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1. Shard Calculation

Query:

Response:

Calculated Shard: 185 (from laksa168)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa185.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.cuemath.com/trigonometry/sin-of-2-pi/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.cuemath.com/trigonometry/sin-of-2-pi/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/","crawl_time":1775511924,"first_indexed_time":1636610793,"http_code":200,"src_unparsed":"com,cuemath!www,\/trigonometry\/sin-of-2-pi\/ s443","src_root_hash":"6203651625873809585","history_drop_reason":null,"meta_title":"Sin 2pi - Find Value of Sin 2pi | Sin 2π","meta_descriptions":["On the unit circle, 2π is one complete rotation and hence 2π is equivalent to the angle 0 radians. We know that 0 radians on the unit circle correspond to the point (1, 0). Thus, sin 2π = sin 0 = y-coordinate of (1, 0) = 0. Hence, sin 2π = 0."],"attrs_boilerpipe_text":"Before going to find the value of sin of 2pi, let us recollect the values of sine function of different standard angles from the trigonometric table. sin 0 = 0, sin π\/6 = 1\/2, sin π\/4 = √2\/2, sin π\/3 = √3\/2, and sin π\/2 = 1. Of course, this table does not include the value of sin 2pi in it. We will find that sin of 2pi is 0 using various methods here. Also, we will solve some example problems using this.\n1.\nWhat is Sin of 2pi?\n2.\nSin of 2pi Using Double Angle Formula\n3.\nSin of 2pi Using Reference Angles\n4.\nSin of 2pi Using Unit Circle\n5.\nFAQs on Sin of 2pi\nWhat is Sin of 2pi?\nThe value of\nsin\nof 2pi is 0. i.e., sin 2\nπ\n= 0. From\ntrigonometric table\n, we know the trigonometric ratios of standard angles 0, π\/6, π\/4, π\/3, and π\/2. So this table doesn't give us the value of sin of 2pi. Usually, to find the value of any trigonometric ratio of a non-standard angle, we use the\nreference angles\nand the quadrant in which the angle lies in. We can do the same to find sin of 2pi also. The value of sin of 2pi can be easily found by using several other methods like\nUsing double angle formula\nUsing reference angle\nUsing unit circle\nWe will prove that sin 2π = 0 in each of these methods.\nSin of 2pi Using Double Angle Formula\nWe can find the value of sin of 2pi using the\ndouble angle formula\nof sine which is sin 2x = 2 sin x cos x. Since we have to find the value of sin(2π), we have to substitute x = π in the above formula. Then we get:\nsin 2π = 2 sin π cos π ... (1)\nSince π is also a non-standard angle, we find the values of sin π and cos π using the\nsum and difference formulas\n. Then we get\nsin π = sin (π\/2 + π\/2) = sin π\/2 cos π\/2 + cos π\/2 sin π\/2 = (1)(0) + (0)(1) = 0\ncos π = cos (π\/2 + π\/2) = cos π\/2 cos π\/2 - sin π\/2 sin π\/2 = (0)(0) - (1)(1) = -1\nSubstitute these values in (1),\nsin 2π = 2 (0) (-1) = 0\nHence, sin of 2pi = 0.\nSin of 2pi Using Reference Angles\nIf we\nconvert\n2π into degrees, we get 360°. Since 360° lies in the interval [0°, 360°], its coterminal angle itself is the reference angle. To find its\ncoterminal angle\n, we subtract 360° from it. Then we get 360° - 360° = 0°. So the coterminal angle of 360° is 0°. Also, we know that 360° means one full rotation and thus it comes either in the first quadrant or in the fourth quadrant. Let us consider both cases.\nFirst Quadrant:\nWe know that in the first quadrant, sin is positive.\nThen sin 360° = + sin 0° = 0 (because sin 0° = 0)\nFourth Quadrant:\nWe know that in the fourth quadrant, sin is negative.\nThen sin 360° = - sin 0° = 0 (because sin 0° = 0)\nFrom both the cases, sin 360° = sin 2π = 0.\nHence, sin 2π = 0.\nSin of 2pi Using Unit Circle\nBefore finding the value of sin of 2pi using\nunit circle\n, let us recollect a few points about the unit circle.\nA unit circle is a\ncircle\nof radius centered at the origin.\nEvery point on the unit circle corresponds to an angle.\nThis angle is made by the line joining the origin and the point with the positive direction of the x-axis in the anti-clockwise direction.\nIf P(x, y) corresponds to some angle θ, then x = cos θ and y = sin θ. i.e., the y-coordinate of the point represents the sine of the corresponding angle.\nAs 2π (which is nothing but 360°) represents one full rotation, it is nothing but the angle made by the x-axis with itself and thus, it is equivalent to 0° on the unit circle. We also know that 0° corresponds to the point (1, 0) on the unit circle (as it is a point on the unit circle present on the x-axis). Thus,\nsin 2π = sin 0° = y-coordinate of (1, 0) = 0.\nHence, sin 2π = 0.\nYou can visualize it from the following figure.\nImportant Notes:\nHere are some important notes that are related to sin of 2pi.\nsin 2π = 0\ncos 2π = 1\ntan 2π = 0\nsin of any multiple of π is 0. i.e., sin nπ = 0, for any\ninteger\n'n'.\nFor example, sin(-2π) = 0, sin π = 0, \nRelated Topics:\nHere are some topics are related to sin of 2pi.\nTrigonometric Ratios\nTrigonometric Formulas\nSine Law\nCosine Law\nExamples Using Sin of 2pi\nExample 1: Find the value of cos of 2pi using the fact that sin of 2pi = 0.\nSolution:\nUsing a\ntrigonometric identity\n,\nsin\n2\nx + cos\n2\nx = 1\nSubstitute x = 2π here,\nsin\n2\n2π + cos\n2\n2π = 1\nWe know that sin 2π = 0. Substitute it here,\n0\n2\n+ cos\n2\n2π = 1\ncos\n2\n2π = 1\ncos 2π = ± 1\nBut 2π is in first or fourth quadrant and in each of the cases, cos is positive.\nThus, cos 2π = 1.\nAnswer:\ncos 2π = 1.\nExample 2: Find the value of sin of 4pi using the fact that sin of 2pi = 0.\nSolution:\nUsing the double angle formula of sin,\nsin 2x = 2 sin x cos x\nSubstitute x = 2π here,\nsin 2(2π) = 2 sin 2π cos 2π\nsin 4π = 2 (0) cos 2π (because sin 2π = 0)\nsin 4π = 0\nAnswer:\nsin 4π = 0.\nExample 3: Find the value of sin 5π\/2 using the sum and difference formulas.\nSolution:\nUsing the sum and difference formulas,\nsin (A + B) = sin A cos B + cos A sin B\nSubstitute A = 2π and B = π\/2 here,\nsin (2π + π\/2) = sin 2π cos π\/2 + cos 2π sin π\/2\nWe know that sin of 2pi = 0 and cos of 2pi = 1. Also, sin π\/2 = 1 and cos π\/2 = 0. Thus,\nsin 5π\/2 = (0)(0) + (1) (1) = 1.\nAnswer:\nsin 5π\/2 = 1.\ngo to slide\ngo to slide\ngo to slide\nGreat learning in high school using simple cues\nIndulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.\nBook a Free Trial Class\nPractice Questions on Sin of 2pi\ngo to slide\ngo to slide\nFAQs on Sin of 2pi\nWhat Is Sin of 2pi?\nThe value of sin of 2 pi is 0. This is because 2π is nothing but 0 degrees on the unit circle. Thus, sin 2π = sin 0 = 0.\nWhat Is the Value of Square of Sin of 2pi?\nWe have sin(2π) = 0. From this, sin\n2\n(2π) = (sin 2π)\n2\n= 0\n2\n= 0. Thus, the value of square of sin of 2pi is 0.\nWhat is Sin^2(pi\/2)?\nWe know that sin π\/2 = 1. So sin\n2\nπ\/2 = (sin π\/2)\n2\n = 1\n2\n = 1.\nIs Sin of 2pi Undefined?\nNo, sin(2π) is NOT undefined, instead sin(2π) = 0. This is because the coterminal angle of 2π is 0, from this we can conclude the value of sin of 2pi as, sin 2π = sin 0 = 0.\nWhy Does Sin of 2pi Equal To Zero?\nSince 2π is an angle that is in the interval [0, 2π], its reference angle is nothing but its coterminal angle, which is 0. Thus, sin of 2pi can be given as, sin 2π = sin 0 = 0.\nHow To Prove that Sin of 2pi is 0?\nOn the unit circle, 2π is one complete rotation and hence 2π is equivalent to the angle 0\nradians\n. We know that 0 radians on the unit circle correspond to the point (1, 0). Thus, sin 2π = sin 0 = y-coordinate of (1, 0) = 0. Hence, sin 2π = 0.\nWhat is the Value of cos(-2pi)?\nWe know that cos (-x) = cos x. So cos (-2π) = cos 2π. On unit circle, cos 2π is same as cos 0 and hence it is 1. Thus, cos(-2pi) = 1.\nHow To Find the Value of Sin of 2pi Using Double Angle Formula?\nWe have sin 2x = 2 sin x cos x. Substitute x = π in the above formula. Then for sin of 2pi, we get: sin 2π = 2 sin π cos π ... (1).\nNow,\nsin π = sin (π\/2 + π\/2) = sin π\/2 cos π\/2 + cos π\/2 sin π\/2 = (1)(0) + (0)(1) = 0\ncos π = cos (π\/2 + π\/2) = cos π\/2 cos π\/2 - sin π\/2 sin π\/2 = (0)(0) - (1)(1) = -1\nSubstitute these values in (1),\nsin 2π = 2 (0) (-1) = 0\nHence, sin 2π = 0.\nWhat Is Tan of 2pi Using Sin of 2pi?\nWe know that sin of 2pi is equal to zero, i.e., sin(2π) = 0. Also we know that\ntan\nx = (sin x) \/ (cos x) for any x. Thus, tan(2π) = [sin(2π)] \/ [cos(2π)] = (0) \/ [cos(2π)] = 0. Hence, tan(2π) = 0.\nWhat is the Value of Sin 2pi\/3?\nLet us convert 2π\/3 into degrees. Then we get 2π\/3 = 120°. Using unit circle, sin 120° = √3\/2. Thus, sin 2pi\/3 = √3\/2.","attrs_markdown":"[Math Classes](https:\/\/www.cuemath.com\/en-eu\/math-online-classes)\n\n[Year K-2](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-1-3)\n\n[Kindergarten](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/kindergarten)\n\n[Year 1](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-1)\n\n[Year 2](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-2)\n\n[Year 3-5](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-4-6)\n\n[Year 3](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-3)\n\n[Year 4](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-4)\n\n[Year 5](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-5)\n\n[Year 6-8](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-7-8)\n\n[Year 6](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-6)\n\n[Year 7](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-7)\n\n[Year 8](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-8)\n\n[Pricing](https:\/\/www.cuemath.com\/en-eu\/pricing)\n\n[Resources]()\n\n[MathFit Evaluation](https:\/\/www.cuemath.com\/mathfit-evaluation)\n\n[Blogs](https:\/\/www.cuemath.com\/blog)\n\n[Cuemath App](https:\/\/www.cuemath.com\/apps\/cuemath?utm_source=website_navbar)\n\n[About Us]()\n\n[Our Impact](https:\/\/www.cuemath.com\/our-impact)\n\n[Our Tutors](https:\/\/www.cuemath.com\/our-tutors)\n\n[Our Story & Mission](https:\/\/www.cuemath.com\/about-us)\n\n[Our Reviews](https:\/\/www.cuemath.com\/our-reviews)\n\n[Cuemath Vs Others](https:\/\/www.cuemath.com\/blog\/tag\/cuemath-vs-others)\n\n[What is MathFit](https:\/\/www.cuemath.com\/blog\/what-does-it-mean-to-be-mathfit)\n\n[FAQs](https:\/\/www.cuemath.com\/frequently-asked-questions)\n\nGet started\n\n[Math Classes](https:\/\/www.cuemath.com\/en-eu\/math-online-classes)\n\n[Year K-2](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-1-3)\n\n[Kindergarten](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/kindergarten)\n\n[Year 1](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-1)\n\n[Year 2](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-2)\n\n[Year 3-5](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-4-6)\n\n[Year 3](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-3)\n\n[Year 4](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-4)\n\n[Year 5](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-5)\n\n[Year 6-8](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-7-8)\n\n[Year 6](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-6)\n\n[Year 7](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-7)\n\n[Year 8](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-8)\n\n[Pricing](https:\/\/www.cuemath.com\/en-eu\/pricing)\n\nResources\n\n[MathFit Evaluation](https:\/\/www.cuemath.com\/mathfit-evaluation)\n\n[Blogs](https:\/\/www.cuemath.com\/blog)\n\n[Cuemath App](https:\/\/www.cuemath.com\/apps\/cuemath?utm_source=website_navbar)\n\nAbout Us\n\n[Our Impact](https:\/\/www.cuemath.com\/our-impact)\n\n[Our Tutors](https:\/\/www.cuemath.com\/our-tutors)\n\n[Our Story & Mission](https:\/\/www.cuemath.com\/about-us)\n\n[Our Reviews](https:\/\/www.cuemath.com\/our-reviews)\n\n[Cuemath Vs Others](https:\/\/www.cuemath.com\/blog\/tag\/cuemath-vs-others)\n\n[What is MathFit](https:\/\/www.cuemath.com\/blog\/what-does-it-mean-to-be-mathfit)\n\n[FAQs](https:\/\/www.cuemath.com\/frequently-asked-questions)\n\n[Learn]()[Practice]()[Download]()\n\n# Sin of 2pi\nBefore going to find the value of sin of 2pi, let us recollect the values of sine function of different standard angles from the trigonometric table. sin 0 = 0, sin π\/6 = 1\/2, sin π\/4 = √2\/2, sin π\/3 = √3\/2, and sin π\/2 = 1. Of course, this table does not include the value of sin 2pi in it. We will find that sin of 2pi is 0 using various methods here. Also, we will solve some example problems using this.\n\n|   |   |\n|---|---|\n| 1\\. | [What is Sin of 2pi?](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#what-is-sin-of-2pi) |\n| 2\\. | [Sin of 2pi Using Double Angle Formula](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#sin-of-2pi-using-double-angle-formula) |\n| 3\\. | [Sin of 2pi Using Reference Angles](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#sin-of-2pi-using-reference-angles) |\n| 4\\. | [Sin of 2pi Using Unit Circle](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#sin-of-2pi-using-unit-circle) |\n| 5\\. | [FAQs on Sin of 2pi](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#faqs) |\n## What is Sin of 2pi?\nThe value of [sin](https:\/\/www.cuemath.com\/trigonometry\/sine-function\/) of 2pi is 0. i.e., sin 2[π](https:\/\/www.cuemath.com\/geometry\/what-is-pi\/) = 0. From [trigonometric table](https:\/\/www.cuemath.com\/trigonometry\/trigonometric-table\/), we know the trigonometric ratios of standard angles 0, π\/6, π\/4, π\/3, and π\/2. So this table doesn't give us the value of sin of 2pi. Usually, to find the value of any trigonometric ratio of a non-standard angle, we use the [reference angles](https:\/\/www.cuemath.com\/geometry\/reference-angle\/) and the quadrant in which the angle lies in. We can do the same to find sin of 2pi also. The value of sin of 2pi can be easily found by using several other methods like\n\n- Using double angle formula\n- Using reference angle\n- Using unit circle\n\nWe will prove that sin 2π = 0 in each of these methods.\n\n![Sin of 2pi is 0.](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/sin-of-2pi-1628225803.png)\n\n## Sin of 2pi Using Double Angle Formula\nWe can find the value of sin of 2pi using the [double angle formula](https:\/\/www.cuemath.com\/double-angle-formulas\/) of sine which is sin 2x = 2 sin x cos x. Since we have to find the value of sin(2π), we have to substitute x = π in the above formula. Then we get:\n\nsin 2π = 2 sin π cos π ... (1)\n\nSince π is also a non-standard angle, we find the values of sin π and cos π using the [sum and difference formulas](https:\/\/www.cuemath.com\/trigonometry\/sum-and-difference-identities\/). Then we get\n\nsin π = sin (π\/2 + π\/2) = sin π\/2 cos π\/2 + cos π\/2 sin π\/2 = (1)(0) + (0)(1) = 0\n\ncos π = cos (π\/2 + π\/2) = cos π\/2 cos π\/2 - sin π\/2 sin π\/2 = (0)(0) - (1)(1) = -1\n\nSubstitute these values in (1),\n\nsin 2π = 2 (0) (-1) = 0\n\nHence, sin of 2pi = 0.\n\n## Sin of 2pi Using Reference Angles\nIf we [convert](https:\/\/www.cuemath.com\/geometry\/radians-to-degrees\/) 2π into degrees, we get 360°. Since 360° lies in the interval \\[0°, 360°\\], its coterminal angle itself is the reference angle. To find its [coterminal angle](https:\/\/www.cuemath.com\/geometry\/coterminal-angles\/), we subtract 360° from it. Then we get 360° - 360° = 0°. So the coterminal angle of 360° is 0°. Also, we know that 360° means one full rotation and thus it comes either in the first quadrant or in the fourth quadrant. Let us consider both cases.\n\n- **First Quadrant:** We know that in the first quadrant, sin is positive.  \n   Then sin 360° = + sin 0° = 0 (because sin 0° = 0)\n- **Fourth Quadrant:** We know that in the fourth quadrant, sin is negative.  \n   Then sin 360° = - sin 0° = 0 (because sin 0° = 0)\n\nFrom both the cases, sin 360° = sin 2π = 0.\n\nHence, sin 2π = 0.\n\n## Sin of 2pi Using Unit Circle\nBefore finding the value of sin of 2pi using [unit circle](https:\/\/www.cuemath.com\/geometry\/unit-circle\/), let us recollect a few points about the unit circle.\n\n- A unit circle is a [circle](https:\/\/www.cuemath.com\/geometry\/circles\/) of radius centered at the origin.\n- Every point on the unit circle corresponds to an angle.\n- This angle is made by the line joining the origin and the point with the positive direction of the x-axis in the anti-clockwise direction.\n- If P(x, y) corresponds to some angle θ, then x = cos θ and y = sin θ. i.e., the y-coordinate of the point represents the sine of the corresponding angle.\n\nAs 2π (which is nothing but 360°) represents one full rotation, it is nothing but the angle made by the x-axis with itself and thus, it is equivalent to 0° on the unit circle. We also know that 0° corresponds to the point (1, 0) on the unit circle (as it is a point on the unit circle present on the x-axis). Thus,\n\nsin 2π = sin 0° = y-coordinate of (1, 0) = 0.\n\nHence, sin 2π = 0.\n\nYou can visualize it from the following figure.\n\n![Sin of 2 pi Using Unit Circle. Sin of 2 pi is equal to the y-coordinate of the point (1, 0). Hence sin of 2 pi is 0.](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/sin-of-2pi-using-unit-circle-1630134016.png)\n\n**Important Notes:**\n\nHere are some important notes that are related to sin of 2pi.\n\n- sin 2π = 0\n- cos 2π = 1\n- tan 2π = 0\n- sin of any multiple of π is 0. i.e., sin nπ = 0, for any [integer](https:\/\/www.cuemath.com\/numbers\/integers\/) 'n'.  \n   For example, sin(-2π) = 0, sin π = 0,\n\n**Related Topics:**\n\nHere are some topics are related to sin of 2pi.\n\n- [Trigonometric Ratios](https:\/\/www.cuemath.com\/trigonometry\/trigonometric-ratios\/)\n- [Trigonometric Formulas](https:\/\/www.cuemath.com\/trigonometry\/trigonometry-formula\/)\n- [Sine Law](https:\/\/www.cuemath.com\/trigonometry\/sine-law\/)\n- [Cosine Law](https:\/\/www.cuemath.com\/trigonometry\/law-of-cosine\/)\n\n## Examples Using Sin of 2pi\n1. **Example 1: Find the value of cos of 2pi using the fact that sin of 2pi = 0.**\n   \n   **Solution:**\n   \n   Using a [trigonometric identity](https:\/\/www.cuemath.com\/trigonometry\/trigonometric-identities\/),\n   \n   sin2x + cos2x = 1\n   \n   Substitute x = 2π here,\n   \n   sin22π + cos22π = 1\n   \n   We know that sin 2π = 0. Substitute it here,\n   \n   02 + cos22π = 1\n   \n   cos22π = 1\n   \n   cos 2π = ± 1\n   \n   But 2π is in first or fourth quadrant and in each of the cases, cos is positive.\n   \n   Thus, cos 2π = 1.\n   \n   **Answer:** cos 2π = 1.\n2. **Example 2: Find the value of sin of 4pi using the fact that sin of 2pi = 0.**\n   \n   **Solution:**\n   \n   Using the double angle formula of sin,\n   \n   sin 2x = 2 sin x cos x\n   \n   Substitute x = 2π here,\n   \n   sin 2(2π) = 2 sin 2π cos 2π\n   \n   sin 4π = 2 (0) cos 2π (because sin 2π = 0)\n   \n   sin 4π = 0\n   \n   **Answer:** sin 4π = 0.\n3. **Example 3: Find the value of sin 5π\/2 using the sum and difference formulas.**\n   \n   **Solution:**\n   \n   Using the sum and difference formulas,\n   \n   sin (A + B) = sin A cos B + cos A sin B\n   \n   Substitute A = 2π and B = π\/2 here,\n   \n   sin (2π + π\/2) = sin 2π cos π\/2 + cos 2π sin π\/2\n   \n   We know that sin of 2pi = 0 and cos of 2pi = 1. Also, sin π\/2 = 1 and cos π\/2 = 0. Thus,\n   \n   sin 5π\/2 = (0)(0) + (1) (1) = 1.\n   \n   **Answer:** sin 5π\/2 = 1.\n\nView Answer \\>\n\n[go to slide](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#slide-430)[go to slide](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#slide-699)[go to slide](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#slide-750)\n\n![](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/add-laptop-mexican-teacher-with-mexican-student-grade8-10-1619247480.png)\n\nGreat learning in high school using simple cues\n\nIndulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.\n\n[Book a Free Trial Class](https:\/\/www.cuemath.com\/parent\/signup\/)\n\n## Practice Questions on Sin of 2pi\nCheck Answer \\>\n\n[go to slide](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#slide-282)[go to slide](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#slide-339)\n\n## FAQs on Sin of 2pi\n### What Is Sin of 2pi?\nThe value of sin of 2 pi is 0. This is because 2π is nothing but 0 degrees on the unit circle. Thus, sin 2π = sin 0 = 0.\n\n### What Is the Value of Square of Sin of 2pi?\nWe have sin(2π) = 0. From this, sin2(2π) = (sin 2π)2 = 02 = 0. Thus, the value of square of sin of 2pi is 0.\n\n### What is Sin^2(pi\/2)?\nWe know that sin π\/2 = 1. So sin2π\/2 = (sin π\/2)2 = 12 = 1.\n\n### Is Sin of 2pi Undefined?\nNo, sin(2π) is NOT undefined, instead sin(2π) = 0. This is because the coterminal angle of 2π is 0, from this we can conclude the value of sin of 2pi as, sin 2π = sin 0 = 0.\n\n### Why Does Sin of 2pi Equal To Zero?\nSince 2π is an angle that is in the interval \\[0, 2π\\], its reference angle is nothing but its coterminal angle, which is 0. Thus, sin of 2pi can be given as, sin 2π = sin 0 = 0.\n\n### How To Prove that Sin of 2pi is 0?\nOn the unit circle, 2π is one complete rotation and hence 2π is equivalent to the angle 0 [radians](https:\/\/www.cuemath.com\/trigonometry\/what-is-a-radian\/). We know that 0 radians on the unit circle correspond to the point (1, 0). Thus, sin 2π = sin 0 = y-coordinate of (1, 0) = 0. Hence, sin 2π = 0.\n\n### What is the Value of cos(-2pi)?\nWe know that cos (-x) = cos x. So cos (-2π) = cos 2π. On unit circle, cos 2π is same as cos 0 and hence it is 1. Thus, cos(-2pi) = 1.\n\n### How To Find the Value of Sin of 2pi Using Double Angle Formula?\nWe have sin 2x = 2 sin x cos x. Substitute x = π in the above formula. Then for sin of 2pi, we get: sin 2π = 2 sin π cos π ... (1).\n\nNow,\n\n- sin π = sin (π\/2 + π\/2) = sin π\/2 cos π\/2 + cos π\/2 sin π\/2 = (1)(0) + (0)(1) = 0\n- cos π = cos (π\/2 + π\/2) = cos π\/2 cos π\/2 - sin π\/2 sin π\/2 = (0)(0) - (1)(1) = -1\n\nSubstitute these values in (1),\n\nsin 2π = 2 (0) (-1) = 0\n\nHence, sin 2π = 0.\n\n### What Is Tan of 2pi Using Sin of 2pi?\nWe know that sin of 2pi is equal to zero, i.e., sin(2π) = 0. Also we know that [tan](https:\/\/www.cuemath.com\/trigonometry\/tangent-function\/) x = (sin x) \/ (cos x) for any x. Thus, tan(2π) = \\[sin(2π)\\] \/ \\[cos(2π)\\] = (0) \/ \\[cos(2π)\\] = 0. Hence, tan(2π) = 0.\n\n### What is the Value of Sin 2pi\/3?\nLet us convert 2π\/3 into degrees. Then we get 2π\/3 = 120°. Using unit circle, sin 120° = √3\/2. 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Logo](https:\/\/d138zd1ktt9iqe.cloudfront.net\/static\/website-v3\/cuemath-logo.svg)\n\n[Terms & conditions](https:\/\/www.cuemath.com\/terms)\n\n[Privacy Policy](https:\/\/www.cuemath.com\/privacy)","attrs_readable_markdown":"Before going to find the value of sin of 2pi, let us recollect the values of sine function of different standard angles from the trigonometric table. sin 0 = 0, sin π\/6 = 1\/2, sin π\/4 = √2\/2, sin π\/3 = √3\/2, and sin π\/2 = 1. Of course, this table does not include the value of sin 2pi in it. We will find that sin of 2pi is 0 using various methods here. Also, we will solve some example problems using this.\n\n|   |   |\n|---|---|\n| 1\\. | [What is Sin of 2pi?](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#what-is-sin-of-2pi) |\n| 2\\. | [Sin of 2pi Using Double Angle Formula](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#sin-of-2pi-using-double-angle-formula) |\n| 3\\. | [Sin of 2pi Using Reference Angles](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#sin-of-2pi-using-reference-angles) |\n| 4\\. | [Sin of 2pi Using Unit Circle](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#sin-of-2pi-using-unit-circle) |\n| 5\\. | [FAQs on Sin of 2pi](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#faqs) |\n\nWhat is Sin of 2pi?\n\nThe value of [sin](https:\/\/www.cuemath.com\/trigonometry\/sine-function\/) of 2pi is 0. i.e., sin 2[π](https:\/\/www.cuemath.com\/geometry\/what-is-pi\/) = 0. From [trigonometric table](https:\/\/www.cuemath.com\/trigonometry\/trigonometric-table\/), we know the trigonometric ratios of standard angles 0, π\/6, π\/4, π\/3, and π\/2. So this table doesn't give us the value of sin of 2pi. Usually, to find the value of any trigonometric ratio of a non-standard angle, we use the [reference angles](https:\/\/www.cuemath.com\/geometry\/reference-angle\/) and the quadrant in which the angle lies in. We can do the same to find sin of 2pi also. The value of sin of 2pi can be easily found by using several other methods like\n\n- Using double angle formula\n- Using reference angle\n- Using unit circle\n\nWe will prove that sin 2π = 0 in each of these methods.\n\n![Sin of 2pi is 0.](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/sin-of-2pi-1628225803.png)\n\nSin of 2pi Using Double Angle Formula\n\nWe can find the value of sin of 2pi using the [double angle formula](https:\/\/www.cuemath.com\/double-angle-formulas\/) of sine which is sin 2x = 2 sin x cos x. Since we have to find the value of sin(2π), we have to substitute x = π in the above formula. Then we get:\n\nsin 2π = 2 sin π cos π ... (1)\n\nSince π is also a non-standard angle, we find the values of sin π and cos π using the [sum and difference formulas](https:\/\/www.cuemath.com\/trigonometry\/sum-and-difference-identities\/). Then we get\n\nsin π = sin (π\/2 + π\/2) = sin π\/2 cos π\/2 + cos π\/2 sin π\/2 = (1)(0) + (0)(1) = 0\n\ncos π = cos (π\/2 + π\/2) = cos π\/2 cos π\/2 - sin π\/2 sin π\/2 = (0)(0) - (1)(1) = -1\n\nSubstitute these values in (1),\n\nsin 2π = 2 (0) (-1) = 0\n\nHence, sin of 2pi = 0.\n\nSin of 2pi Using Reference Angles\n\nIf we [convert](https:\/\/www.cuemath.com\/geometry\/radians-to-degrees\/) 2π into degrees, we get 360°. Since 360° lies in the interval \\[0°, 360°\\], its coterminal angle itself is the reference angle. To find its [coterminal angle](https:\/\/www.cuemath.com\/geometry\/coterminal-angles\/), we subtract 360° from it. Then we get 360° - 360° = 0°. So the coterminal angle of 360° is 0°. Also, we know that 360° means one full rotation and thus it comes either in the first quadrant or in the fourth quadrant. Let us consider both cases.\n\n- **First Quadrant:** We know that in the first quadrant, sin is positive.  \n   Then sin 360° = + sin 0° = 0 (because sin 0° = 0)\n- **Fourth Quadrant:** We know that in the fourth quadrant, sin is negative.  \n   Then sin 360° = - sin 0° = 0 (because sin 0° = 0)\n\nFrom both the cases, sin 360° = sin 2π = 0.\n\nHence, sin 2π = 0.\n\nSin of 2pi Using Unit Circle\n\nBefore finding the value of sin of 2pi using [unit circle](https:\/\/www.cuemath.com\/geometry\/unit-circle\/), let us recollect a few points about the unit circle.\n\n- A unit circle is a [circle](https:\/\/www.cuemath.com\/geometry\/circles\/) of radius centered at the origin.\n- Every point on the unit circle corresponds to an angle.\n- This angle is made by the line joining the origin and the point with the positive direction of the x-axis in the anti-clockwise direction.\n- If P(x, y) corresponds to some angle θ, then x = cos θ and y = sin θ. i.e., the y-coordinate of the point represents the sine of the corresponding angle.\n\nAs 2π (which is nothing but 360°) represents one full rotation, it is nothing but the angle made by the x-axis with itself and thus, it is equivalent to 0° on the unit circle. We also know that 0° corresponds to the point (1, 0) on the unit circle (as it is a point on the unit circle present on the x-axis). Thus,\n\nsin 2π = sin 0° = y-coordinate of (1, 0) = 0.\n\nHence, sin 2π = 0.\n\nYou can visualize it from the following figure.\n\n![Sin of 2 pi Using Unit Circle. Sin of 2 pi is equal to the y-coordinate of the point (1, 0). Hence sin of 2 pi is 0.](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/sin-of-2pi-using-unit-circle-1630134016.png)\n\n**Important Notes:**\n\nHere are some important notes that are related to sin of 2pi.\n\n- sin 2π = 0\n- cos 2π = 1\n- tan 2π = 0\n- sin of any multiple of π is 0. i.e., sin nπ = 0, for any [integer](https:\/\/www.cuemath.com\/numbers\/integers\/) 'n'.  \n   For example, sin(-2π) = 0, sin π = 0,\n\n**Related Topics:**\n\nHere are some topics are related to sin of 2pi.\n\n- [Trigonometric Ratios](https:\/\/www.cuemath.com\/trigonometry\/trigonometric-ratios\/)\n- [Trigonometric Formulas](https:\/\/www.cuemath.com\/trigonometry\/trigonometry-formula\/)\n- [Sine Law](https:\/\/www.cuemath.com\/trigonometry\/sine-law\/)\n- [Cosine Law](https:\/\/www.cuemath.com\/trigonometry\/law-of-cosine\/)\n\n## Examples Using Sin of 2pi\n1. **Example 1: Find the value of cos of 2pi using the fact that sin of 2pi = 0.**\n   \n   **Solution:**\n   \n   Using a [trigonometric identity](https:\/\/www.cuemath.com\/trigonometry\/trigonometric-identities\/),\n   \n   sin2x + cos2x = 1\n   \n   Substitute x = 2π here,\n   \n   sin22π + cos22π = 1\n   \n   We know that sin 2π = 0. Substitute it here,\n   \n   02 + cos22π = 1\n   \n   cos22π = 1\n   \n   cos 2π = ± 1\n   \n   But 2π is in first or fourth quadrant and in each of the cases, cos is positive.\n   \n   Thus, cos 2π = 1.\n   \n   **Answer:** cos 2π = 1.\n2. **Example 2: Find the value of sin of 4pi using the fact that sin of 2pi = 0.**\n   \n   **Solution:**\n   \n   Using the double angle formula of sin,\n   \n   sin 2x = 2 sin x cos x\n   \n   Substitute x = 2π here,\n   \n   sin 2(2π) = 2 sin 2π cos 2π\n   \n   sin 4π = 2 (0) cos 2π (because sin 2π = 0)\n   \n   sin 4π = 0\n   \n   **Answer:** sin 4π = 0.\n3. **Example 3: Find the value of sin 5π\/2 using the sum and difference formulas.**\n   \n   **Solution:**\n   \n   Using the sum and difference formulas,\n   \n   sin (A + B) = sin A cos B + cos A sin B\n   \n   Substitute A = 2π and B = π\/2 here,\n   \n   sin (2π + π\/2) = sin 2π cos π\/2 + cos 2π sin π\/2\n   \n   We know that sin of 2pi = 0 and cos of 2pi = 1. Also, sin π\/2 = 1 and cos π\/2 = 0. Thus,\n   \n   sin 5π\/2 = (0)(0) + (1) (1) = 1.\n   \n   **Answer:** sin 5π\/2 = 1.\n\n[go to slide](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#slide-430)[go to slide](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#slide-699)[go to slide](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#slide-750)\n\n![](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/add-laptop-mexican-teacher-with-mexican-student-grade8-10-1619247480.png)\n\nGreat learning in high school using simple cues\n\nIndulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.\n\n[Book a Free Trial Class](https:\/\/www.cuemath.com\/parent\/signup\/)\n\n## Practice Questions on Sin of 2pi\n[go to slide](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#slide-282)[go to slide](https:\/\/www.cuemath.com\/trigonometry\/sin-of-2-pi\/#slide-339)\n\nFAQs on Sin of 2pi\n\n### What Is Sin of 2pi?\nThe value of sin of 2 pi is 0. This is because 2π is nothing but 0 degrees on the unit circle. Thus, sin 2π = sin 0 = 0.\n\n### What Is the Value of Square of Sin of 2pi?\nWe have sin(2π) = 0. From this, sin2(2π) = (sin 2π)2 = 02 = 0. Thus, the value of square of sin of 2pi is 0.\n\n### What is Sin^2(pi\/2)?\nWe know that sin π\/2 = 1. So sin2π\/2 = (sin π\/2)2 = 12 = 1.\n\n### Is Sin of 2pi Undefined?\nNo, sin(2π) is NOT undefined, instead sin(2π) = 0. This is because the coterminal angle of 2π is 0, from this we can conclude the value of sin of 2pi as, sin 2π = sin 0 = 0.\n\n### Why Does Sin of 2pi Equal To Zero?\nSince 2π is an angle that is in the interval \\[0, 2π\\], its reference angle is nothing but its coterminal angle, which is 0. Thus, sin of 2pi can be given as, sin 2π = sin 0 = 0.\n\n### How To Prove that Sin of 2pi is 0?\nOn the unit circle, 2π is one complete rotation and hence 2π is equivalent to the angle 0 [radians](https:\/\/www.cuemath.com\/trigonometry\/what-is-a-radian\/). We know that 0 radians on the unit circle correspond to the point (1, 0). Thus, sin 2π = sin 0 = y-coordinate of (1, 0) = 0. Hence, sin 2π = 0.\n\n### What is the Value of cos(-2pi)?\nWe know that cos (-x) = cos x. So cos (-2π) = cos 2π. On unit circle, cos 2π is same as cos 0 and hence it is 1. Thus, cos(-2pi) = 1.\n\n### How To Find the Value of Sin of 2pi Using Double Angle Formula?\nWe have sin 2x = 2 sin x cos x. Substitute x = π in the above formula. Then for sin of 2pi, we get: sin 2π = 2 sin π cos π ... (1).\n\nNow,\n\n- sin π = sin (π\/2 + π\/2) = sin π\/2 cos π\/2 + cos π\/2 sin π\/2 = (1)(0) + (0)(1) = 0\n- cos π = cos (π\/2 + π\/2) = cos π\/2 cos π\/2 - sin π\/2 sin π\/2 = (0)(0) - (1)(1) = -1\n\nSubstitute these values in (1),\n\nsin 2π = 2 (0) (-1) = 0\n\nHence, sin 2π = 0.\n\n### What Is Tan of 2pi Using Sin of 2pi?\nWe know that sin of 2pi is equal to zero, i.e., sin(2π) = 0. Also we know that [tan](https:\/\/www.cuemath.com\/trigonometry\/tangent-function\/) x = (sin x) \/ (cos x) for any x. Thus, tan(2π) = \\[sin(2π)\\] \/ \\[cos(2π)\\] = (0) \/ \\[cos(2π)\\] = 0. Hence, tan(2π) = 0.\n\n### What is the Value of Sin 2pi\/3?\nLet us convert 2π\/3 into degrees. Then we get 2π\/3 = 120°. Using unit circle, sin 120° = √3\/2. Thus, sin 2pi\/3 = √3\/2.","meta_canonical":null}

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Boilerpipe Text	Before going to find the value of sin of 2pi, let us recollect the values of sine function of different standard angles from the trigonometric table. sin 0 = 0, sin π/6 = 1/2, sin π/4 = √2/2, sin π/3 = √3/2, and sin π/2 = 1. Of course, this table does not include the value of sin 2pi in it. We will find that sin of 2pi is 0 using various methods here. Also, we will solve some example problems using this. 1. What is Sin of 2pi? 2. Sin of 2pi Using Double Angle Formula 3. Sin of 2pi Using Reference Angles 4. Sin of 2pi Using Unit Circle 5. FAQs on Sin of 2pi What is Sin of 2pi? The value of sin of 2pi is 0. i.e., sin 2 π = 0. From trigonometric table , we know the trigonometric ratios of standard angles 0, π/6, π/4, π/3, and π/2. So this table doesn't give us the value of sin of 2pi. Usually, to find the value of any trigonometric ratio of a non-standard angle, we use the reference angles and the quadrant in which the angle lies in. We can do the same to find sin of 2pi also. The value of sin of 2pi can be easily found by using several other methods like Using double angle formula Using reference angle Using unit circle We will prove that sin 2π = 0 in each of these methods. Sin of 2pi Using Double Angle Formula We can find the value of sin of 2pi using the double angle formula of sine which is sin 2x = 2 sin x cos x. Since we have to find the value of sin(2π), we have to substitute x = π in the above formula. Then we get: sin 2π = 2 sin π cos π ... (1) Since π is also a non-standard angle, we find the values of sin π and cos π using the sum and difference formulas . Then we get sin π = sin (π/2 + π/2) = sin π/2 cos π/2 + cos π/2 sin π/2 = (1)(0) + (0)(1) = 0 cos π = cos (π/2 + π/2) = cos π/2 cos π/2 - sin π/2 sin π/2 = (0)(0) - (1)(1) = -1 Substitute these values in (1), sin 2π = 2 (0) (-1) = 0 Hence, sin of 2pi = 0. Sin of 2pi Using Reference Angles If we convert 2π into degrees, we get 360°. Since 360° lies in the interval [0°, 360°], its coterminal angle itself is the reference angle. To find its coterminal angle , we subtract 360° from it. Then we get 360° - 360° = 0°. So the coterminal angle of 360° is 0°. Also, we know that 360° means one full rotation and thus it comes either in the first quadrant or in the fourth quadrant. Let us consider both cases. First Quadrant: We know that in the first quadrant, sin is positive. Then sin 360° = + sin 0° = 0 (because sin 0° = 0) Fourth Quadrant: We know that in the fourth quadrant, sin is negative. Then sin 360° = - sin 0° = 0 (because sin 0° = 0) From both the cases, sin 360° = sin 2π = 0. Hence, sin 2π = 0. Sin of 2pi Using Unit Circle Before finding the value of sin of 2pi using unit circle , let us recollect a few points about the unit circle. A unit circle is a circle of radius centered at the origin. Every point on the unit circle corresponds to an angle. This angle is made by the line joining the origin and the point with the positive direction of the x-axis in the anti-clockwise direction. If P(x, y) corresponds to some angle θ, then x = cos θ and y = sin θ. i.e., the y-coordinate of the point represents the sine of the corresponding angle. As 2π (which is nothing but 360°) represents one full rotation, it is nothing but the angle made by the x-axis with itself and thus, it is equivalent to 0° on the unit circle. We also know that 0° corresponds to the point (1, 0) on the unit circle (as it is a point on the unit circle present on the x-axis). Thus, sin 2π = sin 0° = y-coordinate of (1, 0) = 0. Hence, sin 2π = 0. You can visualize it from the following figure. Important Notes: Here are some important notes that are related to sin of 2pi. sin 2π = 0 cos 2π = 1 tan 2π = 0 sin of any multiple of π is 0. i.e., sin nπ = 0, for any integer 'n'. For example, sin(-2π) = 0, sin π = 0, Related Topics: Here are some topics are related to sin of 2pi. Trigonometric Ratios Trigonometric Formulas Sine Law Cosine Law Examples Using Sin of 2pi Example 1: Find the value of cos of 2pi using the fact that sin of 2pi = 0. Solution: Using a trigonometric identity , sin 2 x + cos 2 x = 1 Substitute x = 2π here, sin 2 2π + cos 2 2π = 1 We know that sin 2π = 0. Substitute it here, 0 2 + cos 2 2π = 1 cos 2 2π = 1 cos 2π = ± 1 But 2π is in first or fourth quadrant and in each of the cases, cos is positive. Thus, cos 2π = 1. Answer: cos 2π = 1. Example 2: Find the value of sin of 4pi using the fact that sin of 2pi = 0. Solution: Using the double angle formula of sin, sin 2x = 2 sin x cos x Substitute x = 2π here, sin 2(2π) = 2 sin 2π cos 2π sin 4π = 2 (0) cos 2π (because sin 2π = 0) sin 4π = 0 Answer: sin 4π = 0. Example 3: Find the value of sin 5π/2 using the sum and difference formulas. Solution: Using the sum and difference formulas, sin (A + B) = sin A cos B + cos A sin B Substitute A = 2π and B = π/2 here, sin (2π + π/2) = sin 2π cos π/2 + cos 2π sin π/2 We know that sin of 2pi = 0 and cos of 2pi = 1. Also, sin π/2 = 1 and cos π/2 = 0. Thus, sin 5π/2 = (0)(0) + (1) (1) = 1. Answer: sin 5π/2 = 1. go to slide go to slide go to slide Great learning in high school using simple cues Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes. Book a Free Trial Class Practice Questions on Sin of 2pi go to slide go to slide FAQs on Sin of 2pi What Is Sin of 2pi? The value of sin of 2 pi is 0. This is because 2π is nothing but 0 degrees on the unit circle. Thus, sin 2π = sin 0 = 0. What Is the Value of Square of Sin of 2pi? We have sin(2π) = 0. From this, sin 2 (2π) = (sin 2π) 2 = 0 2 = 0. Thus, the value of square of sin of 2pi is 0. What is Sin^2(pi/2)? We know that sin π/2 = 1. So sin 2 π/2 = (sin π/2) 2 = 1 2 = 1. Is Sin of 2pi Undefined? No, sin(2π) is NOT undefined, instead sin(2π) = 0. This is because the coterminal angle of 2π is 0, from this we can conclude the value of sin of 2pi as, sin 2π = sin 0 = 0. Why Does Sin of 2pi Equal To Zero? Since 2π is an angle that is in the interval [0, 2π], its reference angle is nothing but its coterminal angle, which is 0. Thus, sin of 2pi can be given as, sin 2π = sin 0 = 0. How To Prove that Sin of 2pi is 0? On the unit circle, 2π is one complete rotation and hence 2π is equivalent to the angle 0 radians . We know that 0 radians on the unit circle correspond to the point (1, 0). Thus, sin 2π = sin 0 = y-coordinate of (1, 0) = 0. Hence, sin 2π = 0. What is the Value of cos(-2pi)? We know that cos (-x) = cos x. So cos (-2π) = cos 2π. On unit circle, cos 2π is same as cos 0 and hence it is 1. Thus, cos(-2pi) = 1. How To Find the Value of Sin of 2pi Using Double Angle Formula? We have sin 2x = 2 sin x cos x. Substitute x = π in the above formula. Then for sin of 2pi, we get: sin 2π = 2 sin π cos π ... (1). Now, sin π = sin (π/2 + π/2) = sin π/2 cos π/2 + cos π/2 sin π/2 = (1)(0) + (0)(1) = 0 cos π = cos (π/2 + π/2) = cos π/2 cos π/2 - sin π/2 sin π/2 = (0)(0) - (1)(1) = -1 Substitute these values in (1), sin 2π = 2 (0) (-1) = 0 Hence, sin 2π = 0. What Is Tan of 2pi Using Sin of 2pi? We know that sin of 2pi is equal to zero, i.e., sin(2π) = 0. Also we know that tan x = (sin x) / (cos x) for any x. Thus, tan(2π) = [sin(2π)] / [cos(2π)] = (0) / [cos(2π)] = 0. Hence, tan(2π) = 0. What is the Value of Sin 2pi/3? Let us convert 2π/3 into degrees. Then we get 2π/3 = 120°. Using unit circle, sin 120° = √3/2. Thus, sin 2pi/3 = √3/2.
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Of course, this table does not include the value of sin 2pi in it. We will find that sin of 2pi is 0 using various methods here. Also, we will solve some example problems using this. \| \| \| \|---\|---\| \| 1\. \| [What is Sin of 2pi?](https://www.cuemath.com/trigonometry/sin-of-2-pi/#what-is-sin-of-2pi) \| \| 2\. \| [Sin of 2pi Using Double Angle Formula](https://www.cuemath.com/trigonometry/sin-of-2-pi/#sin-of-2pi-using-double-angle-formula) \| \| 3\. \| [Sin of 2pi Using Reference Angles](https://www.cuemath.com/trigonometry/sin-of-2-pi/#sin-of-2pi-using-reference-angles) \| \| 4\. \| [Sin of 2pi Using Unit Circle](https://www.cuemath.com/trigonometry/sin-of-2-pi/#sin-of-2pi-using-unit-circle) \| \| 5\. \| [FAQs on Sin of 2pi](https://www.cuemath.com/trigonometry/sin-of-2-pi/#faqs) \| ## What is Sin of 2pi? The value of [sin](https://www.cuemath.com/trigonometry/sine-function/) of 2pi is 0. i.e., sin 2[π](https://www.cuemath.com/geometry/what-is-pi/) = 0. From [trigonometric table](https://www.cuemath.com/trigonometry/trigonometric-table/), we know the trigonometric ratios of standard angles 0, π/6, π/4, π/3, and π/2. So this table doesn't give us the value of sin of 2pi. Usually, to find the value of any trigonometric ratio of a non-standard angle, we use the [reference angles](https://www.cuemath.com/geometry/reference-angle/) and the quadrant in which the angle lies in. We can do the same to find sin of 2pi also. The value of sin of 2pi can be easily found by using several other methods like - Using double angle formula - Using reference angle - Using unit circle We will prove that sin 2π = 0 in each of these methods. ![Sin of 2pi is 0.](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/sin-of-2pi-1628225803.png) ## Sin of 2pi Using Double Angle Formula We can find the value of sin of 2pi using the [double angle formula](https://www.cuemath.com/double-angle-formulas/) of sine which is sin 2x = 2 sin x cos x. Since we have to find the value of sin(2π), we have to substitute x = π in the above formula. Then we get: sin 2π = 2 sin π cos π ... (1) Since π is also a non-standard angle, we find the values of sin π and cos π using the [sum and difference formulas](https://www.cuemath.com/trigonometry/sum-and-difference-identities/). Then we get sin π = sin (π/2 + π/2) = sin π/2 cos π/2 + cos π/2 sin π/2 = (1)(0) + (0)(1) = 0 cos π = cos (π/2 + π/2) = cos π/2 cos π/2 - sin π/2 sin π/2 = (0)(0) - (1)(1) = -1 Substitute these values in (1), sin 2π = 2 (0) (-1) = 0 Hence, sin of 2pi = 0. ## Sin of 2pi Using Reference Angles If we [convert](https://www.cuemath.com/geometry/radians-to-degrees/) 2π into degrees, we get 360°. Since 360° lies in the interval \[0°, 360°\], its coterminal angle itself is the reference angle. To find its [coterminal angle](https://www.cuemath.com/geometry/coterminal-angles/), we subtract 360° from it. Then we get 360° - 360° = 0°. So the coterminal angle of 360° is 0°. Also, we know that 360° means one full rotation and thus it comes either in the first quadrant or in the fourth quadrant. Let us consider both cases. - First Quadrant: We know that in the first quadrant, sin is positive. Then sin 360° = + sin 0° = 0 (because sin 0° = 0) - Fourth Quadrant: We know that in the fourth quadrant, sin is negative. Then sin 360° = - sin 0° = 0 (because sin 0° = 0) From both the cases, sin 360° = sin 2π = 0. Hence, sin 2π = 0. ## Sin of 2pi Using Unit Circle Before finding the value of sin of 2pi using [unit circle](https://www.cuemath.com/geometry/unit-circle/), let us recollect a few points about the unit circle. - A unit circle is a [circle](https://www.cuemath.com/geometry/circles/) of radius centered at the origin. - Every point on the unit circle corresponds to an angle. - This angle is made by the line joining the origin and the point with the positive direction of the x-axis in the anti-clockwise direction. - If P(x, y) corresponds to some angle θ, then x = cos θ and y = sin θ. i.e., the y-coordinate of the point represents the sine of the corresponding angle. As 2π (which is nothing but 360°) represents one full rotation, it is nothing but the angle made by the x-axis with itself and thus, it is equivalent to 0° on the unit circle. We also know that 0° corresponds to the point (1, 0) on the unit circle (as it is a point on the unit circle present on the x-axis). Thus, sin 2π = sin 0° = y-coordinate of (1, 0) = 0. Hence, sin 2π = 0. You can visualize it from the following figure. ![Sin of 2 pi Using Unit Circle. Sin of 2 pi is equal to the y-coordinate of the point (1, 0). Hence sin of 2 pi is 0.](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/sin-of-2pi-using-unit-circle-1630134016.png) Important Notes: Here are some important notes that are related to sin of 2pi. - sin 2π = 0 - cos 2π = 1 - tan 2π = 0 - sin of any multiple of π is 0. i.e., sin nπ = 0, for any [integer](https://www.cuemath.com/numbers/integers/) 'n'. For example, sin(-2π) = 0, sin π = 0, Related Topics: Here are some topics are related to sin of 2pi. - [Trigonometric Ratios](https://www.cuemath.com/trigonometry/trigonometric-ratios/) - [Trigonometric Formulas](https://www.cuemath.com/trigonometry/trigonometry-formula/) - [Sine Law](https://www.cuemath.com/trigonometry/sine-law/) - [Cosine Law](https://www.cuemath.com/trigonometry/law-of-cosine/) ## Examples Using Sin of 2pi 1. Example 1: Find the value of cos of 2pi using the fact that sin of 2pi = 0. Solution: Using a [trigonometric identity](https://www.cuemath.com/trigonometry/trigonometric-identities/), sin2x + cos2x = 1 Substitute x = 2π here, sin22π + cos22π = 1 We know that sin 2π = 0. Substitute it here, 02 + cos22π = 1 cos22π = 1 cos 2π = ± 1 But 2π is in first or fourth quadrant and in each of the cases, cos is positive. Thus, cos 2π = 1. Answer: cos 2π = 1. 2. Example 2: Find the value of sin of 4pi using the fact that sin of 2pi = 0. Solution: Using the double angle formula of sin, sin 2x = 2 sin x cos x Substitute x = 2π here, sin 2(2π) = 2 sin 2π cos 2π sin 4π = 2 (0) cos 2π (because sin 2π = 0) sin 4π = 0 Answer: sin 4π = 0. 3. Example 3: Find the value of sin 5π/2 using the sum and difference formulas. Solution: Using the sum and difference formulas, sin (A + B) = sin A cos B + cos A sin B Substitute A = 2π and B = π/2 here, sin (2π + π/2) = sin 2π cos π/2 + cos 2π sin π/2 We know that sin of 2pi = 0 and cos of 2pi = 1. Also, sin π/2 = 1 and cos π/2 = 0. Thus, sin 5π/2 = (0)(0) + (1) (1) = 1. Answer: sin 5π/2 = 1. View Answer \> [go to slide](https://www.cuemath.com/trigonometry/sin-of-2-pi/#slide-430)[go to slide](https://www.cuemath.com/trigonometry/sin-of-2-pi/#slide-699)[go to slide](https://www.cuemath.com/trigonometry/sin-of-2-pi/#slide-750) ![](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/add-laptop-mexican-teacher-with-mexican-student-grade8-10-1619247480.png) Great learning in high school using simple cues Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes. [Book a Free Trial Class](https://www.cuemath.com/parent/signup/) ## Practice Questions on Sin of 2pi Check Answer \> [go to slide](https://www.cuemath.com/trigonometry/sin-of-2-pi/#slide-282)[go to slide](https://www.cuemath.com/trigonometry/sin-of-2-pi/#slide-339) ## FAQs on Sin of 2pi ### What Is Sin of 2pi? The value of sin of 2 pi is 0. This is because 2π is nothing but 0 degrees on the unit circle. Thus, sin 2π = sin 0 = 0. ### What Is the Value of Square of Sin of 2pi? We have sin(2π) = 0. From this, sin2(2π) = (sin 2π)2 = 02 = 0. Thus, the value of square of sin of 2pi is 0. ### What is Sin^2(pi/2)? We know that sin π/2 = 1. So sin2π/2 = (sin π/2)2 = 12 = 1. ### Is Sin of 2pi Undefined? No, sin(2π) is NOT undefined, instead sin(2π) = 0. This is because the coterminal angle of 2π is 0, from this we can conclude the value of sin of 2pi as, sin 2π = sin 0 = 0. ### Why Does Sin of 2pi Equal To Zero? Since 2π is an angle that is in the interval \[0, 2π\], its reference angle is nothing but its coterminal angle, which is 0. Thus, sin of 2pi can be given as, sin 2π = sin 0 = 0. ### How To Prove that Sin of 2pi is 0? On the unit circle, 2π is one complete rotation and hence 2π is equivalent to the angle 0 [radians](https://www.cuemath.com/trigonometry/what-is-a-radian/). We know that 0 radians on the unit circle correspond to the point (1, 0). Thus, sin 2π = sin 0 = y-coordinate of (1, 0) = 0. Hence, sin 2π = 0. ### What is the Value of cos(-2pi)? We know that cos (-x) = cos x. So cos (-2π) = cos 2π. On unit circle, cos 2π is same as cos 0 and hence it is 1. Thus, cos(-2pi) = 1. ### How To Find the Value of Sin of 2pi Using Double Angle Formula? We have sin 2x = 2 sin x cos x. Substitute x = π in the above formula. Then for sin of 2pi, we get: sin 2π = 2 sin π cos π ... (1). Now, - sin π = sin (π/2 + π/2) = sin π/2 cos π/2 + cos π/2 sin π/2 = (1)(0) + (0)(1) = 0 - cos π = cos (π/2 + π/2) = cos π/2 cos π/2 - sin π/2 sin π/2 = (0)(0) - (1)(1) = -1 Substitute these values in (1), sin 2π = 2 (0) (-1) = 0 Hence, sin 2π = 0. ### What Is Tan of 2pi Using Sin of 2pi? We know that sin of 2pi is equal to zero, i.e., sin(2π) = 0. Also we know that [tan](https://www.cuemath.com/trigonometry/tangent-function/) x = (sin x) / (cos x) for any x. Thus, tan(2π) = \[sin(2π)\] / \[cos(2π)\] = (0) / \[cos(2π)\] = 0. Hence, tan(2π) = 0. ### What is the Value of Sin 2pi/3? Let us convert 2π/3 into degrees. Then we get 2π/3 = 120°. Using unit circle, sin 120° = √3/2. Thus, sin 2pi/3 = √3/2. Explore math program Download FREE Study Materials Trigonometry Worksheet Explore math program Math worksheets and visual curriculum Book a FREE Class Become MathFit™: Boost math skills with daily fun challenges and puzzles. Download the app STRATEGY GAMES LOGIC PUZZLES MENTAL MATH Become MathFit™: Boost math skills with daily fun challenges and puzzles. 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Readable Markdown	Before going to find the value of sin of 2pi, let us recollect the values of sine function of different standard angles from the trigonometric table. sin 0 = 0, sin π/6 = 1/2, sin π/4 = √2/2, sin π/3 = √3/2, and sin π/2 = 1. Of course, this table does not include the value of sin 2pi in it. We will find that sin of 2pi is 0 using various methods here. Also, we will solve some example problems using this. \| \| \| \|---\|---\| \| 1\. \| [What is Sin of 2pi?](https://www.cuemath.com/trigonometry/sin-of-2-pi/#what-is-sin-of-2pi) \| \| 2\. \| [Sin of 2pi Using Double Angle Formula](https://www.cuemath.com/trigonometry/sin-of-2-pi/#sin-of-2pi-using-double-angle-formula) \| \| 3\. \| [Sin of 2pi Using Reference Angles](https://www.cuemath.com/trigonometry/sin-of-2-pi/#sin-of-2pi-using-reference-angles) \| \| 4\. \| [Sin of 2pi Using Unit Circle](https://www.cuemath.com/trigonometry/sin-of-2-pi/#sin-of-2pi-using-unit-circle) \| \| 5\. \| [FAQs on Sin of 2pi](https://www.cuemath.com/trigonometry/sin-of-2-pi/#faqs) \| What is Sin of 2pi? The value of [sin](https://www.cuemath.com/trigonometry/sine-function/) of 2pi is 0. i.e., sin 2[π](https://www.cuemath.com/geometry/what-is-pi/) = 0. From [trigonometric table](https://www.cuemath.com/trigonometry/trigonometric-table/), we know the trigonometric ratios of standard angles 0, π/6, π/4, π/3, and π/2. So this table doesn't give us the value of sin of 2pi. Usually, to find the value of any trigonometric ratio of a non-standard angle, we use the [reference angles](https://www.cuemath.com/geometry/reference-angle/) and the quadrant in which the angle lies in. We can do the same to find sin of 2pi also. The value of sin of 2pi can be easily found by using several other methods like - Using double angle formula - Using reference angle - Using unit circle We will prove that sin 2π = 0 in each of these methods. ![Sin of 2pi is 0.](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/sin-of-2pi-1628225803.png) Sin of 2pi Using Double Angle Formula We can find the value of sin of 2pi using the [double angle formula](https://www.cuemath.com/double-angle-formulas/) of sine which is sin 2x = 2 sin x cos x. Since we have to find the value of sin(2π), we have to substitute x = π in the above formula. Then we get: sin 2π = 2 sin π cos π ... (1) Since π is also a non-standard angle, we find the values of sin π and cos π using the [sum and difference formulas](https://www.cuemath.com/trigonometry/sum-and-difference-identities/). Then we get sin π = sin (π/2 + π/2) = sin π/2 cos π/2 + cos π/2 sin π/2 = (1)(0) + (0)(1) = 0 cos π = cos (π/2 + π/2) = cos π/2 cos π/2 - sin π/2 sin π/2 = (0)(0) - (1)(1) = -1 Substitute these values in (1), sin 2π = 2 (0) (-1) = 0 Hence, sin of 2pi = 0. Sin of 2pi Using Reference Angles If we [convert](https://www.cuemath.com/geometry/radians-to-degrees/) 2π into degrees, we get 360°. Since 360° lies in the interval \[0°, 360°\], its coterminal angle itself is the reference angle. To find its [coterminal angle](https://www.cuemath.com/geometry/coterminal-angles/), we subtract 360° from it. Then we get 360° - 360° = 0°. So the coterminal angle of 360° is 0°. Also, we know that 360° means one full rotation and thus it comes either in the first quadrant or in the fourth quadrant. Let us consider both cases. - First Quadrant: We know that in the first quadrant, sin is positive. Then sin 360° = + sin 0° = 0 (because sin 0° = 0) - Fourth Quadrant: We know that in the fourth quadrant, sin is negative. Then sin 360° = - sin 0° = 0 (because sin 0° = 0) From both the cases, sin 360° = sin 2π = 0. Hence, sin 2π = 0. Sin of 2pi Using Unit Circle Before finding the value of sin of 2pi using [unit circle](https://www.cuemath.com/geometry/unit-circle/), let us recollect a few points about the unit circle. - A unit circle is a [circle](https://www.cuemath.com/geometry/circles/) of radius centered at the origin. - Every point on the unit circle corresponds to an angle. - This angle is made by the line joining the origin and the point with the positive direction of the x-axis in the anti-clockwise direction. - If P(x, y) corresponds to some angle θ, then x = cos θ and y = sin θ. i.e., the y-coordinate of the point represents the sine of the corresponding angle. As 2π (which is nothing but 360°) represents one full rotation, it is nothing but the angle made by the x-axis with itself and thus, it is equivalent to 0° on the unit circle. We also know that 0° corresponds to the point (1, 0) on the unit circle (as it is a point on the unit circle present on the x-axis). Thus, sin 2π = sin 0° = y-coordinate of (1, 0) = 0. Hence, sin 2π = 0. You can visualize it from the following figure. ![Sin of 2 pi Using Unit Circle. Sin of 2 pi is equal to the y-coordinate of the point (1, 0). Hence sin of 2 pi is 0.](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/sin-of-2pi-using-unit-circle-1630134016.png) Important Notes: Here are some important notes that are related to sin of 2pi. - sin 2π = 0 - cos 2π = 1 - tan 2π = 0 - sin of any multiple of π is 0. i.e., sin nπ = 0, for any [integer](https://www.cuemath.com/numbers/integers/) 'n'. For example, sin(-2π) = 0, sin π = 0, Related Topics: Here are some topics are related to sin of 2pi. - [Trigonometric Ratios](https://www.cuemath.com/trigonometry/trigonometric-ratios/) - [Trigonometric Formulas](https://www.cuemath.com/trigonometry/trigonometry-formula/) - [Sine Law](https://www.cuemath.com/trigonometry/sine-law/) - [Cosine Law](https://www.cuemath.com/trigonometry/law-of-cosine/) ## Examples Using Sin of 2pi 1. Example 1: Find the value of cos of 2pi using the fact that sin of 2pi = 0. Solution: Using a [trigonometric identity](https://www.cuemath.com/trigonometry/trigonometric-identities/), sin2x + cos2x = 1 Substitute x = 2π here, sin22π + cos22π = 1 We know that sin 2π = 0. Substitute it here, 02 + cos22π = 1 cos22π = 1 cos 2π = ± 1 But 2π is in first or fourth quadrant and in each of the cases, cos is positive. Thus, cos 2π = 1. Answer: cos 2π = 1. 2. Example 2: Find the value of sin of 4pi using the fact that sin of 2pi = 0. Solution: Using the double angle formula of sin, sin 2x = 2 sin x cos x Substitute x = 2π here, sin 2(2π) = 2 sin 2π cos 2π sin 4π = 2 (0) cos 2π (because sin 2π = 0) sin 4π = 0 Answer: sin 4π = 0. 3. Example 3: Find the value of sin 5π/2 using the sum and difference formulas. Solution: Using the sum and difference formulas, sin (A + B) = sin A cos B + cos A sin B Substitute A = 2π and B = π/2 here, sin (2π + π/2) = sin 2π cos π/2 + cos 2π sin π/2 We know that sin of 2pi = 0 and cos of 2pi = 1. Also, sin π/2 = 1 and cos π/2 = 0. Thus, sin 5π/2 = (0)(0) + (1) (1) = 1. Answer: sin 5π/2 = 1. [go to slide](https://www.cuemath.com/trigonometry/sin-of-2-pi/#slide-430)[go to slide](https://www.cuemath.com/trigonometry/sin-of-2-pi/#slide-699)[go to slide](https://www.cuemath.com/trigonometry/sin-of-2-pi/#slide-750) ![](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/add-laptop-mexican-teacher-with-mexican-student-grade8-10-1619247480.png) Great learning in high school using simple cues Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes. [Book a Free Trial Class](https://www.cuemath.com/parent/signup/) ## Practice Questions on Sin of 2pi [go to slide](https://www.cuemath.com/trigonometry/sin-of-2-pi/#slide-282)[go to slide](https://www.cuemath.com/trigonometry/sin-of-2-pi/#slide-339) FAQs on Sin of 2pi ### What Is Sin of 2pi? The value of sin of 2 pi is 0. This is because 2π is nothing but 0 degrees on the unit circle. Thus, sin 2π = sin 0 = 0. ### What Is the Value of Square of Sin of 2pi? We have sin(2π) = 0. From this, sin2(2π) = (sin 2π)2 = 02 = 0. Thus, the value of square of sin of 2pi is 0. ### What is Sin^2(pi/2)? We know that sin π/2 = 1. So sin2π/2 = (sin π/2)2 = 12 = 1. ### Is Sin of 2pi Undefined? No, sin(2π) is NOT undefined, instead sin(2π) = 0. This is because the coterminal angle of 2π is 0, from this we can conclude the value of sin of 2pi as, sin 2π = sin 0 = 0. ### Why Does Sin of 2pi Equal To Zero? Since 2π is an angle that is in the interval \[0, 2π\], its reference angle is nothing but its coterminal angle, which is 0. Thus, sin of 2pi can be given as, sin 2π = sin 0 = 0. ### How To Prove that Sin of 2pi is 0? On the unit circle, 2π is one complete rotation and hence 2π is equivalent to the angle 0 [radians](https://www.cuemath.com/trigonometry/what-is-a-radian/). We know that 0 radians on the unit circle correspond to the point (1, 0). Thus, sin 2π = sin 0 = y-coordinate of (1, 0) = 0. Hence, sin 2π = 0. ### What is the Value of cos(-2pi)? We know that cos (-x) = cos x. So cos (-2π) = cos 2π. On unit circle, cos 2π is same as cos 0 and hence it is 1. Thus, cos(-2pi) = 1. ### How To Find the Value of Sin of 2pi Using Double Angle Formula? We have sin 2x = 2 sin x cos x. Substitute x = π in the above formula. Then for sin of 2pi, we get: sin 2π = 2 sin π cos π ... (1). Now, - sin π = sin (π/2 + π/2) = sin π/2 cos π/2 + cos π/2 sin π/2 = (1)(0) + (0)(1) = 0 - cos π = cos (π/2 + π/2) = cos π/2 cos π/2 - sin π/2 sin π/2 = (0)(0) - (1)(1) = -1 Substitute these values in (1), sin 2π = 2 (0) (-1) = 0 Hence, sin 2π = 0. ### What Is Tan of 2pi Using Sin of 2pi? We know that sin of 2pi is equal to zero, i.e., sin(2π) = 0. Also we know that [tan](https://www.cuemath.com/trigonometry/tangent-function/) x = (sin x) / (cos x) for any x. Thus, tan(2π) = \[sin(2π)\] / \[cos(2π)\] = (0) / \[cos(2π)\] = 0. Hence, tan(2π) = 0. ### What is the Value of Sin 2pi/3? Let us convert 2π/3 into degrees. Then we get 2π/3 = 120°. Using unit circle, sin 120° = √3/2. Thus, sin 2pi/3 = √3/2.
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