus* 'sine of the [complementary angle](https://en.wikipedia.org/wiki/Complementary_angle "Complementary angle")' as *cosinus* in [Edmund Gunter](https://en.wikipedia.org/wiki/Edmund_Gunter "Edmund Gunter")'s *Canon triangulorum* (1620), which also includes a similar definition of *cotangens*.[\[48\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEGunter1620-49)
[](https://en.wikipedia.org/wiki/File:Khalili_Collection_Islamic_Art_sci_0040.1_CROP.jpg)
Quadrant from 1840s [Ottoman Turkey](https://en.wikipedia.org/wiki/Ottoman_Empire "Ottoman Empire") with axes for looking up the sine and [versine](https://en.wikipedia.org/wiki/Versine "Versine") of angles
While the early study of trigonometry can be traced to antiquity, the [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions") as they are in use today were developed in the medieval period. The [chord](https://en.wikipedia.org/wiki/Chord_\(geometry\) "Chord (geometry)") function was discovered by [Hipparchus](https://en.wikipedia.org/wiki/Hipparchus "Hipparchus") of [Nicaea](https://en.wikipedia.org/wiki/%C4%B0znik "İznik") (180–125 BCE) and [Ptolemy](https://en.wikipedia.org/wiki/Ptolemy "Ptolemy") of [Roman Egypt](https://en.wikipedia.org/wiki/Egypt_\(Roman_province\) "Egypt (Roman province)") (90–165 CE).[\[49\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-50)
The sine and cosine functions are closely related to the [*jyā* and *koṭi-jyā*](https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81 "Jyā, koti-jyā and utkrama-jyā") functions used in [Indian astronomy](https://en.wikipedia.org/wiki/Indian_astronomy "Indian astronomy") during the [Gupta period](https://en.wikipedia.org/wiki/Gupta_period "Gupta period") (*[Aryabhatiya](https://en.wikipedia.org/wiki/Aryabhatiya "Aryabhatiya")* and *[Surya Siddhanta](https://en.wikipedia.org/wiki/Surya_Siddhanta "Surya Siddhanta")*), via translation from Sanskrit to Arabic and then from Arabic to Latin.[\[42\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEMerzbachBoyer2011-43)[\[50\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-51)
All six trigonometric functions in current use were known in [Islamic mathematics](https://en.wikipedia.org/wiki/Islamic_mathematics "Islamic mathematics") by the 9th century, as was the [law of sines](https://en.wikipedia.org/wiki/Law_of_sines "Law of sines"), used in [solving triangles](https://en.wikipedia.org/wiki/Solving_triangles "Solving triangles").[\[51\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-Gingerich_1986-52) [Al-Khwārizmī](https://en.wikipedia.org/wiki/Al-Khw%C4%81rizm%C4%AB "Al-Khwārizmī") (c. 780–850) produced tables of sines, cosines and tangents.[\[52\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-Sesiano-53)[\[53\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-Britannica-54) [Muhammad ibn Jābir al-Harrānī al-Battānī](https://en.wikipedia.org/wiki/Muhammad_ibn_J%C4%81bir_al-Harr%C4%81n%C4%AB_al-Batt%C4%81n%C4%AB "Muhammad ibn Jābir al-Harrānī al-Battānī") (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.[\[53\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-Britannica-54)
In the early 17th-century, the French mathematician [Albert Girard](https://en.wikipedia.org/wiki/Albert_Girard "Albert Girard") published the first use of the abbreviations *sin*, *cos*, and *tan*; these were further promulgated by Euler (see below). The *Opus palatinum de triangulis* of [Georg Joachim Rheticus](https://en.wikipedia.org/wiki/Georg_Joachim_Rheticus "Georg Joachim Rheticus"), a student of [Copernicus](https://en.wikipedia.org/wiki/Copernicus "Copernicus"), was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.
In a paper published in 1682, [Leibniz](https://en.wikipedia.org/wiki/Gottfried_Leibniz "Gottfried Leibniz") proved that sin *x* is not an [algebraic function](https://en.wikipedia.org/wiki/Algebraic_function "Algebraic function") of *x*.[\[54\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-55) [Roger Cotes](https://en.wikipedia.org/wiki/Roger_Cotes "Roger Cotes") computed the derivative of sine in his *Harmonia Mensurarum* (1722).[\[55\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-56) [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler")'s *Introductio in analysin infinitorum* (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "[Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula")", as well as the near-modern abbreviations *sin.*, *cos.*, *tang.*, *cot.*, *sec.*, and *cosec.*[\[42\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEMerzbachBoyer2011-43)
## Software implementations
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=24 "Edit section: Software implementations")\]
There is no standard algorithm for calculating sine and cosine. [IEEE 754](https://en.wikipedia.org/wiki/IEEE_754 "IEEE 754"), the most widely used standard for the specification of reliable floating-point computation, does not address calculating trigonometric functions such as sine. The reason is that no efficient algorithm is known for computing sine and cosine with a specified accuracy, especially for large inputs.[\[56\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-FOOTNOTEZimmermann2006-57)
Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. `sin(1022)`.
A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest pre-calculated value, or [linearly interpolate](https://en.wikipedia.org/wiki/Linear_interpolation "Linear interpolation") between the 2 closest values to approximate it. This allows results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage.\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
The [CORDIC](https://en.wikipedia.org/wiki/CORDIC "CORDIC") algorithm is commonly used in scientific calculators.
The sine and cosine functions, along with other trigonometric functions, are widely available across programming languages and platforms. In computing, they are typically abbreviated to `sin` and `cos`.
Some CPU architectures have a built-in instruction for sine, including the Intel [x87](https://en.wikipedia.org/wiki/X87 "X87") FPUs since the 80387.
In programming languages, `sin` and `cos` are typically either a built-in function or found within the language's standard math library. For example, the [C standard library](https://en.wikipedia.org/wiki/C_standard_library "C standard library") defines sine functions within [math.h](https://en.wikipedia.org/wiki/C_mathematical_functions "C mathematical functions"): `sin(double)`, `sinf(float)`, and `sinl(long double)`. The parameter of each is a [floating point](https://en.wikipedia.org/wiki/Floating_point "Floating point") value, specifying the angle in radians. Each function returns the same [data type](https://en.wikipedia.org/wiki/Data_type "Data type") as it accepts. Many other trigonometric functions are also defined in [math.h](https://en.wikipedia.org/wiki/C_mathematical_functions "C mathematical functions"), such as for cosine, arc sine, and hyperbolic sine (sinh). Similarly, [Python](https://en.wikipedia.org/wiki/Python_\(programming_language\) "Python (programming language)") defines `math.sin(x)` and `math.cos(x)` within the built-in `math` module. Complex sine and cosine functions are also available within the `cmath` module, e.g. `cmath.sin(z)`. [CPython](https://en.wikipedia.org/wiki/CPython "CPython")'s math functions call the [C](https://en.wikipedia.org/wiki/C_\(programming_language\) "C (programming language)") `math` library, and use a [double-precision floating-point format](https://en.wikipedia.org/wiki/Double-precision_floating-point_format "Double-precision floating-point format").
### Turns based implementations
\[[edit](https://en.wikipedia.org/w/index.php?title=Sine_and_cosine&action=edit§ion=25 "Edit section: Turns based implementations")\]
"sinpi" redirects here. For the township in Pingtung County, Taiwan, see [Xinpi](https://en.wikipedia.org/wiki/Xinpi "Xinpi").
"cospi" redirects here. For the 17th-century Bolognese nobleman, see [Ferdinando Cospi](https://en.wikipedia.org/wiki/Ferdinando_Cospi "Ferdinando Cospi").
Some software libraries provide implementations of sine and cosine using the input angle in half-[turns](https://en.wikipedia.org/wiki/Turn_\(angle\) "Turn (angle)"), a half-turn being an angle of 180 degrees or  radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.[\[57\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-matlab-58)[\[58\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-r-59) These functions are called `sinpi` and `cospi` in MATLAB,[\[57\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-matlab-58) [OpenCL](https://en.wikipedia.org/wiki/OpenCL "OpenCL"),[\[59\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-60) R,[\[58\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-r-59) Julia,[\[60\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-61) [CUDA](https://en.wikipedia.org/wiki/CUDA "CUDA"),[\[61\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-62) and ARM.[\[62\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-63) For example, `sinpi(x)` would evaluate to  where *x* is expressed in half-turns, and consequently the final input to the function, *πx* can be interpreted in radians by sin. [SciPy](https://en.wikipedia.org/wiki/SciPy "SciPy") provides similar functions `sindg` and `cosdg` with input in degrees.[\[63\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-64)
The accuracy advantage stems from the ability to perfectly represent key angles like full-turn, half-turn, and quarter-turn losslessly in binary floating-point or fixed-point. In contrast, representing , , and  in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits.
Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing modulo  involves inaccuracies in representing .
For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to 4096 over one complete revolution.[\[64\]](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_note-65) If half-turns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to  would be incurred.
- [Āryabhaṭa's sine table](https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa%27s_sine_table "Āryabhaṭa's sine table")
- [Bhaskara I's sine approximation formula](https://en.wikipedia.org/wiki/Bhaskara_I%27s_sine_approximation_formula "Bhaskara I's sine approximation formula")
- [Discrete sine transform](https://en.wikipedia.org/wiki/Discrete_sine_transform "Discrete sine transform")
- [Dixon elliptic functions](https://en.wikipedia.org/wiki/Dixon_elliptic_functions "Dixon elliptic functions")
- [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula")
- [Generalized trigonometry](https://en.wikipedia.org/wiki/Generalized_trigonometry "Generalized trigonometry")
- [Hyperbolic function](https://en.wikipedia.org/wiki/Hyperbolic_function "Hyperbolic function")
- [Lemniscate elliptic functions](https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions "Lemniscate elliptic functions")
- [Law of sines](https://en.wikipedia.org/wiki/Law_of_sines "Law of sines")
- [List of periodic functions](https://en.wikipedia.org/wiki/List_of_periodic_functions "List of periodic functions")
- [List of trigonometric identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities "List of trigonometric identities")
- [Madhava series](https://en.wikipedia.org/wiki/Madhava_series "Madhava series")
- [Madhava's sine table](https://en.wikipedia.org/wiki/Madhava%27s_sine_table "Madhava's sine table")
- [Optical sine theorem](https://en.wikipedia.org/wiki/Optical_sine_theorem "Optical sine theorem")
- [Polar sine](https://en.wikipedia.org/wiki/Polar_sine "Polar sine")—a generalization to vertex angles
- [Proofs of trigonometric identities](https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities "Proofs of trigonometric identities")
- [Sinc function](https://en.wikipedia.org/wiki/Sinc_function "Sinc function")
- [Sine and cosine transforms](https://en.wikipedia.org/wiki/Sine_and_cosine_transforms "Sine and cosine transforms")
- [Sine integral](https://en.wikipedia.org/wiki/Sine_integral "Sine integral")
- [Sine quadrant](https://en.wikipedia.org/wiki/Sine_quadrant "Sine quadrant")
- [Sine wave](https://en.wikipedia.org/wiki/Sine_wave "Sine wave")
- [Sine–Gordon equation](https://en.wikipedia.org/wiki/Sine%E2%80%93Gordon_equation "Sine–Gordon equation")
- [Sinusoidal model](https://en.wikipedia.org/wiki/Sinusoidal_model "Sinusoidal model")
- [SOH-CAH-TOA](https://en.wikipedia.org/wiki/Mnemonics_in_trigonometry#SOH-CAH-TOA "Mnemonics in trigonometry")
- [Trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions")
- [Trigonometric integral](https://en.wikipedia.org/wiki/Trigonometric_integral "Trigonometric integral")
1. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-1) [***c***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA27_27]_1-2) [Young (2017)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2017), p. [27](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27).
2. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA36_36]_2-0)** [Young (2017)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2017), p. [36](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA36).
3. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200742_3-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200742_3-1) [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 42.
4. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA37_37],_[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA78_78]_4-0)** [Young (2017)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2017), p. [37](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA37), [78](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA78).
5. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]_5-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA634_634]_5-1) [Axler (2012)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFAxler2012), p. [634](https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA634).
6. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAxler2012[httpsbooksgooglecombooksidB5RxDwAAQBAJpgPA632_632]_6-0)** [Axler (2012)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFAxler2012), p. [632](https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA632).
7. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-7)**
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Cross Product"](https://mathworld.wolfram.com/CrossProduct.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*. Retrieved 5 June 2025.
8. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-8)**
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Dot Product"](https://mathworld.wolfram.com/DotProduct.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*. Retrieved 5 June 2025.
9. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200741_9-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 41.
10. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA68_68]_10-0)** [Young (2017)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2017), p. [68](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA68).
11. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200747_11-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 47.
12. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200741–42_12-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 41–42.
13. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200741,_43_13-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 41, 43.
14. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2012[httpsbooksgooglecombooksidOMrcN0a3LxICpgRA1-PA165_165]_14-0)** [Young (2012)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2012), p. [165](https://books.google.com/books?id=OMrcN0a3LxIC&pg=RA1-PA165).
15. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon200742,_47_15-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 42, 47.
16. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-16)**
["OEIS A003957"](https://oeis.org/A003957). *oeis.org*. Retrieved 2019-05-26.
17. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]_17-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEBourchteinBourchtein2022[httpsbooksgooglecombooksidnGxOEAAAQBAJpgPA294_294]_17-1) [Bourchtein & Bourchtein (2022)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFBourchteinBourchtein2022), p. [294](https://books.google.com/books?id=nGxOEAAAQBAJ&pg=PA294).
18. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007115_18-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 115.
19. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007155_19-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 155.
20. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007157_20-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 157.
21. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007199_21-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 199.
22. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVince2023[httpsbooksgooglecombooksidGnW6EAAAQBAJpgPA162_162]_22-0)** [Vince (2023)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVince2023), p. [162](https://books.google.com/books?id=GnW6EAAAQBAJ&pg=PA162).
23. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAdlaj2012_23-0)** [Adlaj (2012)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFAdlaj2012).
24. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-24)**
OEIS
[sequence A105419 (Decimal expansion of the arc length of the sine or cosine curve for one full period.)](https://oeis.org/A105419)
25. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007365_25-0)** [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 365.
26. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEYoung2017[httpsbooksgooglecombooksid476ZDwAAQBAJpgPA99_99]_26-0)** [Young (2017)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFYoung2017), p. [99](https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA99).
27. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-28)**
Dennis G. Zill (2013). *Precalculus with Calculus Previews*. Jones & Bartlett Publishers. p. 238. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-4496-4515-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4496-4515-1 "Special:BookSources/978-1-4496-4515-1")
.
[Extract of page 238](https://books.google.com/books?id=dtS5M4lx7scC&pg=PA238)
28. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-29)**
["Sine-squared function"](https://calculus.subwiki.org/wiki/Sine-squared_function#Identities). Retrieved August 9, 2019.
29. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007491–492_30-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEVarbergPurcellRigdon2007491–492_30-1) [Varberg, Purcell & Rigdon (2007)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFVarbergPurcellRigdon2007), p. 491–492.
30. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]_31-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEAbramowitzStegun1970[httpsbooksgooglecombooksidMtU8uP7XMvoCpgPA74_74]_31-1) [Abramowitz & Stegun (1970)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFAbramowitzStegun1970), p. [74](https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA74).
31. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEPowell1981150_32-0)** [Powell (1981)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFPowell1981), p. 150.
32. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTERudin198788_33-0)** [Rudin (1987)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFRudin1987), p. 88.
33. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEZygmund19681_34-0)** [Zygmund (1968)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFZygmund1968), p. 1.
34. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEZygmund196811_35-0)** [Zygmund (1968)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFZygmund1968), p. 11.
35. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]_36-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]_36-1) [***c***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_24]_36-2) [Howie (2003)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFHowie2003), p. [24](https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24).
36. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTERudin19872_37-0)** [Rudin (1987)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFRudin1987), p. 2.
37. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-38)**
Brown, James Ward; [Churchill, Ruel](https://en.wikipedia.org/wiki/Ruel_Vance_Churchill "Ruel Vance Churchill") (2014). *Complex Variables and Applications* (9th ed.). [McGraw-Hill](https://en.wikipedia.org/wiki/McGraw-Hill "McGraw-Hill"). p. 105. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-338317-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-338317-0 "Special:BookSources/978-0-07-338317-0")
.
38. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEHowie2003[httpsbooksgooglecombooksid0FZDBAAAQBAJpgPA24_23–24]_39-0)** [Howie (2003)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFHowie2003), p. [23–24](https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24).
39. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-40)**
["Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?"](https://math.stackexchange.com/q/220418). *math.stackexchange.com*. Retrieved 2019-08-12.
40. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEPlofker2009[httpsbooksgooglecombooksidDHvThPNp9yMCpgPA257_257]_41-0)** [Plofker (2009)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFPlofker2009), p. [257](https://books.google.com/books?id=DHvThPNp9yMC&pg=PA257).
41. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEMaor1998[httpsbooksgooglecombooksidr9aMrneWFpUCpgPA35_35]_42-0)** [Maor (1998)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFMaor1998), p. [35](https://books.google.com/books?id=r9aMrneWFpUC&pg=PA35).
42. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEMerzbachBoyer2011_43-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEMerzbachBoyer2011_43-1) [***c***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEMerzbachBoyer2011_43-2) [Merzbach & Boyer (2011)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFMerzbachBoyer2011).
43. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEMaor199835–36_44-0)** [Maor (1998)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFMaor1998), p. 35–36.
44. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEKatz2008253_45-0)** [Katz (2008)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFKatz2008), p. 253.
45. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTESmith1958202_46-0)** [Smith (1958)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFSmith1958), p. 202.
46. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-47)**
Various sources credit the first use of *sinus* to either
- [Plato Tiburtinus](https://en.wikipedia.org/wiki/Plato_Tiburtinus "Plato Tiburtinus")'s 1116 translation of the *Astronomy* of [Al-Battani](https://en.wikipedia.org/wiki/Al-Battani "Al-Battani")
- [Gerard of Cremona](https://en.wikipedia.org/wiki/Gerard_of_Cremona "Gerard of Cremona")'s translation of the *Algebra* of [al-Khwārizmī](https://en.wikipedia.org/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB "Muḥammad ibn Mūsā al-Khwārizmī")
- [Robert of Chester](https://en.wikipedia.org/wiki/Robert_of_Chester "Robert of Chester")'s 1145 translation of the tables of al-Khwārizmī
See [Merlet (2004)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFMerlet2004). See [Maor (1998)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFMaor1998), Chapter 3, for an earlier etymology crediting Gerard. See [Katz (2008)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFKatz2008), p. 210.
47. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-48)**
Fale's book alternately uses the spellings "sine", "signe", or "sign".
Fale, Thomas (1593). [*Horologiographia. The Art of Dialling: Teaching, an Easie and Perfect Way to make all Kindes of Dials ...*](https://archive.org/details/b30333106/page/19/mode/1up) London: F. Kingston. p. 11, for example.
48. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEGunter1620_49-0)** [Gunter (1620)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFGunter1620).
49. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-50)**
Brendan, T. (February 1965). "How Ptolemy constructed trigonometry tables". *The Mathematics Teacher*. **58** (2): 141–149\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.5951/MT.58.2.0141](https://doi.org/10.5951%2FMT.58.2.0141). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [27967990](https://www.jstor.org/stable/27967990).
50. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-51)**
[Van Brummelen, Glen](https://en.wikipedia.org/wiki/Glen_Van_Brummelen "Glen Van Brummelen") (2009). "India". *The Mathematics of the Heavens and the Earth*. Princeton University Press. Ch. 3, pp. 94–134. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-691-12973-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-12973-0 "Special:BookSources/978-0-691-12973-0")
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51. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-Gingerich_1986_52-0)**
Gingerich, Owen (1986). ["Islamic Astronomy"](https://web.archive.org/web/20131019140821/http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm). *[Scientific American](https://en.wikipedia.org/wiki/Scientific_American "Scientific American")*. Vol. 254. p. 74. Archived from [the original](http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm) on 2013-10-19. Retrieved 2010-07-13.
52. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-Sesiano_53-0)**
Jacques Sesiano, "Islamic mathematics", p. 157, in
[Selin, Helaine](https://en.wikipedia.org/wiki/Helaine_Selin "Helaine Selin"); [D'Ambrosio, Ubiratan](https://en.wikipedia.org/wiki/Ubiratan_D%27Ambrosio "Ubiratan D'Ambrosio"), eds. (2000). *Mathematics Across Cultures: The History of Non-western Mathematics*. [Springer Science+Business Media](https://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Media "Springer Science+Business Media"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-4020-0260-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-0260-1 "Special:BookSources/978-1-4020-0260-1")
.
53. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-Britannica_54-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-Britannica_54-1)
["trigonometry"](http://www.britannica.com/EBchecked/topic/605281/trigonometry). Encyclopedia Britannica. 17 June 2024.
54. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-55)**
Nicolás Bourbaki (1994). [*Elements of the History of Mathematics*](https://archive.org/details/elementsofhistor0000bour). Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9783540647676](https://en.wikipedia.org/wiki/Special:BookSources/9783540647676 "Special:BookSources/9783540647676")
.
55. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-56)** "[Why the sine has a simple derivative](http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf) [Archived](https://web.archive.org/web/20110720102700/http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf) 2011-07-20 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine")", in *[Historical Notes for Calculus Teachers](http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm) [Archived](https://web.archive.org/web/20110720102613/http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm) 2011-07-20 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine")* by [V. Frederick Rickey](http://www.math.usma.edu/people/rickey/) [Archived](https://web.archive.org/web/20110720102654/http://www.math.usma.edu/people/rickey/) 2011-07-20 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine")
56. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-FOOTNOTEZimmermann2006_57-0)** [Zimmermann (2006)](https://en.wikipedia.org/wiki/Sine_and_cosine#CITEREFZimmermann2006).
57. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-matlab_58-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-matlab_58-1)
["sinpi - Compute sin(X\*pi) accurately"](https://www.mathworks.com/help/matlab/ref/double.sinpi.html). *www.mathworks.com*. [Archived](http://web.archive.org/web/20251123002724/https://www.mathworks.com/help/matlab/ref/double.sinpi.html) from the original on 2025-11-23. Retrieved 2026-01-08.
58. ^ [***a***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-r_59-0) [***b***](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-r_59-1)
["Trig function - RDocumentation"](https://www.rdocumentation.org/packages/base/versions/3.5.3/topics/Trig). *www.rdocumentation.org*. Retrieved 2026-02-17.
59. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-60)**
["sin, sincos, sinh, sinpi"](https://registry.khronos.org/OpenCL/sdk/1.0/docs/man/xhtml/sin.html). *registry.khronos.org*. Retrieved 2026-02-17.
60. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-61)**
["sinpi » Julia Functions"](http://www.jlhub.com/julia/manual/en/function/sinpi). *www.jlhub.com*. Retrieved 2026-02-17.
61. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-62)**
["Double Precision Mathematical Functions"](http://web.archive.org/web/20240723062728/https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__DOUBLE.html). *docs.nvidia.com*. Archived from [the original](https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__DOUBLE.html) on 2024-07-23. Retrieved 2026-01-08.
62. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-63)**
["Documentation – Arm Developer"](https://developer.arm.com/documentation/100614/latest/b-opencl-built-in-functions/b2-math-functions). *developer.arm.com*. Retrieved 2026-02-17.
63. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-64)**
["Special functions (scipy.special) — SciPy v1.17.0 Manual"](https://docs.scipy.org/doc/scipy/reference/special.html#convenience-functions). *docs.scipy.org*. Retrieved 25 February 2026.
64. **[^](https://en.wikipedia.org/wiki/Sine_and_cosine#cite_ref-65)**
["AAS33051: Precision Angle Sensor IC with Incremental and Motor Commutation Outputs and On-Chip Linearization"](http://web.archive.org/web/20190417143715/https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx). *www.allegromicro.com*. Archived from [the original](https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx) on 2019-04-17. Retrieved 2026-02-17.
- [Abramowitz, Milton](https://en.wikipedia.org/wiki/Milton_Abramowitz "Milton Abramowitz"); [Stegun, Irene A.](https://en.wikipedia.org/wiki/Irene_Stegun "Irene Stegun") (1970), *[Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun "Abramowitz and Stegun")*, New York: [Dover Publications](https://en.wikipedia.org/wiki/Dover_Publications "Dover Publications"), Ninth printing
- Adlaj, Semjon (2012), ["An Eloquent Formula for the Perimeter of an Ellipse"](https://www.ams.org/notices/201208/rtx120801094p.pdf) (PDF), *American Mathematical Society*, **59** (8): 1097
- Axler, Sheldon (2012), *Algebra and Trigonometry*, [John Wiley & Sons](https://en.wikipedia.org/wiki/John_Wiley_%26_Sons "John Wiley & Sons"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0470-58579-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0470-58579-5 "Special:BookSources/978-0470-58579-5")
- [Bourchtein, Ludmila](https://en.wikipedia.org/wiki/Ludmila_Bourchtein "Ludmila Bourchtein"); Bourchtein, Andrei (2022), *Theory of Infinite Sequences and Series*, Springer, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-030-79431-6](https://doi.org/10.1007%2F978-3-030-79431-6), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-030-79431-6](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-79431-6 "Special:BookSources/978-3-030-79431-6")
- [Gunter, Edmund](https://en.wikipedia.org/wiki/Edmund_Gunter "Edmund Gunter") (1620), *Canon triangulorum*
- Howie, John M. (2003), *Complex Analysis*, Springer Undergraduate Mathematics Series, Springer, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-1-4471-0027-0](https://doi.org/10.1007%2F978-1-4471-0027-0), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-4471-0027-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4471-0027-0 "Special:BookSources/978-1-4471-0027-0")
- Traupman, Ph.D., John C. (1966), [*The New College Latin & English Dictionary*](https://archive.org/details/boysgirlsbookabo00gard_0), Toronto: Bantam, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-553-27619-0](https://en.wikipedia.org/wiki/Special:BookSources/0-553-27619-0 "Special:BookSources/0-553-27619-0")
- Katz, Victor J. (2008), [*A History of Mathematics*](http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf) (PDF) (3rd ed.), Boston: Addison-Wesley, "The English word "sine" comes from a series of mistranslations of the Sanskrit *jyā-ardha* (chord-half). Āryabhaṭa frequently abbreviated this term to *jyā* or its synonym *jīvá*. When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an otherwise meaningless Arabic word *jiba*. But since Arabic is written without vowels, later writers interpreted the consonants *jb* as *jaib*, which means bosom or breast. In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word *sinus*, which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf."
- Maor, Eli (1998), *Trigonometric Delights*, [Princeton University Press](https://en.wikipedia.org/wiki/Princeton_University_Press "Princeton University Press"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[1-4008-4282-4](https://en.wikipedia.org/wiki/Special:BookSources/1-4008-4282-4 "Special:BookSources/1-4008-4282-4")
- Merlet, Jean-Pierre (2004), "A Note on the History of the Trigonometric Functions", in Ceccarelli, Marco (ed.), *International Symposium on History of Machines and Mechanisms*, Springer, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/1-4020-2204-2](https://doi.org/10.1007%2F1-4020-2204-2), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-4020-2203-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-2203-6 "Special:BookSources/978-1-4020-2203-6")
- [Merzbach, Uta C.](https://en.wikipedia.org/wiki/Uta_Merzbach "Uta Merzbach"); [Boyer, Carl B.](https://en.wikipedia.org/wiki/Carl_B._Boyer "Carl B. Boyer") (2011), *A History of Mathematics* (3rd ed.), [John Wiley & Sons](https://en.wikipedia.org/wiki/John_Wiley_%26_Sons "John Wiley & Sons"), "It was Robert of Chester's translation from Arabic that resulted in our word "sine". The Hindus had given the name jiva to the half-chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language, there is also the word jaib meaning "bay" or "inlet". When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence, he used the word sinus, the Latin word for "bay" or "inlet"."
- Plofker (2009), [*Mathematics in India*](https://en.wikipedia.org/wiki/Mathematics_in_India_\(book\) "Mathematics in India (book)"), [Princeton University Press](https://en.wikipedia.org/wiki/Princeton_University_Press "Princeton University Press")
- [Powell, Michael J. D.](https://en.wikipedia.org/wiki/Michael_J._D._Powell "Michael J. D. Powell") (1981), *Approximation Theory and Methods*, [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-521-29514-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-29514-7 "Special:BookSources/978-0-521-29514-7")
- [Rudin, Walter](https://en.wikipedia.org/wiki/Walter_Rudin "Walter Rudin") (1987), *Real and complex analysis* (3rd ed.), New York: [McGraw-Hill](https://en.wikipedia.org/wiki/McGraw-Hill "McGraw-Hill"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-054234-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-054234-1 "Special:BookSources/978-0-07-054234-1")
, [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0924157](https://mathscinet.ams.org/mathscinet-getitem?mr=0924157)
- Smith, D. E. (1958) \[1925\], *History of Mathematics*, vol. I, [Dover Publications](https://en.wikipedia.org/wiki/Dover_Publications "Dover Publications"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-486-20429-4](https://en.wikipedia.org/wiki/Special:BookSources/0-486-20429-4 "Special:BookSources/0-486-20429-4")
- Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), [*Calculus*](https://archive.org/details/matematika-a-purcell-calculus-9th-ed/mode/2up) (9th ed.), [Pearson Prentice Hall](https://en.wikipedia.org/wiki/Pearson_Prentice_Hall "Pearson Prentice Hall"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0131469686](https://en.wikipedia.org/wiki/Special:BookSources/978-0131469686 "Special:BookSources/978-0131469686")
- Vince, John (2023), *Calculus for Computer Graphics*, Springer, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-031-28117-4](https://doi.org/10.1007%2F978-3-031-28117-4), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-031-28117-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-031-28117-4 "Special:BookSources/978-3-031-28117-4")
- [Young, Cynthia](https://en.wikipedia.org/wiki/Cynthia_Y._Young "Cynthia Y. Young") (2012), *Trigonometry* (3rd ed.), John Wiley & Sons, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-119-32113-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-119-32113-2 "Special:BookSources/978-1-119-32113-2")
- ——— (2017), *Trigonometry* (4th ed.), John Wiley & Sons, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-119-32113-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-119-32113-2 "Special:BookSources/978-1-119-32113-2")
- Zimmermann, Paul (2006), "Can we trust floating-point numbers?", [*Grand Challenges of Informatics*](http://www.jaist.ac.jp/~bjorner/ae-is-budapest/talks/Sept20pm2_Zimmermann.pdf) (PDF), p. 14/31
- [Zygmund, Antoni](https://en.wikipedia.org/wiki/Antoni_Zygmund "Antoni Zygmund") (1968), [*Trigonometric Series*](https://archive.org/details/trigonometricser0012azyg/) (2nd, reprinted ed.), [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"), [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0236587](https://mathscinet.ams.org/mathscinet-getitem?mr=0236587)
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Look up ***[sine](https://en.wiktionary.org/wiki/sine "wiktionary:sine")*** in Wiktionary, the free dictionary.
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| Shard | 152 (laksa) |
| Root Hash | 17790707453426894952 |
| Unparsed URL | org,wikipedia!en,/wiki/Sine_and_cosine s443 |