{{Short description|Fundamental trigonometric functions}}
{{redirect2|Sine|Cosine|other uses|Sine (disambiguation)|and|Cosine (disambiguation)|text="Sine" is not to be confused with [[Sign]], [[Sign (mathematics)]] or the [[sign function]].}}
{{Infobox mathematical function
| name = Sine and cosine
| image = Sine_cosine_one_period.svg
| general_definition = <math>
\begin{align}
&\sin(\theta) = \frac {\textrm{opposite}} {\textrm{hypotenuse}} \\[8pt]
&\cos(\theta) = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} \\[8pt]
\end{align}</math>
| fields_of_application= [[Trigonometry]], [[Fourier series]], [[Mathematical analysis]].
| domain = real number <math> \mathbb{R} </math>
| range = <math> [-1,1] </math>
}}

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 <math>\theta</math>, the sine and cosine functions are denoted as <math>\sin(\theta)</math> and <math>\cos(\theta)</math>.

The definitions of sine and cosine have been extended to any [[real number|real]] value in terms of the lengths of certain line segments in a [[unit circle]]. More modern definitions express the sine and cosine as [[Series (mathematics)|infinite series]], or as the solutions of certain [[differential equation]]s, allowing their extension to arbitrary positive and negative values and even to [[complex number]]s.

The sine and cosine functions are commonly used to model [[periodic function|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, koti-jyā and utkrama-jyā|{{tlit|sa|jyā}} and {{tlit|sa|koṭi-jyā}}]] functions used in [[Indian astronomy]] during the [[Gupta period]].

== Elementary descriptions ==
=== Right-angled triangle definition ===
[[File:Trigono sine en2.svg|right|thumb|upright=1|For the angle {{math|1=''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 <math> \alpha </math>, start with a [[right triangle]] that contains an angle of measure <math> \alpha </math>; in the accompanying figure, angle <math> \alpha </math> in a right triangle <math> ABC </math> is the angle of interest. The three sides of the triangle are named as follows:{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27 27]}}
* The ''opposite side'' is the side opposite to the angle of interest; in this case, it is <math> a </math>.
* The ''hypotenuse'' is the side opposite the right angle; in this case, it is <math> h </math>. The hypotenuse is always the longest side of a right-angled triangle.
* The ''adjacent side'' is the remaining side; in this case, it is <math> b </math>. 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:{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27 27]}}
<math display="block">
 \sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}, \qquad
 \cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}.
</math>

The other trigonometric functions of the angle can be defined similarly; for example, the [[Trigonometric functions#Right-angled triangle definitions|tangent]] is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and cosine functions. The [[multiplicative inverse|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:{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27 27]}}
<math display="block"> \begin{align}
 \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{align} </math>

=== Special angle measures ===
As stated, the values <math>\sin(\alpha)</math> and <math>\cos(\alpha)</math> appear to depend on the choice of a right triangle containing an angle of measure <math> \alpha </math>. However, this is not the case as all such triangles are [[similarity (geometry)|similar]], and so the ratios are the same for each of them. For example, each [[Catheti|leg]] of the 45-45-90 right triangle is 1 unit, and its hypotenuse is <math> \sqrt{2} </math>; therefore, <math display="inline"> \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2} </math>.{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA36 36]}} The following table shows the special value of each input for both sine and cosine with the domain between <math display="inline"> 0 \le x \le \frac{\pi}{2} </math>. 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.{{sfnp|Varberg|Purcell|Rigdon|2007|p=42}}{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA37 37], [https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA78 78]}}
{| class="wikitable" style="text-align:center;"
|-----
! colspan="4" style="background:#ffdead;" | Angle, {{Mvar|x}}
! colspan="2" style="background:#ffdead;" | {{Math|sin(''x'')}}
! colspan="2" style="background:#ffdead;" | {{Math|cos(''x'')}}
|-----
! style="background:#efefef;" | [[Degree (angle)|Degree]]s
! style="background:#efefef;" | [[Radian]]s
! style="background:#efefef;" | [[Gradian]]s
! style="background:#efefef;" | [[Turn (geometry)|Turn]]s
! style="background:#efefef;" | Exact
! style="background:#efefef;" | Decimal
! style="background:#efefef;" | Exact
! style="background:#efefef;" | Decimal
|-----
| 0°
| 0
| <math>0^g</math>
| 0
| 0
| 0
| 1
| 1
|-----
| 30°
| <math>\frac {1}{6} \pi</math>
| <math>33\frac{1}{3}^g</math>
| <math>\frac {1}{12}</math>
| <math>\frac {1}{2}</math>
| 0.5
| <math>\frac{\sqrt{3}}{2}</math>
| ≈ 0.866
|-----
| 45°
| <math>\frac{1}{4} \pi</math>
| <math>50^g</math>
| <math>\frac {1}{8}</math>
| <math>\frac{\sqrt{2}}{2}</math>
| ≈ 0.707
| <math>\frac{\sqrt{2}}{2}</math>
| ≈ 0.707
|-----
| 60°
| <math>\frac {1}{3} \pi</math>
| <math>66 \frac {2}{3}^g</math>
| <math>\frac {1}{6}</math>
| <math>\frac{\sqrt{3}}{2}</math>
| ≈ 0.866
| <math>\frac {1}{2}</math>
| 0.5
|-----
| 90°
| <math>\frac {1}{2}\pi</math>
| <math>100^g</math>
| <math>\frac {1}{4}</math>
| 1
| 1
| 0
| 0
|}

=== Laws ===
{{Main|Law of sines|Law of cosines}}
[[File:Law of sines (simple).svg|thumb|upright=1|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.{{sfnp|Axler|2012|p=[https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA634 634]}} Given a triangle <math> ABC </math> with sides <math> a </math>, <math> b </math>, and <math> c</math>, and angles opposite those sides <math> \alpha </math>, <math> \beta </math>, and <math> \gamma </math>, the law states,
<math display="block"> \frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c}.</math>
This is equivalent to the equality of the first three expressions below:
<math display="block">\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} = 2R,</math>
where <math> R </math> is the triangle's [[circumcircle|circumradius]].

The [[law of cosines]] is useful for computing the length of an unknown side if two other sides and an angle are known.{{sfnp|Axler|2012|p=[https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA634 634]}} The law states,
<math display="block"> a^2 + b^2 - 2ab\cos(\gamma) = c^2</math>
In the case where <math>\gamma = \pi/2</math> from which <math>\cos(\gamma) = 0</math>, the resulting equation becomes the [[Pythagorean theorem]].{{sfnp|Axler|2012|p=[https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA632 632]}}

=== Vector definition ===
The [[cross product]] and [[dot product]] are operations on two [[Vector (mathematics and physics)|vectors]] in [[Euclidean vector space]]. The sine and cosine functions can be defined in terms of the cross product and dot product. If <math> \mathbf{a} </math> and <math> \mathbf{b} </math> are vectors, and <math>\theta</math> is the angle between <math> \mathbf{a} </math> and <math> \mathbf{b} </math>, then sine and cosine can be defined as:<ref>{{MathWorld|CrossProduct|Cross Product|access-date=5 June 2025}}</ref><ref>{{MathWorld|DotProduct|Dot Product|access-date=5 June 2025}}</ref>
<math display="block"> \begin{align}
 \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{align} </math>

== Analytic descriptions ==
=== Unit circle definition ===
The sine and cosine functions may also be defined in a more general way by using [[Unit circle#Trigonometric functions on the unit circle|unit circle]], a circle of radius one centered at the origin <math> (0,0) </math>, formulated as the equation of <math> x^2 + y^2 = 1 </math> in the [[Cartesian coordinate system]]. A ray from the origin making an angle of <math> \theta </math> with the positive half of the {{nowrap|1=<math> x </math>-}}axis intersects the unit circle at exactly one point. The {{nowrap|1=<math> x </math>-}} and {{nowrap|1=<math> y </math>-}}coordinates of this point of intersection are equal to <math> \cos (\theta) </math> and <math> \sin (\theta) </math>, respectively; that is,{{sfnp|Varberg|Purcell|Rigdon|2007|p=41}}
<math display="block"> \sin (\theta) = y, \qquad \cos (\theta) = x. </math>

This definition is consistent with the right-angled triangle definition of sine and cosine when <math display="inline"> 0 < \theta < \frac{\pi}{2} </math> 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 {{nowrap|1=<math> y </math>-}}coordinate. A similar argument can be made for the cosine function to show that the cosine of an angle when <math display="inline"> 0 < \theta < \frac{\pi}{2} </math>, even under the new definition using the unit circle.{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA68 68]}}{{sfnp|Varberg|Purcell|Rigdon|2007|p=47}}

==== Graph of a function and its elementary properties ====
[[File:Circle cos sin.gif|thumb|upright=2|Animation demonstrating how the sine function (in red) is graphed from the {{nowrap|1={{math|1=''y''}}-}}coordinate (red dot) of a point on the [[unit circle]] (in green), at an angle of {{math|1=''θ''}}. The cosine (in blue) is the {{nowrap|1={{math|1=''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 <math> \theta > 0 </math>. In a sine function, if the input is <math display="inline"> \theta = \frac{\pi}{2} </math>, the point is rotated counterclockwise and stopped exactly on the {{nowrap|1=<math> y </math>-}}axis. If <math> \theta = \pi </math>, the point is at the circle's halfway point. If <math> \theta = 2\pi </math>, the point returns to its origin. This results in both sine and cosine functions having the [[Range of a function|range]] between <math> -1 \le y \le 1 </math>.{{sfnp|Varberg|Purcell|Rigdon|2007|p=41&ndash;42}}

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 {{nowrap|1=<math> y </math>-}}coordinate. In other words, both sine and cosine functions are [[periodic function|periodic]], meaning any angle added by the circle's circumference is the angle itself. Mathematically,{{sfnp|Varberg|Purcell|Rigdon|2007|p=41, 43}}
<math display="block"> \begin{align}
 \sin(\theta + 2\pi) &= \sin(\theta), \\
 \cos(\theta + 2\pi) &= \cos (\theta).
\end{align}
</math>

A function <math> f </math> is said to be [[Odd function|odd]] if <math> f(-x) = -f(x) </math>, and is said to be [[Even function|even]] if <math> f(-x) = f(x) </math>. The sine function is odd, whereas the cosine function is even.{{sfnp|Young|2012|p=[https://books.google.com/books?id=OMrcN0a3LxIC&pg=RA1-PA165 165]}} Both sine and cosine functions are similar, with their difference being [[Translation (geometry)|shifted]] by <math display="inline"> \frac{\pi}{2} </math>. This phase shift can be expressed as 
<math display="block"> \begin{align}
 \sin(\theta) &= \cos\left(\theta - \frac{\pi}{2}\right), \\
 \cos(\theta) &= \sin\left(\theta + \frac{\pi}{2}\right).
\end{align}
</math>

This is distinct from the cofunction identities that follow below, which arise from right-triangle geometry and are not phase shifts: {{sfnp|Varberg|Purcell|Rigdon|2007|p=42, 47}}
<math display="block"> \begin{align}
 \sin(\theta) &= \cos\left(\frac{\pi}{2} - \theta \right), \\
 \cos(\theta) &= \sin\left(\frac{\pi}{2} - \theta \right).
\end{align}
</math>

[[File:Cosine_fixed_point.svg|thumb|The fixed point iteration {{math|1=''x''<sub>''n''+1</sub>&nbsp;=&nbsp;cos(''x<sub>n</sub>'')}} with initial value {{math|1=''x''<sub>0</sub>&nbsp;=&nbsp;−1}} converges to the Dottie number.]]
Zero is the only real [[Fixed point (mathematics)|fixed point]] of the sine function; in other words the only intersection of the sine function and the [[identity function]] is <math>\sin(0)=0</math>. 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 <math> \cos (x) = x</math>. The decimal expansion of the Dottie number is approximately 0.739085.<ref>{{Cite web|url=https://oeis.org/A003957|title=OEIS A003957|website=oeis.org|access-date=2019-05-26}}</ref>

==== Continuity and differentiation ====
{{main|Differentiation of trigonometric functions}}
[[File:Sine quads 01 Pengo.svg|thumb|390px|The quadrants of the unit circle and of {{math|sin(''x'')}}, using the [[Cartesian coordinate system]]]]
The sine and cosine functions are infinitely differentiable.{{sfnp|Bourchtein|Bourchtein|2022|p=[https://books.google.com/books?id=nGxOEAAAQBAJ&pg=PA294 294]}} The derivative of sine is cosine, and the derivative of cosine is negative sine:{{sfnp|Varberg|Purcell|Rigdon|2007|p=115}}
<math display="block"> \frac{d}{dx}\sin(x) = \cos(x), \qquad  \frac{d}{dx}\cos(x) = -\sin(x).</math>
Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself.{{sfnp|Bourchtein|Bourchtein|2022|p=[https://books.google.com/books?id=nGxOEAAAQBAJ&pg=PA294 294]}} These derivatives can be applied to the [[first derivative test]], according to which the [[Monotone function|monotonicity]] of a function can be defined as the inequality of function's first derivative greater or less than equal to zero.{{sfnp|Varberg|Purcell|Rigdon|2007|p=155}} It can also be applied to [[second derivative test]], according to which the [[Concave function|concavity]] of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero.{{sfnp|Varberg|Purcell|Rigdon|2007|p=157}} The following table shows that both sine and cosine functions have concavity and monotonicity&mdash;the positive sign (<math> + </math>) denotes a graph is increasing (going upward) and the negative sign (<math> - </math>) is decreasing (going downward)&mdash;in certain intervals.{{sfnp|Varberg|Purcell|Rigdon|2007|p=42}} This information can be represented as a Cartesian coordinates system divided into four quadrants.

{| class="wikitable" style="text-align:center;"
|-
! rowspan=2 | [[Cartesian coordinate system#Quadrants and octants|Quadrant]]
! colspan=2 | Angle
! colspan=3 | Sine
! colspan=3 | Cosine
|-
! [[Degree (angle)|Degrees]]
! [[Radian]]s
! [[Sign (mathematics)|Sign]]
! [[Monotonic function|Monotony]]
! [[Convex function|Convexity]]
! [[Sign (mathematics)|Sign]]
! [[Monotonic function|Monotony]]
! [[Convex function|Convexity]]
|-
|-
| style="text-align:left;" | 1st quadrant, I
|<math>0^\circ < x< 90^\circ</math>
|<math>0 < x < \frac{\pi}{2}</math>
|<math>+</math>
|Increasing
|Concave
|<math>+</math>
|Decreasing
|Concave
|-
| style="text-align:left;" | 2nd quadrant, II
|<math>90^\circ < x < 180^\circ</math>
|<math>\frac{\pi}{2} < x < \pi</math>
|<math>+</math>
|Decreasing
|Concave
|<math>-</math>
|Decreasing
|Convex
|-
| style="text-align:left;" | 3rd quadrant, III
|<math>180^\circ < x < 270^\circ</math>
|<math>\pi < x < \frac{3\pi}{2}</math>
|<math>-</math>
|Decreasing
|Convex
|<math>-</math>
|Increasing
|Convex
|-
| style="text-align:left;" | 4th quadrant, IV
|<math>270^\circ < x < 360^\circ</math>
|<math>\frac{3\pi}{2} < x < 2\pi</math>
|<math>-</math>
|Increasing
|Convex
|<math>+</math>
|Increasing
|Concave
|}

Both sine and cosine functions can be defined by using differential equations. The pair of <math>(\cos \theta, \sin \theta)</math> is the solution <math>(x(\theta), y(\theta))</math> to the two-dimensional system of [[differential equation]]s <math>y'(\theta) = x(\theta)</math> and <math>x'(\theta) = -y(\theta)</math> with the [[initial conditions]] <math>y(0) = 0</math> and <math>x(0) = 1</math>. 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 <math>y'(\theta) = x(\theta)</math> and <math>x'(\theta) = -y(\theta)</math> starting from the initial conditions <math>y(0) = 0</math> and <math>x(0) = 1</math>.{{citation needed|date=August 2024}}

==== Integral and the usage in mensuration ====
{{main|List of integrals of trigonometric functions}}
Their area under a curve can be obtained by using the [[integral]] with a certain bounded interval. Their antiderivatives are:
<math display="block"> \int \sin(x)\,dx = -\cos(x) + C \qquad \int \cos(x)\,dx = \sin(x) + C, </math>
where <math> C </math> denotes the [[constant of integration]].{{sfnp|Varberg|Purcell|Rigdon|2007|p=199}} 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 <math>0</math> and <math>t</math> is
<math display="block"> \int_0^t\!\sqrt{1+\cos^2(x)}\, dx =\sqrt{2} \operatorname{E} \left(t, \frac{1}{\sqrt{2}} \right),</math>
where <math>\operatorname{E}(\varphi,k)</math> is the [[Elliptic integral#Incomplete elliptic integral of the second kind|incomplete elliptic integral of the second kind]] with modulus <math>k</math>. It cannot be expressed using [[elementary function]]s.{{sfnp|Vince|2023|p=[https://books.google.com/books?id=GnW6EAAAQBAJ&pg=PA162 162]}} In the case of a full period, its arc length is
<math display="block"> 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</math>
where <math>\Gamma</math> is the [[gamma function]] and <math>\varpi</math> is the [[lemniscate constant]].{{sfnp|Adlaj|2012}}<ref>{{OEIS el|A105419|Decimal expansion of the arc length of the sine or cosine curve for one full period.}}</ref>

==== Inverse functions ====
[[File:Arcsine_Arccosine.svg|thumb|upright=1|The usual principal values of the {{math|arcsin(''x'')}} and {{math|arccos(''x'')}} functions graphed on the Cartesian plane]]
The functions <math display="inline">\sin : \mathbb{R}\to \mathbb{R}
</math> and <math>\cos : \mathbb{R}\to \mathbb{R}</math> (as well as those functions with the same function rule and domain whose codomain is a subset of <math>\mathbb{R}</math> containing the interval <math>\left[ -1,1\right]</math>) are not bijective and therefore do not have inverse functions. For example, <math> \sin (0) = 0 </math>, but also <math> \sin (\pi) = 0 </math>,  <math> \sin (2\pi) = 0 </math>. Sine's "inverse", called arcsine, can then be described not as a function but a relation (for example, all integer multiples of <math>\pi</math> would have an arcsine of zero). To define the inverse functions of sine and cosine, they must be restricted to their [[principal branch|principal branches]] by restricting their domain and codomain; the standard functions used to define arcsine and arccosine are then  <math>\sin : \left[-\pi/2, \pi/2 \right]\to \left[-1, 1 \right]</math> and <math>\cos : \left[0,\pi\right]\to \left[-1,1 \right]</math>.{{sfnp|Varberg|Purcell|Rigdon|2007|p=365}} These are bijective and have inverses: <math>\arcsin : \left[-1,1 \right]\to \left[-\pi/2,\pi/2 \right]</math> and <math>\arccos : \left[-1,1 \right]\to \left[ 0, \pi\right]</math>. Alternative notation is <math>\sin^{-1}</math> for arcsine and <math>\cos^{-1}</math> for arccosine.

Using these definitions, one obtains the identity maps and equations:
<math display="block">\begin{align}
\sin\circ\arcsin\,(x)&= \sin(\arcsin(x)) = x & x&\in \left[ -1,1\right],\\
\arcsin\circ\sin\, (\theta)&= \arcsin(\sin(\theta)) = \theta & \theta&\in \left[ -\pi/2, \pi/2\right],
\end{align}</math>and 
<math display="block">\begin{align}
\cos\circ\arccos\, (x)&= \cos(\arccos(x)) = x & x&\in \left[-1,1\right],\\
\arccos\circ\cos\,(\theta)&= \arccos(\cos(\theta)) = \theta & \theta&\in \left[0,\pi\right].
\end{align}</math>

An acute angle <math>\theta</math> is given by:
<math display="block"> \theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \arccos \left( \frac{\text{adjacent}}{\text{hypotenuse}} \right),</math>
where for some integer <math> k </math>,
<math display="block">\begin{align}
 \sin(y) = x \iff & \begin{cases}
   y = \hphantom{\pi -} \arcsin(x) + 2\pi k , &\text{ or }\\
   y = \pi - \arcsin(x) + 2\pi k \\
 \end{cases}\\
 \cos(y) = x \iff & \begin{cases}
   y = \hphantom{-} \arccos(x) + 2\pi k , &\text{ or }\\
   y = - \arccos(x) + 2\pi k
 \end{cases}
 \end{align}</math>

==== Other identities ====
{{Main|List of trigonometric identities}}

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:{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA99 99]}}{{efn|1=Here, <math> \sin^2 (x) </math> means the squared sine function <math> \sin (x) \cdot \sin (x) </math>.}}
<math display="block"> \sin^2 (\theta) + \cos^2(\theta) = 1. </math>

Sine and cosine satisfy the following double-angle formulas:<ref>{{cite book |title=Precalculus with Calculus Previews |author1=Dennis G. Zill |edition= |publisher=Jones & Bartlett Publishers |year=2013 |isbn=978-1-4496-4515-1 |page=238 }} [https://books.google.com/books?id=dtS5M4lx7scC&pg=PA238 Extract of page 238]</ref>
<math display="block"> \begin{align}
 \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{align} </math>

<!-- This section is the target of a redirect -->
{{anchor|Sine squared function}}
[[File:SinSquared.png|thumb|Sine function in blue and sine squared function in red. The {{nowrap|1={{math|1=''x''}}-}}axis is in radians.]]
The cosine double angle formula implies that {{math|sin<sup>2</sup>}} and {{math|cos<sup>2</sup>}} are, themselves, shifted and scaled sine waves.  Specifically,<ref>{{cite web |title=Sine-squared function |url=https://calculus.subwiki.org/wiki/Sine-squared_function#Identities |access-date=August 9, 2019}}</ref>
<math display="block"> \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}\qquad\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}</math>
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|date=August 2024}}

=== Series and polynomials ===
[[File:Sine.gif|thumb|right|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 {{section link||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 <math> \sin(x) </math> are <math> \cos(x) </math>, <math> -\sin(x) </math>, <math> -\cos(x) </math>, <math> \sin(x) </math>, continuing to repeat those four functions. The {{nowrap|1=<math> (4n + k) </math>-}}th derivative, evaluated at the point 0:
<math display="block"> \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}</math>
where the superscript represents repeated differentiation. This implies the following Taylor series expansion at <math> x = 0 </math>. One can then use the theory of [[Taylor series]] to show that the following identities hold for all [[real number]]s <math> x </math>&mdash;where <math> x </math> is the angle in radians.{{sfnp|Varberg|Purcell|Rigdon|2007|p=491&ndash;492}} More generally, for all [[complex number]]s:{{sfnp|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA74 74]}}
<math display="block">\begin{align}
\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{align}</math>
Taking the derivative of each term gives the Taylor series for cosine:{{sfnp|Varberg|Purcell|Rigdon|2007|p=491&ndash;492}}{{sfnp|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA74 74]}}
<math display="block">\begin{align}
\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{align}</math>

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 [[Trigonometric interpolation|its interpolation]], and its extension of a periodic function known as the [[Fourier series]]. Let <math> a_n </math> and <math> b_n </math> be any coefficients, then the trigonometric polynomial of a degree <math> N </math>&mdash;denoted as <math> T(x) </math>&mdash;is defined as:{{sfnp|Powell|1981|p=150}}{{sfnp|Rudin|1987|p=88 }}
<math display="block"> T(x) = a_0 + \sum_{n=1}^N a_n \cos (nx) + \sum_{n=1}^N b_n \sin (nx). </math>

The [[trigonometric series]] can be defined similarly analogous to the trigonometric polynomial, its infinite inversion. Let <math> A_n </math> and <math> B_n </math> be any coefficients, then the trigonometric series can be defined as:{{sfnp|Zygmund|1968|p=1}}
<math display="block"> \frac{1}{2} A_0 + \sum_{n=1}^\infty A_n \cos (nx) + B_n \sin (nx). </math>
In the case of a Fourier series with a given integrable function <math> f </math>, the coefficients of a trigonometric series are:{{sfnp|Zygmund|1968|p=11}}
<math display="block"> \begin{align}
 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{align} </math>

== Complex numbers relationship ==
{{more citations needed section|date=August 2024}}

=== Complex exponential function definitions ===
Both sine and cosine can be extended further via [[complex number]], a set of numbers composed of both [[Real number|real]] and [[imaginary number]]s. For real number <math> \theta </math>, the definition of both sine and cosine functions can be extended in a [[complex plane]] in terms of an [[exponential function]] as follows:{{sfnp|Howie|2003|p=[https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24 24]}}
<math display="block"> \begin{align}
 \sin (\theta) &= \frac{e^{i \theta} - e^{-i \theta}}{2i}, \\
 \cos (\theta) &= \frac{e^{i \theta} + e^{-i \theta}}{2},
\end{align} </math>

Alternatively, both functions can be defined in terms of [[Euler's formula]]:{{sfnp|Howie|2003|p=[https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24 24]}}
<math display="block"> \begin{align}
 e^{i \theta} &= \cos (\theta) + i \sin (\theta), \\
 e^{-i \theta} &= \cos (\theta) - i \sin (\theta).
\end{align} </math>

When plotted on the [[complex plane]], the function <math>e^{ix}</math> for real values of <math>x</math> traces out the [[unit circle]] in the complex plane. Both sine and cosine functions may be simplified to the imaginary and real parts of <math>e^{i \theta}</math> as:{{sfnp|Rudin|1987|p=2}}
<math display="block"> \begin{align}
 \sin \theta &= \operatorname{Im}(e^{i \theta}), \\
 \cos \theta &= \operatorname{Re}(e^{i \theta}).
\end{align} </math>

When <math>z=x+iy</math> for real values <math>x</math> and <math>y</math>, where <math> i = \sqrt{-1} </math>, both sine and cosine functions can be expressed in terms of real sines, cosines, and [[hyperbolic function]]s as:<ref>{{cite book |last=Brown |first=James Ward  |last2=Churchill |first2=Ruel |author2-link=Ruel Vance Churchill |date=2014 |title=Complex Variables and Applications |edition=9th |publisher=[[McGraw-Hill]] |isbn=978-0-07-338317-0 |page=105}}</ref>
<math display="block"> \begin{align}
 \sin z &= \sin x \cosh y + i \cos x \sinh y, \\
 \cos z &= \cos x \cosh y - i \sin x \sinh y.
\end{align}</math>

=== Polar coordinates ===
[[File:Sinus und Kosinus am Einheitskreis 3.svg|thumb|Both functions {{math|cos(''&theta;'')}} and {{math|sin(''&theta;'')}} are the real and imaginary parts of {{math|''e''<sup>''i&theta;''</sup>}}.]]
Sine and cosine are used to connect the real and imaginary parts of a [[complex number]] with its [[polar coordinates]] <math> (r, \theta) </math>:
<math display="block"> z = r(\cos(\theta) + i\sin(\theta)) = r\cos(\theta) + ir\sin(\theta), </math>
and the real and imaginary parts are
<math display="block"> \begin{align}
 \operatorname{Re}(z) &= r \cos(\theta), \\
 \operatorname{Im}(z) &= r \sin(\theta),
\end{align} </math>
where <math> r </math> and <math> \theta </math> represent the magnitude and angle of the complex number <math> z </math>.{{sfnp|Howie|2003|p=[https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24 23&ndash;24]}}

For any real number <math> \theta </math>, Euler's formula in terms of polar coordinates is stated as <math display="inline">z = re^{i\theta}</math>.{{sfnp|Howie|2003|p=[https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24 24]}}

===Complex arguments===
[[File:Complex_sin.jpg|thumb|[[Domain coloring]] of {{math|sin(''z'')}} in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.]]
[[File:Sin z vector field 02 Pengo.svg|thumb|Vector field rendering of {{math|sin(''z'')}}]]

Applying the series definition of the sine and cosine to a complex argument, <math> z </math>, gives:
<math display="block">\begin{align}
\sin(z)& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}z^{2n+1} \\
& = \frac{e^{i z} - e^{-i z}}{2i} \\
& = \frac{\sinh \left( i z\right) }{i} \\
& = -i \sinh \left(i z\right)\\
\cos(z)& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}z^{2n} \\
& = \frac{e^{i z} + e^{-i z}}{2} \\
& = \cosh( i z) \\
\end{align}</math>
where <math>\sinh</math> and <math>\cosh</math> are the [[hyperbolic function|hyperbolic sine and cosine]]. These are [[entire function]]s.

It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument:
<math display="block">\begin{align}
\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{align}</math>

==== Partial fraction and product expansions of complex sine ====
Using the partial fraction expansion technique in [[complex analysis]], one can find that the infinite series
<math display="block"> \sum_{n = -\infty}^{\infty}\frac{(-1)^n}{z-n} = \frac{1}{z} -2z \sum_{n = 1}^{\infty}\frac{(-1)^n}{n^2 - z^2}</math>
both converge and are equal to <math display="inline">\frac{\pi}{\sin (\pi z)}</math>. Similarly, one can show that
<math display="block">\frac{\pi^2}{\sin^2 (\pi z)} = \sum_{n=-\infty}^\infty \frac{1}{(z - n)^2}.</math>

Using product expansion technique, one can derive
<math display="block">\sin(\pi z) = \pi z \prod_{n = 1}^\infty \left( 1 - \frac{z^2}{n^2} \right).</math>

==== Usage of complex sine ====
{{math|sin(''z'')}} is found in the [[functional equation]] for the [[Gamma function]],
<math display="block">\Gamma(s)\Gamma(1 - s) = {\pi\over\sin(\pi s)},</math>
which in turn is found in the [[functional equation]] for the [[Riemann zeta-function]],
<math display="block">\zeta(s) = 2(2\pi)^{s-1}\Gamma(1 - s)\sin\left(\frac{\pi}{2} s\right)\zeta(1 - s).</math>

As a [[holomorphic function]], {{math|sin ''z''}} is a 2D solution of [[Laplace's equation]]:
<math display="block">\Delta u(x_1, x_2) = 0.</math>

The complex sine function is also related to the level curves of [[pendulums]].{{how|reason=This does not explain how sine is related to pendulums.|date=August 2019}}<ref>{{cite web|url=https://math.stackexchange.com/q/220418|title=Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?|website=math.stackexchange.com|access-date=2019-08-12}}</ref>{{better source needed|date=August 2019}}

=== Complex graphs ===
<gallery mode="packed"
caption="Sine function in the complex plane" widths="1000px" heights="130px">
Complex sin real 01 Pengo.svg|Real component
Complex sin imag 01 Pengo.svg|Imaginary component
Complex sin abs 01 Pengo.svg|Magnitude
</gallery>
<br />
<gallery mode="packed"
caption="Arcsine function in the complex plane" widths="1000px" heights="130px">
Complex arcsin real 01 Pengo.svg|Real component
Complex arcsin imag 01 Pengo.svg|Imaginary component
Complex arcsin abs 01 Pengo.svg|Magnitude
</gallery>

== Background ==
=== Etymology ===
{{main|History of trigonometry#Etymology}}

The word ''sine'' is derived, indirectly, from the [[Sanskrit]] word {{lang|sa|jyā}} 'bow-string' or more specifically its synonym {{lang|sa|jīvá}} (both adopted from [[Ancient Greek language|Ancient Greek]] {{lang|grc|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#Corollaries|Ptolemy's Theorem]]). This was [[transliteration|transliterated]] in [[Arabic language|Arabic]] as {{tlit|ar|jība}}, which is meaningless in that language and written as {{tlit|ar|jb}} ({{lang|ar|جب}}). Since Arabic is written without short vowels, {{tlit|ar|jb}} was interpreted as the [[homograph]] {{tlit|ar|jayb}} ([[:wikt:جيب|{{lang|ar|جيب|cat=no}}]]), which means 'bosom', 'pocket', or 'fold'.{{sfnp|Plofker|2009|p=[https://books.google.com/books?id=DHvThPNp9yMC&pg=PA257 257]}}{{sfnp|Maor|1998|p=[https://books.google.com/books?id=r9aMrneWFpUC&pg=PA35 35]}} When the Arabic texts of [[Al-Battani]] and [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]] were translated into [[Medieval Latin]] in the 12th century by [[Gerard of Cremona]], he used the Latin equivalent [[wikt:sinus|{{lang|la|sinus|cat=no}}]] (which also means 'bay' or 'fold', and more specifically 'the hanging fold of a [[toga]] over the breast').{{sfnp|Merzbach|Boyer|2011}}{{sfnp|Maor|1998|p=35&ndash;36}}{{sfnp|Katz|2008|p=253}} 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.{{sfnp|Smith|1958|p=202}}<ref>Various sources credit the first use of {{Lang|la-x-medieval|sinus}} to either 
* [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]
* [[Gerard of Cremona]]'s translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]
* [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī
See {{harvp|Merlet|2004}}. See {{harvp|Maor|1998}}, Chapter 3, for an earlier etymology crediting Gerard. See {{harvp|Katz|2008|p=210}}.</ref> The English form ''sine'' was introduced in [[Thomas Fale]]'s 1593 ''Horologiographia''.<ref>Fale's book alternately uses the spellings "sine", "signe", or "sign". {{pb}} {{cite book |last=Fale |first=Thomas |title=Horologiographia. The Art of Dialling: Teaching, an Easie and Perfect Way to make all Kindes of Dials .... |page=11, for example |url=https://archive.org/details/b30333106/page/19/mode/1up |year=1593 |place=London |publisher=F. Kingston }}</ref>

The word ''cosine'' derives from an abbreviation of the Latin {{lang|la|complementi sinus}} 'sine of the [[complementary angle]]' as {{lang|la|cosinus}} in [[Edmund Gunter]]'s ''Canon triangulorum'' (1620), which also includes a similar definition of ''cotangens''.{{sfnp|Gunter|1620}}

=== History ===
{{Main|History of trigonometry}}

[[File:Khalili Collection Islamic Art sci 0040.1 CROP.jpg|right|thumb|Quadrant from 1840s [[Ottoman Empire|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 (geometry)|chord]] function was discovered by [[Hipparchus]] of [[İznik|Nicaea]] (180–125 BCE) and [[Ptolemy]] of [[Egypt (Roman province)|Roman Egypt]] (90–165 CE).<ref>{{Cite journal |last=Brendan |first=T. |date=February 1965 |title=How Ptolemy constructed trigonometry tables |journal=The Mathematics Teacher |volume=58 |issue=2 |pages=141–149 |doi=10.5951/MT.58.2.0141 |jstor=27967990 }}</ref>

The sine and cosine functions are closely related to the [[Jyā, koti-jyā and utkrama-jyā|{{tlit|sa|jyā}} and {{tlit|sa|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.{{sfnp|Merzbach|Boyer|2011}}<ref>{{cite book |last=Van Brummelen |first=Glen |author-link=Glen Van Brummelen |year=2009 |title=The Mathematics of the Heavens and the Earth |chapter=India |at=Ch. 3, {{pgs|94–134}} |publisher=Princeton University Press |isbn=978-0-691-12973-0}}</ref>

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]].<ref name="Gingerich_1986">{{cite magazine |title=Islamic Astronomy |author-first=Owen |author-last=Gingerich |magazine=[[Scientific American]] |date=1986 |volume=254 |page=74 |url=http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm |access-date=2010-07-13 |archive-url=https://web.archive.org/web/20131019140821/http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm |archive-date=2013-10-19}}</ref> [[Al-Khwārizmī]] (c. 780–850) produced tables of sines, cosines and tangents.<ref name="Sesiano">Jacques Sesiano, "Islamic mathematics", p. 157, in {{Cite book |title=Mathematics Across Cultures: The History of Non-western Mathematics |editor1-first=Helaine |editor1-last=Selin |editor1-link=Helaine Selin |editor2-first=Ubiratan |editor2-last=D'Ambrosio |editor2-link=Ubiratan D'Ambrosio |year=2000 |publisher=[[Springer Science+Business Media]] |isbn=978-1-4020-0260-1}}</ref><ref name="Britannica">{{cite web |title=trigonometry |date=17 June 2024 |url=http://www.britannica.com/EBchecked/topic/605281/trigonometry |publisher=Encyclopedia Britannica}}</ref> [[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°.<ref name="Britannica"/>

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, [[Gottfried Leibniz|Leibniz]] proved that {{math|sin ''x''}} is not an [[algebraic function]] of {{mvar|x}}.<ref>{{cite book|title=Elements of the History of Mathematics|url=https://archive.org/details/elementsofhistor0000bour|url-access=registration|author=[[Nicolas Bourbaki]]|publisher=Springer|year=1994|isbn=9783540647676}}</ref> [[Roger Cotes]] computed the derivative of sine in his ''Harmonia Mensurarum'' (1722).<ref>"[http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf Why the sine has a simple derivative] {{webarchive|url=https://web.archive.org/web/20110720102700/http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf |date=2011-07-20 }}", in ''[http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm Historical Notes for Calculus Teachers] {{webarchive|url=https://web.archive.org/web/20110720102613/http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm |date=2011-07-20 }}'' by [http://www.math.usma.edu/people/rickey/ V. Frederick Rickey] {{webarchive|url=https://web.archive.org/web/20110720102654/http://www.math.usma.edu/people/rickey/ |date=2011-07-20 }}</ref> [[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.''{{sfnp|Merzbach|Boyer|2011}}

== Software implementations ==
{{more citations needed section|date=August 2024}}
{{See also|Lookup table#Computing sines}}
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.{{sfnp|Zimmermann|2006}}

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. <code>sin(10{{sup|22}})</code>.

A common programming optimization, used especially in 3D graphics<!-- A quick sine function is useful in many other fields. Citation needed? -->, 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 [[Linear interpolation|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|date=October 2012}} <!-- Looking up in a table essentially involves copying a number from one memory location to another. How can calculating a sine be as quick as that? A: depends on many things, but mainly if the value in memory is not in cache, then the cache-miss can be relatively expensive, whereas a trig function never needs to touch main memory (or alternative, it can be expensive for the rest of the code to have the CPU cache filled with trig tables, causing cache-misses elsewhere). Probably needs a more definitive answer. Have changed 'typically' to 'may' for now. -->

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 <code>sin</code> and <code>cos</code>.

Some CPU architectures have a built-in instruction for sine, including the Intel [[x87]] FPUs since the 80387.

In programming languages, <code>sin</code> and <code>cos</code> are typically either a built-in function (e.g. in [[Fortran]] and MATLAB) or found within the language's standard math library. For example, the [[C standard library]] defines sine functions within [[C mathematical functions|math.h]]: <code>sin([[Double-precision floating-point format|double]])</code>, <code>sinf([[Single-precision floating-point format|float]])</code>, and <code>sinl([[long double]])</code>. 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 [[C mathematical functions|math.h]], such as for cosine, arc sine, and hyperbolic sine (sinh). Similarly, [[Python (programming language)|Python]] defines <code>math.sin(x)</code> and <code>math.cos(x)</code> within the built-in <code>math</code> module. Complex sine and cosine functions are also available within the <code>cmath</code> module, e.g. <code>cmath.sin(z)</code>. [[CPython]]'s math functions call the [[C (programming language)|C]] <code>math</code> library, and use a [[double-precision floating-point format]].

=== Turns based implementations ===
{{redirect|sinpi|the township in Pingtung County, Taiwan|Xinpi}}
{{redirect|cospi|the 17th-century Bolognese nobleman|Ferdinando Cospi}}
Some software libraries provide implementations of sine and cosine using the input angle in half-[[Turn (angle)|turns]], a half-turn being an angle of 180 degrees or <math>\pi</math> radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases. These functions, following the convention set by Fortran, are typically called <code>sinpi</code> and <code>cospi</code>. For example, <code>sinpi(x)</code> would evaluate to <math>\sin(\pi x),</math> where <math>x</math> is expressed in half-turns so that {{tmath|\pi x}} is the angle measure in radians. [[SciPy]] provides similar functions <code>sindg</code> and <code>cosdg</code> with input in degrees, as does Fortran (but named <code>sind</code> and <code>cosd</code>).<ref>Documentation for: {{ulist
| Fortran: {{cite web|url=https://wg5-fortran.org/N2201-N2250/N2212.pdf |format=PDF|title=Sine and cosine|website=Wg5-fortran.org|access-date=2026-06-15}}
| MATLAB: {{Cite web |title=sinpi - Compute sin(X*pi) accurately |url=https://www.mathworks.com/help/matlab/ref/double.sinpi.html |access-date=2026-01-08 |website=Mathworks.com }}
| [[OpenCL]]: {{Cite web |title=sin, sincos, sinh, sinpi |url=https://registry.khronos.org/OpenCL/sdk/1.0/docs/man/xhtml/sin.html |access-date=2026-02-17 |website=registry.khronos.org}}
| R: {{Cite web |title=Trig function - RDocumentation |url=https://www.rdocumentation.org/packages/base/versions/3.5.3/topics/Trig |access-date=2026-02-17 |website=Rdocumentation.org}}
| Julia: {{Cite web |title=sinpi » Julia Functions |url=http://www.jlhub.com/julia/manual/en/function/sinpi |access-date=2026-02-17 |website=Jlhub.com}}
| Python: {{Cite web |title=What's new in Python 3.16 |url=https://docs.python.org/3.16/whatsnew/3.16.html#math |access-date=2026-07-26 |website=docs.python.org}}
| SciPy: {{cite web |title=Special functions (scipy.special) — SciPy v1.17.0 Manual |url=https://docs.scipy.org/doc/scipy/reference/special.html#convenience-functions |website=docs.scipy.org |access-date=25 February 2026}}
| [[CUDA]]: {{Cite web |title=Double Precision Mathematical Functions |url=https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__DOUBLE.html |url-status=dead |archive-url=https://web.archive.org/web/20240723062728/https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__DOUBLE.html |archive-date=2024-07-23 |access-date=2026-01-08 |website=docs.nvidia.com |language=en-us}}
| ARM: {{Cite web |title=Documentation – Arm Developer |url=https://developer.arm.com/documentation/100614/latest/b-opencl-built-in-functions/b2-math-functions |access-date=2026-02-17 |website=developer.arm.com}}
}}</ref>

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 <math>2\pi</math>, <math>\pi</math>, and <math display="inline">\frac{\pi}{2}</math> 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 <math display="inline">\frac{\pi}{2}</math> involves inaccuracies in representing <math display="inline">\frac{\pi}{2}</math>.

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.<ref>{{Cite web |title=AAS33051: Precision Angle Sensor IC with Incremental and Motor Commutation Outputs and On-Chip Linearization |url=https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx |archive-url=https://web.archive.org/web/20190417143715/https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx |archive-date=2019-04-17 |access-date=2026-02-17 |website=www.allegromicro.com}}</ref> 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 <math display="inline">\frac{\pi}{2048}</math> would be incurred. (See also [[Binary angular measurement]].)

== See also ==
{{div col|colwidth=20em}}
* [[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]]
* [[Mnemonics in trigonometry#SOH-CAH-TOA|SOH-CAH-TOA]]
* [[Trigonometric functions]]
* [[Trigonometric integral]]
{{div col end}}

== References ==
=== Footnotes ===
{{notelist|group="efn"}}

=== Citations ===
{{Reflist}}

=== Works cited ===
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}}
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{{refend}}

== External links ==
{{Wiktionary|sine}}
* {{Commons category-inline|Sine function}}
{{Wiktionary}}

{{Trigonometric and hyperbolic functions}}

[[Category:Angle]]
[[Category:Trigonometric functions]]

[[no:Trigonometriske funksjoner#Sinus, cosinus og tangens]]