View source for Sine

{{short description|A trigonometric function of an angle}}
{{other uses}}
{{distinguish|sign|sign (mathematics)}}
{{Infobox mathematics function
| name = Sine
| image = Sine one period.svg
| parity=odd |domain=(−{{lower|0.2em|{{resize|130%|∞}}}}, +{{lower|0.2em|{{resize|130%|∞}}}})&thinsp;{{smallsup|a}} |codomain=[−1, 1]&thinsp;{{smallsup|a}} |period=2{{pi}}
| zero=0 |plusinf= |minusinf= 
|max=(2''k''{{pi}} + {{sfrac|{{pi}}|2}}, 1){{smallsup|b}} |min=(2''k''{{pi}} − {{sfrac|{{pi}}|2}}, −1)
| vr1= |f1= <!--......--> |vr5= |f5=
| asymptote= |root=''k''{{pi}} |critical=''k''{{pi}} + {{sfrac|{{pi}}|2}} |inflection=''k''{{pi}} |fixed=0
| notes = {{ublist |{{sup|a}} For [[Real number|real]] numbers. |{{sup|b}} Variable ''k'' is an [[integer]].}}
}}
{{Trigonometry}}

In [[mathematics]], the '''sine''' is a [[trigonometric function]] of an [[angle]]. The sine of an acute angle is defined in the context of a [[right triangle]]: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the [[hypotenuse]]).

More generally, the definition of sine (and other trigonometric functions) can be extended to any [[real number|real]] value in terms of the length of a certain line segment in a [[unit circle]]. More modern definitions express the sine as an [[Series (mathematics)|infinite series]] or as the solution of certain [[differential equation]]s, allowing their extension to arbitrary positive and negative values and even to [[complex number]]s.

The sine function is 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.

The function sine can be traced to the [[Jyā, koti-jyā and utkrama-jyā|''jyā'' and ''koṭi-jyā '']] functions used in [[Gupta period]] [[Indian astronomy]] (''[[Aryabhatiya]]'', ''[[Surya Siddhanta]]''), via translation from Sanskrit to Arabic and then from Arabic to Latin.<ref name="Boyer, Carl B. 1991 p. 210">[[Uta Merzbach|Uta C. Merzbach]], [[Carl B. Boyer]] (2011), A History of Mathematics, Hoboken, N.J.: John Wiley & Sons, 3rd ed., p. 189.</ref> The word "sine" (Latin "sinus") comes from a [[Latin]] mistranslation of the Arabic ''jiba'', which is a [[transliteration]] of the Sanskrit word for half the chord, ''jya-ardha''.<ref>Victor J. Katz (2008), ''A History of Mathematics'', Boston: Addison-Wesley, 3rd. ed., p. 253, sidebar 8.1. {{cite web |url=http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf |title=Archived copy |accessdate=2015-04-09 |url-status=live |archiveurl=https://web.archive.org/web/20150414065531/http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf |archivedate=2015-04-14 }}</ref>

== Right-angled triangle definition ==
[[File:Trigono sine en2.svg|right|thumb| For the angle ''α'', the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.]]

To define the sine function of an acute angle ''α'', start with a [[right triangle]] that contains an angle of measure ''α''; in the accompanying figure, angle ''A'' in triangle ''ABC'' is the angle of interest. The three sides of the triangle are named as follows:

* The ''opposite side'' is the side opposite to the angle of interest, in this case side&nbsp;''a''.
* The ''hypotenuse'' is the side opposite the right angle, in this case side&nbsp;''h''. The hypotenuse is always the longest side of a right-angled triangle.
* The ''adjacent side'' is the remaining side, in this case side&nbsp;''b''.  It forms a side of (is adjacent to) both the angle of interest (angle ''A'') 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, or:

:<math>\sin(\alpha) = \frac {\textrm{opposite}} {\textrm{hypotenuse}}</math>

The other trigonometric functions of the angle can be defined similarly; for example, the [[Trigonometric functions#cosine|cosine]] of the angle is the ratio between the adjacent side and the hypotenuse, while the [[Trigonometric functions#tangent|tangent]] gives the ratio between the opposite and adjacent sides.

As stated, the value <math>\sin(\alpha)</math> appears to depend on the choice of right triangle containing an angle of measure ''α''.  However, this is not the case: all such triangles are [[similarity (geometry)|similar]], and so the ratio is the same for each of them.

== Unit circle definition ==
[[File:Unit circle.svg|right|thumb|Unit circle: the radius has length 1. The variable ''t'' measures the [[angle]] referred to as ''θ'' in the text.]]
In [[trigonometry]], a unit circle is the circle of radius one centered at the origin (0, 0) in the [[Cartesian coordinate system]].

Let a line through the origin, making an angle of ''θ'' with the positive half of the ''x''-axis, intersect the unit circle. The ''x''- and ''y''-coordinates of this point of intersection are equal to {{math|cos(''θ'')}} and {{math|sin(''θ'')}}, respectively. The point's distance from the origin is always&nbsp;1.

Unlike the definitions with the right triangle or slope, the angle can be extended to the full set of real arguments by using the [[Unit circle#Trigonometric functions on the unit circle|unit circle]]. This can also be achieved by requiring certain symmetries and that sine be a [[periodic function]].

<gallery widths="450px" heights="275px">
File:Circle_cos_sin.gif|Animation showing how the sine function (in red) <math>y = \sin(\theta)</math> is graphed from the ''y''-coordinate (red dot) of a point on the [[unit circle]] (in green) at an angle of ''θ''.
</gallery>

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

Exact identities (using [[radian]]s):

These apply for all values of <math>\theta</math>.

: <math>
\sin(\theta) = \cos\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\csc(\theta)}
</math>

=== Reciprocal ===
The [[multiplicative inverse|reciprocal]] of sine is cosecant, i.e., the reciprocal of {{math|sin(''A'')}} is {{math|csc(''A'')}}, or cosec(''A''). Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side:

:<math>\csc(A) = \frac{1}{\sin(A)} = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac{h}{a}. </math>

=== Inverse ===
[[File:Arcsine.svg|thumb|180px|The usual principal values of the {{math|arcsin(''x'')}} function graphed on the cartesian plane. Arcsin is the inverse of sin.]]
The [[inverse function]] of sine is arcsine (arcsin or asin) or inverse sine ({{math|sin{{sup|-1}}}}). As sine is non-[[injective]], it is not an exact inverse function but a partial inverse function. For example, {{math|sin(0) {{=}} 0}}, but also {{math|sin({{pi}}) {{=}} 0}}, {{math|sin(2{{pi}}) {{=}} 0}} etc. It follows that the arcsine function is multivalued: {{math|arcsin(0) {{=}} 0}}, but also {{math|arcsin(0) {{=}} {{pi}}}}, {{math|arcsin(0) {{=}} 2{{pi}}}}, etc. When only one value is desired, the function may be restricted to its [[principal branch]]. With this restriction, for each ''x'' in the domain the expression {{math|arcsin(''x'')}} will evaluate only to a single value, called its [[principal value]].

:<math>\theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \sin^{-1} \left( \frac {a}{h} \right).</math>

''k'' is some integer:

:<math>\begin{align}
\sin(y) = x \iff & y = \arcsin(x) + 2\pi k , \text{ or }\\
 & y = \pi - \arcsin(x) + 2\pi k
\end{align}</math>

Or in one equation:

:<math>\sin(y) = x \iff  y = (-1)^k \arcsin(x) + \pi k</math>

Arcsin satisfies:

:<math>\sin(\arcsin(x)) = x\!</math>
and
:<math>\arcsin(\sin(\theta)) = \theta\quad\text{for }-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}.</math>

=== Calculus ===
{{See also|List of integrals of trigonometric functions|Differentiation of trigonometric functions}}

For the sine function:

:<math>f(x) = \sin(x) </math>

The derivative is:

:<math>f'(x) = \cos(x) </math>

The antiderivative is:

:<math>\int f(x)\,dx = -\cos(x) + C </math>

''C'' denotes the [[constant of integration]].

=== Other trigonometric functions ===
[[File:Sine cosine one period.svg|right|thumb|The sine and cosine functions are related in multiple ways.  The two functions are out of phase by 90°: <math>\sin(\pi/2 - x)</math> = <math>\cos(x)</math> for all angles ''x''.  Also, the derivative of the function {{math|sin(''x'')}} is {{math|cos(''x'')}}.]]

It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign, or using the [[sign function]]).

Sine in terms of the other common [[trigonometric functions]]:

{| class="wikitable"
|-
!colspan="1" rowspan="3"|
!colspan="1" rowspan="3"|f ''θ''
!colspan="5"|Using plus/minus (±)
!colspan="1"|Using sign function (sgn)
|-
!colspan="1" rowspan=2|f ''θ'' =
!colspan="4"|± per Quadrant
!rowspan="2"|f ''θ'' =
|-
! I
! II
! III
! IV
|-
!rowspan="2"|cos
|<math>\sin(\theta)</math>
|<math>= \pm\sqrt{1 - \cos^2(\theta)}</math>
| +
| +
| −
| −
|<math>= \sgn\left( \cos \left(\theta - \frac{\pi}{2}\right)\right) \sqrt{1 - \cos^2(\theta)}</math>
|-
|<math>\cos(\theta)</math>
|<math>= \pm\sqrt{1 - \sin^2(\theta)}</math>
| +
| −
| −
| +
|<math>= \sgn\left( \sin \left(\theta+ \frac{\pi}{2}\right)\right) \sqrt{1 - \sin^2(\theta)}</math>
|-
!rowspan="2"|cot
| <math>\sin(\theta)</math>
| <math>= \pm\frac{1}{\sqrt{1 + \cot^2(\theta)}}</math>
| +
| +
| −
| −
| <math>=  \sgn\left( \cot\left( \frac{\theta}{2}\right)\right) \frac{1}{\sqrt{1 + \cot^2(\theta)}}</math>
|-
|<math>\cot(\theta)</math>
|<math> = \pm\frac{\sqrt{1 - \sin^2(\theta)}}{\sin(\theta)}</math>
| +
| −
| −
| +
| <math>= \sgn\left( \sin \left(\theta+ \frac{\pi}{2}\right)\right) \frac{\sqrt{1 - \sin^2(\theta)}}{\sin(\theta)}</math>
|-
!rowspan="2"|tan
| <math>\sin(\theta)</math>
| <math>= \pm\frac{\tan(\theta)}{\sqrt{1 + \tan^2(\theta)}} </math>
| +
| −
| −
| +
| <math>= \sgn\left( \tan\left(\frac{2\theta + \pi}{4}\right)\right) \frac{\tan(\theta)}{\sqrt{1 + \tan^2(\theta)}}</math>
|-
| <math>\tan(\theta)</math>
| <math>= \pm\frac{\sin(\theta)}{\sqrt{1 - \sin^2(\theta)}}</math>
| +
| −
| −
| +
| <math>= \sgn\left( \sin \left(\theta+ \frac{\pi}{2}\right)\right) \frac{\sin(\theta)}{\sqrt{1 - \sin^2(\theta)}}</math>
|-
!rowspan="2"|sec
| <math>\sin(\theta)</math>
| <math>= \pm\frac{\sqrt{\sec^2(\theta) - 1}}{\sec(\theta)} </math>
| +
| −
| +
| −
| <math>= \sgn\left( \sec \left( \frac{4 \theta - \pi}{2}\right)\right) \frac{\sqrt{\sec^2(\theta) - 1}}{\sec(\theta)} </math>
|-
| <math>\sec(\theta)</math>
| <math>= \pm\frac{1}{\sqrt{1 - \sin^2(\theta)}}</math>
| +
| −
| −
| +
| <math>= \sgn\left( \sin \left(\theta+ \frac{\pi}{2}\right)\right) \frac{1}{\sqrt{1 - \sin^2(\theta)}}</math>
|}

For all equations which use plus/minus (±), the result is positive for angles in the first quadrant.

The basic relationship between the sine and the cosine can also be expressed as the [[Pythagorean trigonometric identity]]:

:<math>\cos^2(\theta) + \sin^2(\theta) = 1\!</math>

where sin<sup>2</sup>(''x'') means (sin(''x''))<sup>2</sup>.

===Sine squared function===
<!-- This section is the target of a redirect -->
[[File:SinSquared.png|thumb|Sine function in blue and sine squared function in red.  The Y axis is in radians.]]
The graph shows both the sine function and the [[b:Trigonometry/Graph of Sine Squared|sine squared]] function, with the sine in blue and 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.

The sine squared function can be expressed as a modified sine wave from the Pythagorean identity and power reduction by the cosine double-angle formula:<ref>{{cite web |title=Sine-squared function |url=https://calculus.subwiki.org/wiki/Sine-squared_function#Identities |accessdate=August 9, 2019}}</ref>

:<math> \sin^2(\theta)\  =  \frac{1 - \sin(2\theta+\tfrac{\pi}{2})}{2} \ </math>

== Properties relating to the quadrants ==
[[File:Quadrants 01 Pengo.svg|thumb|200px|The four quadrants of a Cartesian coordinate system.]]
The table below displays many of the key properties of the sine function (sign, monotonicity, convexity), arranged by the quadrant of the argument.  For arguments outside those in the table, one may compute the corresponding information by using the periodicity  <math>\sin(\alpha + 360^\circ) = \sin(\alpha)</math> of the sine function.
{| class="wikitable"  style="text-align: center"
|-
![[Cartesian coordinate system#Quadrants and octants|Quadrant]]
![[Degree (angle)|Degrees]]
![[Radian]]s
!Value
![[Sign (mathematics)|Sign]]
![[Monotonic function|Monotony]]
![[Convex function|Convexity]]
|-
|1st Quadrant
|<math>0^\circ<x<90^\circ</math>
|<math>0<x< \frac{\pi}{2}</math>
|<math>0<\sin(x)<1</math>
|<math>+</math>
|increasing
|concave
|-
|2nd Quadrant
|<math>90^\circ<x<180^\circ</math>
|<math>\frac{\pi}{2}<x<\pi</math>
|<math>0<\sin(x)<1</math>
|<math>+</math>
|decreasing
|concave
|-
|3rd Quadrant
|<math>180^\circ<x<270^\circ</math>
|<math>\pi<x<\frac{3\pi}{2}</math>
|<math>-1<\sin(x)<0</math>
|<math>-</math>
|decreasing
|convex
|-
|4th Quadrant
|<math>270^\circ<x<360^\circ</math>
|<math>\frac{3\pi}{2}<x<2\pi</math>
|<math>-1<\sin(x)<0</math>
|<math>-</math>
|increasing
|convex
|}

[[File:Sine quads 01 Pengo.svg|thumb|390px|The quadrants of the unit circle and of sin(''x''), using the [[Cartesian coordinate system]].]]

The following table gives basic information at the boundary of the quadrants.

{| class="wikitable" style="text-align: center"
|-
![[Degree (angle)|Degrees]]
![[Radian]]s
!<math>\sin(x)</math>
!Point type
|-
|<math>0^\circ</math>
|<math>0</math>
|<math>0</math>
|align=left|[[Zero of a function|Root]], [[Inflection point|Inflection]]
|-
|<math>90^\circ</math>
|<math>\frac{\pi}{2}</math>
|<math>1</math>
|align=left|[[Maxima and minima|Maximum]]
|-
|<math>180^\circ</math>
|<math>\pi</math>
|<math>0</math>
|align=left|[[Zero of a function|Root]], [[Inflection point|Inflection]]
|-
|<math>270^\circ</math>
|<math>\frac{3\pi}{2}</math>
|<math>-1</math>
|align=left|[[Maxima and minima|Minimum]]
|}

== Series definition ==
[[Image:Taylorsine.svg|thumb|right|The sine function (blue) is closely approximated by its [[Taylor's theorem|Taylor polynomial]] of degree 7 (pink) for a full cycle centered on the origin.]]

[[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.]]

Using only geometry and properties of [[limit of a function|limits]], it can be shown that the [[derivative]] of sine is cosine, and that the derivative of cosine is the negative of sine.

Using the reflection from the calculated geometric derivation of the sine is with the (4''n''+''k'')-th derivative at the point 0:

:<math>\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>

This gives the following Taylor series expansion at x = 0. One can then use the theory of [[Taylor series]] to show that the following identities hold for all [[real number]]s ''x'' (where x is the angle in radians) :<ref>See Ahlfors, pages 43–44.</ref>

:<math>
\begin{align}
\sin(x) &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\[8pt]
& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\[8pt]
\end{align}
</math>

If ''x'' were expressed in degrees then the series would contain factors involving powers of π/180: if ''x'' is the number of degrees, the number of radians is ''y'' = π''x'' /180, so

:<math>\begin{align}
\sin(x_\mathrm{deg}) & = \sin(y_\mathrm{rad}) \\
& = \frac{\pi}{180} x - \left (\frac{\pi}{180} \right )^3 \frac{x^3}{3!} + \left (\frac{\pi}{180} \right )^5 \frac{x^5}{5!} - \left (\frac{\pi}{180} \right )^7 \frac{x^7}{7!} + \cdots .
\end{align}</math>

The series formulas for the sine and [[cosine]] are uniquely determined, up to the choice of unit for angles, by the requirements that
:<math>
\begin{align}
\sin(0) = 0 & \text{ and } \sin(2x) = 2 \sin(x) \cos(x) \\
\cos^2(x) + \sin^2(x) = 1 & \text{ and } \cos(2x) = \cos^2(x) - \sin^2(x) \\
\end{align}
</math>
The radian is the unit that leads to the expansion with leading coefficient 1 for the sine and is determined by the additional requirement that
:<math>
\sin(x) \approx x \text{ when } x \approx 0.
</math>
The coefficients for both the sine and cosine series may therefore be derived by substituting their expansions into the pythagorean and double angle identities, taking the leading coefficient for the sine to be 1, and matching the remaining coefficients.

In general, mathematically important relationships between the sine and cosine functions and the [[exponential function]] (see, for example, [[Euler's formula]]) are substantially simplified when angles are expressed in radians, rather than in degrees, grads or other units. Therefore, in most branches of mathematics beyond practical geometry, angles are generally assumed to be expressed in radians.

A similar series is [[Gregory's series]] for [[arctan]], which is obtained by omitting the factorials in the denominator.

===Continued fraction===
The sine function can also be represented as a [[generalized continued fraction]]:

:<math> \sin(x) =
\cfrac{x}{1 + \cfrac{x^2}{2\cdot3-x^2 +
\cfrac{2\cdot3 x^2}{4\cdot5-x^2 +
\cfrac{4\cdot5 x^2}{6\cdot7-x^2 + \ddots}}}}.
</math>

The continued fraction representation can be derived from [[Euler's continued fraction formula]] and expresses the [[real number]] values, both [[rational number|rational]] and [[irrational number|irrational]], of the sine function.

==Fixed point==
[[Image:Sine fixed point.svg|thumb|The fixed point iteration ''x''<sub>''n''+1</sub>&nbsp;=&nbsp;sin(''x<sub>n</sub>'') with initial value ''x''<sub>0</sub> = 2 converges to 0.]]
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 sin(0) = 0.
{{clear}}

==Arc length==
The arc length of the sine curve between <math>a</math> and <math>b</math> is <math display="inline"> \int_a^b\!\sqrt{1+\cos^2(x)}\, dx </math>.
This integral is an [[Legendre form|elliptic integral of the second kind]].

The arc length for a full period is <math display="inline">\frac{4\sqrt{2\pi ^3}}{\Gamma(1/4)^2} + \frac{\Gamma(1/4)^2}{\sqrt{2\pi}}=7.640395578\ldots</math>
where <math>\Gamma</math> is the [[Gamma function]].

The arc length of the sine curve from ''0'' to ''x'' is the above number divided by <math>2\pi</math> times ''x'', plus a correction
that varies periodically in ''x'' with period <math>\pi</math>.  The [[Fourier series]] for this correction
can be written in closed form using special functions, but it is perhaps more instructive to write the decimal approximations of the Fourier coefficients.
The sine curve arc length from ''0'' to ''x'' is

:<math>
1.21600672x + 0.10317093 \sin(2x)-0.00220445\sin(4x)+0.00012584\sin(6x)-0.00001011\sin(8 x)+\cdots
</math>

==Law of sines==
{{Main|Law of sines}}
The [[law of sines]] states that for an arbitrary [[triangle]] with sides ''a'', ''b'', and ''c'' and angles opposite those sides ''A'', ''B'' and ''C'':

:<math>\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}.</math>

This is equivalent to the equality of the first three expressions below:

:<math>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,</math>

where ''R'' is the triangle's [[circumscribed circle|circumradius]].

It can be proven by dividing the triangle into two right ones and using the above definition of sine. 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. This is a common situation occurring in ''[[triangulation]]'', a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

== Special values ==
{{See also|Exact trigonometric constants}}
[[Image:Unit circle angles.svg|right|thumb|350px|Some common angles (''θ'') shown on the [[unit circle]]. The angles are given in degrees and radians, together with the corresponding intersection point on the unit circle, (cos(''θ''), sin(''θ'')).]]
For certain integral numbers ''x'' of degrees, the value of sin(''x'') is particularly simple.  A table of some of these values is given below.
{| border="1" cellpadding="5" cellspacing="0"
|-----
! style="background:#ffdead;" colspan="4" | ''x'' (angle)
! colspan="2" style="background:#ffdead;" | sin(''x'')
|-----
| style="background:#efefef;" align=Center | [[Degree (angle)|Degree]]s
| style="background:#efefef;" | [[Radian]]s
| style="background:#efefef;" align=Center | [[Gradian]]s
| style="background:#efefef;" align=Center | [[Turn (geometry)|Turn]]s
| style="background:#efefef;" align=Center | Exact
| style="background:#efefef;" | Decimal
|-----
| align=Center | 0° || align="Center" | 0
| align=Center | 0<sup>g</sup>
| align=Center | 0
| align=Center rowspan=2 | 0 || rowspan=2 | 0
|-----
| align=Center | 180° || align=Center | π
| align=Center | 200<sup>g</sup>
| align=Center | {{sfrac|1|2}}
|-----
| align=Center | 15°
| align=Center | {{sfrac|1|12}}π
| align=Center | {{sfrac|16|2|3}}<sup>g</sup>
| align=Center | {{sfrac|1|24}}
| align=Center rowspan=2 | <math>\frac{\sqrt{6}-\sqrt{2}}{4}</math>
| rowspan=2 | 0.258819045102521
|-----
| align=Center | 165°
| align=Center | {{sfrac|11|12}}π
| align=Center | {{nowrap|{{sfrac|183|1|3}}<sup>g</sup>}}
| align=Center | {{sfrac|11|24}}
|-----
| align=Center | 30°
| align=Center | {{sfrac|1|6}}π
| align=Center | {{nowrap|{{sfrac|33|1|3}}<sup>g</sup>}}
| align=Center | {{sfrac|1|12}}
| align=Center rowspan=2 | {{sfrac|1|2}}
| rowspan=2 | 0.5
|-----
| align=Center | 150°
| align=Center | {{sfrac|5|6}}π
| align=Center | {{nowrap|{{sfrac|166|2|3}}<sup>g</sup>}}
| align=Center | {{sfrac|5|12}}
|-----
| align=Center | 45°
| align=Center | {{sfrac|1|4}}π
| align=Center | {{nowrap|50<sup>g</sup>}}
| align=Center | {{sfrac|1|8}}
| align=Center rowspan=2 | <math>\frac{\sqrt{2}}{2}</math>
| rowspan=2 | 0.707106781186548
|-----
| align=Center | 135°
| align=Center | {{sfrac|3|4}}π
| align=Center | {{nowrap|150<sup>g</sup>}}
| align=Center | {{sfrac|3|8}}
|-----
| align=Center | 60°
| align=Center | {{sfrac|1|3}}π
| align=Center | {{nowrap|{{sfrac|66|2|3}}<sup>g</sup>}}
| align=Center | {{sfrac|1|6}}
| align=Center rowspan=2 | <math>\frac{\sqrt{3}}{2}</math>
| rowspan=2 | 0.866025403784439
|-----
| align=Center | 120°
| align=Center | {{sfrac|2|3}}π
| align=Center | {{nowrap|{{sfrac|133|1|3}}<sup>g</sup>}}
| align=Center | {{sfrac|1|3}}
|-----
| align=Center | 75°
| align=Center | {{sfrac|5|12}}π
| align=Center | {{nowrap|{{sfrac|83|1|3}}<sup>g</sup>}}
| align=Center | {{sfrac|5|24}}
| align=Center rowspan=2 | <math>\frac{\sqrt{6}+\sqrt{2}}{4}</math>
| rowspan=2 | 0.965925826289068
|-----
| align=Center | 105°
| align=Center | {{sfrac|7|12}}π
| align=Center | {{nowrap|{{sfrac|116|2|3}}<sup>g</sup>}}
| align=Center | {{sfrac|7|24}}
|-----
| align=Center | 90°
| align=Center | {{sfrac|1|2}}π
| align=Center | {{nowrap|100<sup>g</sup>}}
| align=Center | {{sfrac|1|4}}
| align=Center | 1
| 1
|}

90 degree increments:

{| border="1" cellpadding="5" cellspacing="0"
|- bgcolor="white" align="center"
|''x'' in degrees || 0° || 90° || 180° || 270° || 360°
|- bgcolor="white" align="center"
|''x'' in radians || 0 || π/2 || π || 3π/2 || 2π
|- bgcolor="white" align="center"
|''x'' in gons || 0 || {{nowrap|100<sup>g</sup>}} || {{nowrap|200<sup>g</sup>}} || {{nowrap|300<sup>g</sup>}} || {{nowrap|400<sup>g</sup>}}
|- bgcolor="white" align="center"
|''x'' in turns || 0 || 1/4 || 1/2 || 3/4 || 1
|- bgcolor="white" align="center"
|sin ''x''|| 0 || 1 || 0 || -1 || 0
|}

Other values not listed above:
: <math>\sin\left(\frac{\pi}{60}\right) = \sin(3^\circ)=\frac{(2-\sqrt{12})\sqrt{5+\sqrt5}+(\sqrt{10}-\sqrt2)(\sqrt3+1)}{16}</math> {{OEIS2C|A019812}}
: <math>\sin\left(\frac{\pi}{30}\right) = \sin(6^\circ)=\frac{\sqrt{30-\sqrt{180}}-\sqrt5-1}{8}</math> {{OEIS2C|A019815}}
: <math>\sin\left(\frac{\pi}{20}\right) = \sin(9^\circ)=\frac{\sqrt{10}+\sqrt2-\sqrt{20-\sqrt{80}}}{8}</math> {{OEIS2C|A019818}}
: <math>\sin\left(\frac{\pi}{15}\right) = \sin(12^\circ)=\frac{\sqrt{10+\sqrt{20}}+\sqrt{3}-\sqrt{15}}{8}</math> {{OEIS2C|A019821}}
: <math>\sin\left(\frac{\pi}{10}\right) = \sin(18^\circ)=\frac{\sqrt5-1}{4}=\tfrac{1}{2}\varphi^{-1}</math> {{OEIS2C|A019827}}
: <math>\sin\left(\frac{7\pi}{60}\right) = \sin(21^\circ)=\frac{(2+\sqrt{12})\sqrt{5-\sqrt5}-(\sqrt{10}+\sqrt2)(\sqrt3-1)}{16}</math> {{OEIS2C|A019830}}
: <math>\sin\left(\frac{\pi}{8}\right) = \sin(22.5^\circ)=\frac{\sqrt{2-\sqrt2}}{2}</math>
: <math>\sin\left(\frac{2\pi}{15}\right) = \sin(24^\circ)=\frac{\sqrt3+\sqrt{15}-\sqrt{10-\sqrt{20}}}{8}</math> {{OEIS2C|A019833}}
: <math>\sin\left(\frac{3\pi}{20}\right) = \sin(27^\circ)=\frac{\sqrt{20+\sqrt{80}}-\sqrt{10}+\sqrt2}{8}</math> {{OEIS2C|A019836}}
: <math>\sin\left(\frac{11\pi}{60}\right) = \sin(33^\circ)=\frac{(\sqrt{12}-2)\sqrt{5+\sqrt5}+(\sqrt{10}-\sqrt2)(\sqrt3+1)}{16}</math> {{OEIS2C|A019842}}
: <math>\sin\left(\frac{\pi}{5}\right) = \sin(36^\circ)=\frac{\sqrt{10-\sqrt{20}}}{4}</math> {{OEIS2C|A019845}}
: <math>\sin\left(\frac{13\pi}{60}\right) = \sin(39^\circ)=\frac{(2-\sqrt{12})\sqrt{5-\sqrt5}+(\sqrt{10}+\sqrt2)(\sqrt3+1)}{16}</math> {{OEIS2C|A019848}}
: <math>\sin\left(\frac{7\pi}{30}\right) = \sin(42^\circ)=\frac{\sqrt{30+\sqrt{180}}-\sqrt5+1}{8}</math> {{OEIS2C|A019851}}

==Relationship to complex numbers==
{{Main|Trigonometric functions#Relationship to exponential function and complex numbers}}
[[File:Complex picture.svg|thumb|An illustration of the [[complex plane]]. The [[imaginary number]]s are on the vertical coordinate axis.]]
Sine is used to determine the [[imaginary part]] of a [[complex number]] given in [[polar coordinates]] (''r'', ''φ''):

:<math> z = r(\cos(\varphi) + i\sin(\varphi))</math>

the imaginary part is:

:<math>\operatorname{Im}(z) = r \sin(\varphi)</math>

''r'' and ''φ'' represent the magnitude and angle of the complex number respectively. ''i'' is the [[imaginary unit]]. ''z'' is a [[complex number]].

Although dealing with complex numbers, sine's parameter in this usage is still a [[real number]]. Sine can also take a complex number as an argument.
{{clr}}

===Sine with a complex argument===
[[File:Complex_sin.jpg|thumb|<math>\sin(z)</math><br/><br/> [[Domain coloring]] of sin(''z'') in the complex plane. Brightness indicates absolute magnitude, saturation represents complex argument.]]
[[File:Sin z vector field 02 Pengo.svg|thumb|sin(''z'') as a vector field]]
[[File:Sinus und Kosinus am Einheitskreis 3.svg|thumb|<math>\sin(\theta)</math> is the imaginary part of <math>e^{i\theta}</math>.]]

The definition of the sine function for complex arguments ''z'':

: <math>\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}
\end{align}</math>

where ''i''<sup>&nbsp;2</sup>&nbsp;=&nbsp;−1, and sinh is [[hyperbolic function|hyperbolic sine]]. This is an [[entire function]]. Also, for purely real ''x'',

: <math>\sin(x) = \operatorname{Im}(e^{i x}). </math>

For purely imaginary numbers:
:<math> \sin(iy) = i \sinh(y). </math>

It is also sometimes useful to express the complex sine function in terms of the real and imaginary parts of its argument:

: <math>\begin{align}
\sin (x + iy) &= \sin(x) \cos(iy) + \cos(x) \sin(iy) \\
&= \sin(x) \cosh(y) + i \cos(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>\begin{align}
\sum_{n = -\infty}^{\infty}\frac{(-1)^n}{z-n} = \frac{1}{z} -2z \sum_{n = 1}^{\infty}\frac{(-1)^n}{n^2-z^2}
\end{align}</math>

both converge and are equal to <math display="inline">\frac{\pi}{\sin (\pi z)}</math>.  Similarly, one can show that

: <math>\begin{align}
\frac{\pi^2}{\sin^2 (\pi z)} = \sum_{n=-\infty}^\infty \frac{1}{(z-n)^2}.
\end{align}</math>

Using product expansion technique, one can derive

: <math>\begin{align}
\sin(\pi z) = \pi z \prod_{n = 1}^\infty \left( 1- \frac{z^2}{n^2} \right).
\end{align}</math>

Alternatively, the infinite product for the sine can be proved using [[Fourier series#Complex-valued_functions|complex Fourier series]].

{{Collapse top|title=Proof of the infinite product for the sine}}

Using complex Fourier series, the function <math>\cos (zx)</math> can be decomposed as

:<math>\cos (zx)=\frac{z\sin (\pi z)}{\pi}\displaystyle\sum_{n=-\infty}^\infty \frac{(-1)^{n} \, e^{inx}}{z^2-n^2}, \, z\in\C\setminus\{\Z\}, \, x\in [-\pi ,\pi].</math>

Setting <math>x=\pi</math> yields

:<math>\cos (\pi z)=\frac{z\sin (\pi z)}{\pi}\displaystyle\sum_{n=-\infty}^\infty \frac{1}{z^2-n^2}=\frac{z\sin (\pi z)}{\pi}\left(\frac{1}{z^2}+2\displaystyle\sum_{n=1}^\infty \frac{1}{z^2-n^2}\right).</math>

Therefore we get

:<math>\pi \cot (\pi z)=\frac{1}{z}+2\displaystyle\sum_{n=1}^\infty \frac{z}{z^2-n^2}.</math>

The function <math>\pi \cot (\pi z)</math> is the derivative of <math>\ln (\sin (\pi z))+C_0</math>. Furthermore, if <math display="inline">\frac{df}{dz}=\frac{z}{z^2-n^2}</math>, then the function <math>f</math> such that the emerged series converges is <math display="inline">f=\frac{1}{2}\ln (1-z^2/n^2)+C_1</math>, which can be proved using the [[Weierstrass M-test]]. It follows that

:<math>\ln (\sin (\pi z))=\ln (z)+\displaystyle\sum_{n=1}^\infty \ln \left(1-\frac{z^2}{n^2}\right)+C.</math>

Exponentiating gives

:<math>\sin (\pi z)=ze^C \displaystyle\prod_{n=1}^\infty \left(1-\frac{z^2}{n^2}\right).</math>

Since <math display="inline">\lim_{z\to 0}\frac{\sin (\pi z)}{z}=\pi</math> and <math display="inline">\lim_{z\to 0}\prod_{n=1}^\infty \left(1-\frac{z^2}{n^2}\right)=1</math>, we have <math>e^C =\pi</math>. Hence

:<math>\sin (\pi z)=\pi z\displaystyle\prod_{n=1}^\infty \left(1-\frac{z^2}{n^2}\right).</math>

Let <math>\textstyle{a_{n}(z)=-\frac{z^2}{n^2}}</math>. Since <math>\textstyle{\sum_{n=1}^\infty |a_{n}(z)|}</math> [[Uniform convergence|converges uniformly]] on any closed disk, <math>\textstyle{\prod_{n=1}^\infty (1+a_{n}(z))}</math> converges uniformly on any closed disk as well. Therefore the infinite product for the sine is valid for all <math>z\in\mathbb C</math>, which completes the proof. <math>\blacksquare </math>
{{Collapse bottom}}

==== Usage of complex sine ====
sin(''z'') is found in the [[functional equation]] for the [[Gamma function]],
:<math>\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>\zeta(s)=2(2\pi)^{s-1}\Gamma(1-s)\sin(\pi s/2)\zeta(1-s).</math>

As a [[holomorphic function]], ''sin z'' is a 2D solution of [[Laplace's equation]]:
:<math>\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===

{| style="text-align:center"
|+ '''Sine function in the complex plane'''
|[[File:Complex sin real 01 Pengo.svg|1000x130px|none]]
|[[File:Complex sin imag 01 Pengo.svg|1000x130px|none]]
|[[File:Complex sin abs 01 Pengo.svg|1000x130px|none]]
|-
|real component
|imaginary component
|magnitude
|}
<br />
{| style="text-align:center"
|+ '''Arcsine function in the complex plane'''
|[[File:Complex arcsin real 01 Pengo.svg|1000x130px|none]]
|[[File:Complex arcsin imag 01 Pengo.svg|1000x130px|none]]
|[[File:Complex arcsin abs 01 Pengo.svg|1000x130px|none]]
|-
|real component
|imaginary component
|magnitude
|}

== History ==
{{Main|Trigonometric functions#History|History of trigonometry}}

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).

The function of sine and [[versine]] (1 - cosine) can be traced to the [[Jyā, koti-jyā and utkrama-jyā|''jyā'' and ''koṭi-jyā '']] functions used in [[Gupta period]] (320 to 550 CE) [[Indian astronomy]] (''[[Aryabhatiya]]'', ''[[Surya Siddhanta]]''), via translation from Sanskrit to Arabic and then from Arabic to Latin.<ref name="Boyer, Carl B. 1991 p. 210"/>

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"/> With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant.<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 |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"/>

The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician [[Albert Girard]]; 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 sin ''x'' is not an [[algebraic function]] of ''x''.<ref>{{cite book|title=Elements of the History of Mathematics|url=https://archive.org/details/elementsofhistor0000bour|url-access=registration|author=Nicolás Bourbaki|publisher=Springer|year=1994}}</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.''<ref name=boyer>See Merzbach, Boyer (2011).</ref>

=== Etymology ===
{{Wiktionary|sine}}
[[Etymology|Etymologically]], the word ''sine'' derives from the [[Sanskrit]] word for chord, ''jiva''*(''jya'' being its more popular synonym). This was [[transliteration|transliterated]] in [[Arabic language|Arabic]] as ''jiba'' جيب, which however is meaningless in that language and abbreviated ''jb'' جب . Since Arabic is written without short vowels, "jb" was interpreted as the word ''jaib'' جيب, which means "bosom". When the Arabic texts were translated in the 12th century into [[Medieval Latin|Latin]] by [[Gerard of Cremona]], he used the Latin equivalent for "bosom", ''[[wikt:sinus|sinus]]'' (which means "bosom" or "bay" or "fold").<ref>Eli Maor (1998), ''Trigonometric Delights'', Princeton: Princeton University Press, p. 35-36.</ref><ref>Victor J. Katz (2008), ''A History of Mathematics'', Boston: Addison-Wesley, 3rd. ed., p. 253, sidebar 8.1. {{cite web |url=http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf |title=Archived copy |accessdate=2015-04-09 |url-status=live |archiveurl=https://web.archive.org/web/20150414065531/http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf |archivedate=2015-04-14}}</ref> 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.<ref>{{citation|first=D.E.|last=Smith|title=History of Mathematics|volume=I|year=1958|origyear=1925|publisher=Dover|page=202|isbn=0-486-20429-4}}</ref> The English form ''sine'' was introduced in the 1590s.

== Software implementations ==
{{See also|Lookup table#Computing sines}}
The sine function, along with other trigonometric functions, is widely available across programming languages and platforms. In computing, it is typically abbreviated to <code>sin</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> is typically either a built-in function or found within the language's standard math library.

For example, the [[C standard library]] defines sine functions within [[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> within the built-in <code>math</code> module. Complex sine 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]].

There is no standard algorithm for calculating sine. [[IEEE 754-2008]], the most widely used standard for floating-point computation, does not address calculating trigonometric functions such as sine.<ref>Grand Challenges of Informatics, Paul Zimmermann. September 20, 2006 – p. 14/31 {{cite web |url=http://www.jaist.ac.jp/~bjorner/ae-is-budapest/talks/Sept20pm2_Zimmermann.pdf |title=Archived copy |accessdate=2010-09-11 |url-status=live |archiveurl=https://web.archive.org/web/20110716205108/http://www.jaist.ac.jp/~bjorner/ae-is-budapest/talks/Sept20pm2_Zimmermann.pdf |archivedate=2011-07-16 }}</ref> 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 once common programming optimization, used especially in 3D graphics, was to pre-calculate a table of sine values, for example one value per degree. This allowed 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.

=== Turns based implementations ===

Some software libraries provide implementations of sine using the input angle in half-Turns. Representing angles in Turns or half-Turns has accuracy advantages and efficiency advantages in some cases.
<ref>"[https://www.mathworks.com/help/matlab/ref/double.sinpi.html MATLAB Documentation sinpi]</ref>
<ref>"[https://www.rdocumentation.org/packages/base/versions/3.5.3/topics/Trig R Documentation sinpi]</ref>

{| class="wikitable sortable"
|-
! Environment !! function name !! angle units
|-
| MATLAB
|| sinpi
<ref>"[https://www.mathworks.com/help/matlab/ref/double.sinpi.html MATLAB Documentation sinpi]</ref>
|| half-Turns
|-
| OpenCL
|| sinpi
<ref>"[https://www.khronos.org/registry/OpenCL/sdk/1.0/docs/man/xhtml/sin.html OpenCL Documentation sinpi]</ref>
|| half-Turns
|-
| R
|| sinpi
<ref>"[https://www.rdocumentation.org/packages/base/versions/3.5.3/topics/Trig R Documentation sinpi]</ref>
|| half-Turns
|-
| Julia
|| sinpi
<ref>"[http://www.jlhub.com/julia/manual/en/function/sinpi Julia Documentation sinpi]</ref>
|| half-Turns
|-
| CUDA
|| sinpi
<ref>"[https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__DOUBLE.html CUDA Documentation sinpi]</ref>
|| half-Turns
|-
| ARM
|| sinpi
<ref>"[https://developer.arm.com/docs/100614/latest/b-opencl-built-in-functions/b2-math-functions ARM Documentation sinpi]</ref>
|| half-Turns
|}

The accuracy advantage stems from the ability to perfectly represent key angles like full-Turn, half-Turn, and quarter-Turn losslessly in binary floating-point or fixed-point. In contrast, representing 2*pi, pi, and pi/2 in binary floating-point or binary scaled fixed-point always involves a loss of accuracy. 

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 2*pi involves inaccuracies in representing 2*pi.

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>"[https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx ALLEGRO Angle Sensor Datasheet]</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 pi/2048 would be incurred.

== See also ==
{{div col|colwidth=20em}}
* [[Āryabhaṭa's sine table]]
* [[Bhaskara I's sine approximation formula]]
* [[Discrete sine transform]]
* [[Euler's formula]]
* [[Generalized trigonometry]]
* [[Hyperbolic function]]
* [[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]]
* [[Trigonometric functions]]
{{div col end}}

== Citations ==
{{Reflist}}

== References ==
* {{citation | first1 = John C. | last1 = Traupman | year = 1966 | title = The New College Latin & English Dictionary | publisher = Bantam | location = Toronto | isbn = 0-553-27619-0 | url-access = registration | url = https://archive.org/details/boysgirlsbookabo00gard_0 }}
* {{citation | title = Webster's Seventh New Collegiate Dictionary | location = Springfield | publisher =G. & C. Merriam Company | date = 1969 | ref = {{harvid|Webster|1969}} }}

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

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

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