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| Boilerpipe Text | The "Unit Circle" is a circle with a radius of 1.
Being so simple, it is a great way to learn and talk about lengths and angles.
The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement here.
Sine, Cosine and Tangent
Since the radius is 1, we can read
sine, cosine and tangent
just from the x and y coordinates.
What happens when the angle, θ, is 0°?
cos 0°
=
1, sin 0°
=
0 and tan 0°
=
0
What happens when θ is 90°?
cos 90°
=
0, sin 90°
=
1 and tan 90° is undefined
Try It Yourself!
Play with the interactive Unit Circle below. See how different angles (in
radians
or
degrees
) affect sine, cosine and tangent:
Can you find an angle where sine and cosine are equal?
The "sides" can be positive or negative according to the rules of
Cartesian coordinates
. This makes the sine, cosine and tangent change between positive and negative values also.
Try the
Interactive Unit Circle
Pythagoras
Pythagoras' Theorem
says for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:
x
2
+ y
2
=
1
2
But 1
2
is just 1, so:
x
2
+ y
2
=
1
equation of the unit circle
Also, since x=cos and y=sin, we get:
(cos(θ))
2
+ (sin(θ))
2
=
1
A useful
identity
Important Angles: 30°, 45° and 60°
You should try to
remember
sin, cos and tan for the angles 30°, 45° and 60°
.
Yes, yes, it is a pain to have to remember things, but it will make life easier when you know them, not just in exams, but other times when you need to do quick estimates, and so on.
These are the values you should remember!
Angle
Cos
Sin
Tan=Sin/Cos
30°
√3
2
1
2
1
√3
=
√3
3
45°
√2
2
√2
2
1
60°
1
2
√3
2
√3
How To Remember?
To help you remember, cos goes
"3,2,1"
cos(30°)
=
√
3
2
cos(45°)
=
√
2
2
cos(60°)
=
√
1
2
=
1
2
And, sin goes
"1,2,3"
:
sin(30°)
=
√
1
2
=
1
2
(because √1
=
1)
sin(45°)
=
√
2
2
sin(60°)
=
√
3
2
Just 3 Numbers
In fact, knowing 3 numbers is enough:
1
2
,
√2
2
and
√3
2
Because they work for both
cos
and
sin
:
Your hand can help you remember:
For example there are 3 fingers above 30°, so cos(30°)
=
√
3
2
What about tan?
Well,
tan
=
sin/cos
, so we can calculate it like this:
tan(30°)
=
sin(30°)
cos(30°)
=
1/2
√3/2
=
1
√3
=
√3
3
*
tan(45°)
=
sin(45°)
cos(45°)
=
√2/2
√2/2
=
1
tan(60°)
=
sin(60°)
cos(60°)
=
√3/2
1/2
=
√3
* Note: writing
1
√3
may cost you marks
so use
√3
3
instead (see
Rational Denominators
to learn more).
Quick Sketch
Another way to help you remember 30° and 60° is to make a quick sketch:
Draw a triangle with side lengths of 2
Cut in half.
Pythagoras
says the new side is √3
a
2
+ b
2
=
c
2
1
2
+ (√3)
2
=
2
2
1 + 3
=
4
Then use
sohcahtoa
for sin, cos or tan
Example: sin(30°)
Sine:
soh
cahtoa
sine is opposite divided by hypotenuse
sin(30°)
=
opposite
hypotenuse
=
1
2
The Whole Circle
For the whole circle we need values in
every quadrant
, with the correct plus or minus sign as per
Cartesian Coordinates
:
Note that
cos
is first and
sin
is second, so it goes
(cos, sin)
:
Save as PDF
Example: What is cos(330°) ?
Make a sketch like this, and we can see it is the "long" value:
√3
2
And this is the same Unit Circle in
radians
.
Example: What is sin(7
π
/6) ?
Think "7
π
/6
=
π +
π
/6", then make a sketch.
We can then see it is
negative
and is the "short" value: −½
Footnote: where do the values come from?
We can use the equation
x
2
+ y
2
=
1
to find the lengths of
x
and
y
(which are equal to
cos
and
sin
when the radius is
1
):
45 Degrees
For 45 degrees, x and y are equal, so
y=x
:
x
2
+ x
2
=
1
2x
2
=
1
x
2
=
½
x
=
y
=
√(½)
60 Degrees
Take an
equilateral triangle
(all sides are equal and all angles are 60°)
and split it down the middle.
The "x" side is now
½
,
And the "y" side is:
(½)
2
+ y
2
=
1
¼ + y
2
=
1
y
2
=
1 − ¼
=
¾
y
=
√(¾)
30 Degrees
30° is just 60° with x and y swapped, so
x
=
√(¾)
and
y
=
½
And:
√
1/2
=
√
2/4
=
√
2
√
4
=
√
2
2
Also:
√
3/4
=
√
3
√
4
=
√
3
2
And here is the result (same as before):
Angle
Cos
Sin
Tan=Sin/Cos
30°
√3
2
1
2
1
√3
=
√3
3
45°
√2
2
√2
2
1
60°
1
2
√3
2
√3 | ||||||||||||||||||
| Markdown | [Circle](https://www.mathsisfun.com/geometry/circle.html)
[Interactive Unit Circle](https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html)
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# Unit Circle
The "Unit Circle" is a circle with a radius of 1.
Being so simple, it is a great way to learn and talk about lengths and angles.
The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement here.
## Sine, Cosine and Tangent
Since the radius is 1, we can read [sine, cosine and tangent](https://www.mathsisfun.com/sine-cosine-tangent.html) just from the x and y coordinates.
What happens when the angle, θ, is 0°?
cos 0° *\=* 1, sin 0° *\=* 0 and tan 0° *\=* 0
What happens when θ is 90°?
cos 90° *\=* 0, sin 90° *\=* 1 and tan 90° is undefined
## Try It Yourself\!
Play with the interactive Unit Circle below. See how different angles (in [radians](https://www.mathsisfun.com/geometry/radians.html) or [degrees](https://www.mathsisfun.com/geometry/degrees.html)) affect sine, cosine and tangent:
../algebra/images/circle-triangle.js
Can you find an angle where sine and cosine are equal?
The "sides" can be positive or negative according to the rules of [Cartesian coordinates](https://www.mathsisfun.com/data/cartesian-coordinates.html). This makes the sine, cosine and tangent change between positive and negative values also.
Try the [Interactive Unit Circle](https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html)
## Pythagoras
[Pythagoras' Theorem](https://www.mathsisfun.com/pythagoras.html) says for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:
x2 + y2 *\=* 12
But 12 is just 1, so:
x2 + y2 *\=* 1 *equation of the unit circle*
Also, since x=cos and y=sin, we get:
(cos(θ))2 + (sin(θ))2 *\=* 1
*A useful **identity***
## Important Angles: 30°, 45° and 60°
You should try to **remember** sin, cos and tan for the angles 30°, 45° and 60°**.**
Yes, yes, it is a pain to have to remember things, but it will make life easier when you know them, not just in exams, but other times when you need to do quick estimates, and so on.
**These are the values you should remember\!**
| Angle | Cos | Sin | *Tan=Sin/Cos* |
|---|---|---|---|
| 30° | *√3***2** | *1***2** | *1* **√3** *\=* *√3* **3** |
| 45° | *√2***2** | *√2***2** | 1 |
| 60° | *1***2** | *√3***2** | √3 |
### How To Remember?
To help you remember, cos goes **"3,2,1"**
cos(30°) *\=* *√**3*****2**
cos(45°) *\=* *√**2*****2**
cos(60°) *\=* *√**1*****2** *\=* *1***2**
And, sin goes **"1,2,3"** :
sin(30°) *\=* *√**1*****2** *\=* *1***2** (because √1 *\=* 1)
sin(45°) *\=* *√**2*****2**
sin(60°) *\=* *√**3*****2**
## Just 3 Numbers
In fact, knowing 3 numbers is enough: *1***2** , *√2***2** and *√3***2**
Because they work for both **cos** and **sin**:
 
Your hand can help you remember:
For example there are 3 fingers above 30°, so cos(30°) *\=* *√3***2**
## What about tan?
Well, **tan *\=* sin/cos**, so we can calculate it like this:
tan(30°) *\=* *sin(30°)***cos(30°)** ***\=*** *1/2***√3/2** *\=* *1***√3** *\=* ***√3***3**** \*
tan(45°) *\=* *sin(45°)***cos(45°)** ***\=*** *√2/2***√2/2** = **1**
tan(60°) *\=* *sin(60°)***cos(60°)** ***\=*** *√3/2***1/2** = **√3**
\* Note: writing *1***√3** **may cost you marks** so use ***√3***3**** instead (see [Rational Denominators](https://www.mathsisfun.com/algebra/rationalize-denominator.html) to learn more).
## Quick Sketch
Another way to help you remember 30° and 60° is to make a quick sketch:
| | | |
|---|---|---|
| Draw a triangle with side lengths of 2 | |  |
| Cut in half. [Pythagoras](https://www.mathsisfun.com/pythagoras.html) says the new side is √3 a2 + b2 *\=* c2 12 + (√3)2 *\=* 22 1 + 3 *\=* 4  | |  |
| Then use [sohcahtoa](https://www.mathsisfun.com/algebra/sohcahtoa.html) for sin, cos or tan | |  |
### Example: sin(30°)
Sine: **soh**cahtoa
sine is opposite divided by hypotenuse
sin(30°) *\=* *opposite* **hypotenuse** *\=* *1* **2**
## The Whole Circle
For the whole circle we need values in [every quadrant](https://www.mathsisfun.com/algebra/trig-four-quadrants.html), with the correct plus or minus sign as per [Cartesian Coordinates](https://www.mathsisfun.com/data/cartesian-coordinates.html):
Note that **cos** is first and **sin** is second, so it goes **(cos, sin)**:
[Save as PDF](https://www.mathsisfun.com/geometry/images/circle-unit.pdf)
### Example: What is cos(330°) ?
Make a sketch like this, and we can see it is the "long" value: *√3***2**
And this is the same Unit Circle in **radians**.
### Example: What is sin(7π/6) ?
Think "7π/6 *\=* π + π/6", then make a sketch.
We can then see it is **negative** and is the "short" value: −½
Mathopolis:[Q1]() [Q2]() [Q3]() [Q4]() [Q5]() [Q6]() [Q7]() [Q8]() [Q9]() [Q10]()
### Footnote: where do the values come from?
We can use the equation x2 + y2 *\=* 1 to find the lengths of **x** and **y** (which are equal to **cos** and **sin** when the radius is **1**):
### 45 Degrees
For 45 degrees, x and y are equal, so **y=x**:
x2 + x2 *\=* 1
2x2 *\=* 1
x2 *\=* ½
x *\=* y *\=* √(½)
### 60 Degrees
Take an [equilateral triangle](https://www.mathsisfun.com/triangle.html) *(all sides are equal and all angles are 60°)* and split it down the middle.
The "x" side is now **½**,
And the "y" side is:
(½)2 + y2 *\=* 1
¼ + y2 *\=* 1
y2 *\=* 1 − ¼ *\=* ¾
y *\=* √(¾)
### 30 Degrees
30° is just 60° with x and y swapped, so **x *\=* √(¾)** and **y *\=* ½**
And:
√1/2 *\=* √2/4 *\=* *√2***√4** *\=* *√2***2**
Also:
√3/4 *\=* *√3***√4** *\=* *√3***2**
And here is the result (same as before):
| Angle | Cos | Sin | *Tan=Sin/Cos* |
|---|---|---|---|
| 30° | *√3***2** | *1***2** | *1* **√3** *\=* *√3* **3** |
| 45° | *√2***2** | *√2***2** | 1 |
| 60° | *1***2** | *√3***2** | √3 |
[Circle](https://www.mathsisfun.com/geometry/circle.html) [Interactive Unit Circle](https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html) [Sine, Cosine and Tangent in Four Quadrants](https://www.mathsisfun.com/algebra/trig-four-quadrants.html) [Trigonometry Index](https://www.mathsisfun.com/algebra/trigonometry-index.html)
[Donate]() ○ [Search](https://www.mathsisfun.com/search/search.html) ○ [Index](https://www.mathsisfun.com/links/index.html) ○ [About](https://www.mathsisfun.com/aboutmathsisfun.html) ○ [Contact](https://www.mathsisfun.com/contact.html) ○ [Cite This Page]() ○ **[Privacy](https://www.mathsisfun.com/about-privacy.html)**
Copyright © 2025 Rod Pierce | ||||||||||||||||||
| Readable Markdown | 
The "Unit Circle" is a circle with a radius of 1.
Being so simple, it is a great way to learn and talk about lengths and angles.
The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement here.
## Sine, Cosine and Tangent
Since the radius is 1, we can read [sine, cosine and tangent](https://www.mathsisfun.com/sine-cosine-tangent.html) just from the x and y coordinates.
What happens when the angle, θ, is 0°?
cos 0° *\=* 1, sin 0° *\=* 0 and tan 0° *\=* 0
What happens when θ is 90°?
cos 90° *\=* 0, sin 90° *\=* 1 and tan 90° is undefined
## Try It Yourself\!
Play with the interactive Unit Circle below. See how different angles (in [radians](https://www.mathsisfun.com/geometry/radians.html) or [degrees](https://www.mathsisfun.com/geometry/degrees.html)) affect sine, cosine and tangent:
Can you find an angle where sine and cosine are equal?
The "sides" can be positive or negative according to the rules of [Cartesian coordinates](https://www.mathsisfun.com/data/cartesian-coordinates.html). This makes the sine, cosine and tangent change between positive and negative values also.
Try the [Interactive Unit Circle](https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html)
## Pythagoras
[Pythagoras' Theorem](https://www.mathsisfun.com/pythagoras.html) says for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:
x2 + y2 *\=* 12
But 12 is just 1, so:
x2 + y2 *\=* 1 *equation of the unit circle*
Also, since x=cos and y=sin, we get:
(cos(θ))2 + (sin(θ))2 *\=* 1
*A useful **identity***
## Important Angles: 30°, 45° and 60°
You should try to **remember** sin, cos and tan for the angles 30°, 45° and 60°**.**
Yes, yes, it is a pain to have to remember things, but it will make life easier when you know them, not just in exams, but other times when you need to do quick estimates, and so on.
**These are the values you should remember\!**
| Angle | Cos | Sin | *Tan=Sin/Cos* |
|---|---|---|---|
| 30° | *√3***2** | *1***2** | *1* **√3** *\=* *√3* **3** |
| 45° | *√2***2** | *√2***2** | 1 |
| 60° | *1***2** | *√3***2** | √3 |
### How To Remember?
To help you remember, cos goes **"3,2,1"**
cos(30°) *\=* *√**3*****2**
cos(45°) *\=* *√**2*****2**
cos(60°) *\=* *√**1*****2** *\=* *1***2**
And, sin goes **"1,2,3"** :
sin(30°) *\=* *√**1*****2** *\=* *1***2** (because √1 *\=* 1)
sin(45°) *\=* *√**2*****2**
sin(60°) *\=* *√**3*****2**
## Just 3 Numbers
In fact, knowing 3 numbers is enough: *1***2** , *√2***2** and *√3***2**
Because they work for both **cos** and **sin**:
 
Your hand can help you remember:
For example there are 3 fingers above 30°, so cos(30°) *\=* *√3***2**
## What about tan?
Well, **tan *\=* sin/cos**, so we can calculate it like this:
tan(30°) *\=* *sin(30°)***cos(30°)** ***\=*** *1/2***√3/2** *\=* *1***√3** *\=* ***√3***3**** \*
tan(45°) *\=* *sin(45°)***cos(45°)** ***\=*** *√2/2***√2/2** = **1**
tan(60°) *\=* *sin(60°)***cos(60°)** ***\=*** *√3/2***1/2** = **√3**
\* Note: writing *1***√3** **may cost you marks** so use ***√3***3**** instead (see [Rational Denominators](https://www.mathsisfun.com/algebra/rationalize-denominator.html) to learn more).
## Quick Sketch
Another way to help you remember 30° and 60° is to make a quick sketch:
| | | |
|---|---|---|
| Draw a triangle with side lengths of 2 | |  |
| Cut in half. [Pythagoras](https://www.mathsisfun.com/pythagoras.html) says the new side is √3 a2 + b2 *\=* c2 12 + (√3)2 *\=* 22 1 + 3 *\=* 4  | |  |
| Then use [sohcahtoa](https://www.mathsisfun.com/algebra/sohcahtoa.html) for sin, cos or tan | |  |
### Example: sin(30°)
Sine: **soh**cahtoa
sine is opposite divided by hypotenuse
sin(30°) *\=* *opposite* **hypotenuse** *\=* *1* **2**
## The Whole Circle
For the whole circle we need values in [every quadrant](https://www.mathsisfun.com/algebra/trig-four-quadrants.html), with the correct plus or minus sign as per [Cartesian Coordinates](https://www.mathsisfun.com/data/cartesian-coordinates.html):
Note that **cos** is first and **sin** is second, so it goes **(cos, sin)**:
[Save as PDF](https://www.mathsisfun.com/geometry/images/circle-unit.pdf)
### Example: What is cos(330°) ?
Make a sketch like this, and we can see it is the "long" value: *√3***2**
And this is the same Unit Circle in **radians**.
### Example: What is sin(7π/6) ?
Think "7π/6 *\=* π + π/6", then make a sketch.
We can then see it is **negative** and is the "short" value: −½
### Footnote: where do the values come from?
We can use the equation x2 + y2 *\=* 1 to find the lengths of **x** and **y** (which are equal to **cos** and **sin** when the radius is **1**):
### 45 Degrees
For 45 degrees, x and y are equal, so **y=x**:
x2 + x2 *\=* 1
2x2 *\=* 1
x2 *\=* ½
x *\=* y *\=* √(½)
### 60 Degrees
Take an [equilateral triangle](https://www.mathsisfun.com/triangle.html) *(all sides are equal and all angles are 60°)* and split it down the middle.
The "x" side is now **½**,
And the "y" side is:
(½)2 + y2 *\=* 1
¼ + y2 *\=* 1
y2 *\=* 1 − ¼ *\=* ¾
y *\=* √(¾)
### 30 Degrees
30° is just 60° with x and y swapped, so **x *\=* √(¾)** and **y *\=* ½**
And:
√1/2 *\=* √2/4 *\=* *√2***√4** *\=* *√2***2**
Also:
√3/4 *\=* *√3***√4** *\=* *√3***2**
And here is the result (same as before):
| Angle | Cos | Sin | *Tan=Sin/Cos* |
|---|---|---|---|
| 30° | *√3***2** | *1***2** | *1* **√3** *\=* *√3* **3** |
| 45° | *√2***2** | *√2***2** | 1 |
| 60° | *1***2** | *√3***2** | √3 | | ||||||||||||||||||
| ML Classification | |||||||||||||||||||
| ML Categories |
Raw JSON{
"/Science": 808,
"/Science/Mathematics": 806,
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| Content Metadata | |||||||||||||||||||
| Language | en | ||||||||||||||||||
| Author | Rod Pierce | ||||||||||||||||||
| Publish Time | not set | ||||||||||||||||||
| Original Publish Time | 2014-12-01 12:49:44 (11 years ago) | ||||||||||||||||||
| Republished | No | ||||||||||||||||||
| Word Count (Total) | 1,063 | ||||||||||||||||||
| Word Count (Content) | 964 | ||||||||||||||||||
| Links | |||||||||||||||||||
| External Links | 3 | ||||||||||||||||||
| Internal Links | 37 | ||||||||||||||||||
| Technical SEO | |||||||||||||||||||
| Meta Nofollow | No | ||||||||||||||||||
| Meta Noarchive | No | ||||||||||||||||||
| JS Rendered | Yes | ||||||||||||||||||
| Redirect Target | null | ||||||||||||||||||
| Performance | |||||||||||||||||||
| Download Time (ms) | 64 | ||||||||||||||||||
| TTFB (ms) | 64 | ||||||||||||||||||
| Download Size (bytes) | 4,666 | ||||||||||||||||||
| Shard | 29 (laksa) | ||||||||||||||||||
| Root Hash | 10041072732158755029 | ||||||||||||||||||
| Unparsed URL | com,mathsisfun!www,/geometry/unit-circle.html s443 | ||||||||||||||||||