Sine and cosine

"Sine" and "Cosine" redirect here. For other uses, seeSine (disambiguation)andCosine (disambiguation)."Sine" is not to be confused withSign,Sign (mathematics)or thesign function.

Sine and cosine
General information
General definition	${\begin{aligned}&\sin(\ Alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\[8pt]&\cos(\ Alpha )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\[8pt]\end{aligned}}$
Fields of application	Trigonometry,Fourier series,etc.

Inmathematics,sineandcosinearetrigonometric functionsof anangle.The sine and cosine of an acuteangleare defined in the context of aright triangle:for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of thetriangle(thehypotenuse), and the cosine is theratioof the length of the adjacent leg to that of thehypotenuse.For an angle$\theta$,the sine and cosine functions are denoted as$\sin \theta$and$\cos \theta$.

The definitions of sine and cosine have been extended to anyrealvalue in terms of the lengths of certain line segments in aunit circle.More modern definitions express the sine and cosine asinfinite series,or as the solutions of certaindifferential equations,allowing their extension to arbitrary positive and negative values and even tocomplex numbers.

The sine and cosine functions are commonly used to modelperiodicphenomena such assoundandlight 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 thejyāandkoṭi-jyāfunctions used inIndian astronomyduring theGupta period.

Notation

Main article:Trigonometric functions § Notation

Sine and cosine are written usingfunctional notationwith the abbreviationssinandcos.

Often, if the argument is simple enough, the function value will be written without parentheses, assinθrather than assin(θ).

Each of sine and cosine is a function of an angle, which is usually expressed in terms ofradiansordegrees.Except where explicitly stated otherwise, this article assumes that the angle is measured in radians.

Definitions

Right-angled triangle definitions

For the angleα,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 angleα,start with aright trianglethat contains an angle of measureα;in the accompanying figure, angleαin triangleABCis the angle of interest. The three sides of the triangle are named as follows:

• Theopposite sideis the side opposite to the angle of interest, in this case sidea.
• Thehypotenuseis the side opposite the right angle, in this case sideh.The hypotenuse is always the longest side of a right-angled triangle.
• Theadjacent sideis the remaining side, in this case sideb.It forms a side of (and is adjacent to) both the angle of interest (angleA) 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:

$\sin(\ Alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\qquad \cos(\ Alpha )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}$

The other trigonometric functions of the angle can be defined similarly; for example, thetangentis the ratio between the opposite and adjacent sides.

As stated, the values$\sin(\ Alpha )$and$\cos(\ Alpha )$appear to depend on the choice of right triangle containing an angle of measureα.However, this is not the case: all such triangles aresimilar,and so the ratios are the same for each of them.

Unit circle definitions

Intrigonometry,aunit circleis the circle of radius one centered at the origin (0, 0) in theCartesian coordinate system.

Unit circle: a circle with radius one

Let a line through the origin intersect the unit circle, making an angle ofθwith the positive half of thex-axis. Thex- andy-coordinates of this point of intersection are equal tocos(θ)andsin(θ),respectively. This definition is consistent with the right-angled triangle definition of sine and cosine when$0<\theta <{\frac {\pi }{2}}$:because the length of the hypotenuse of the unit circle is always 1,$\sin(\theta )={\frac {\text{opposite}}{\text{hypotenuse}}}={\frac {\text{opposite}}{1}}={\text{opposite}}$.The length of the opposite side of the triangle is simply they-coordinate. A similar argument can be made for the cosine function to show that$\cos(\theta )={\frac {\text{adjacent}}{\text{hypotenuse}}}$when$0<\theta <{\frac {\pi }{2}}$,even under the new definition using the unit circle.tan(θ)is then defined as${\frac {\sin(\theta )}{\cos(\theta )}}$,or, equivalently, as the slope of the line segment.

Using the unit circle definition has the advantage that the angle can be extended to any real argument. This can also be achieved by requiring certain symmetries, and that sine be aperiodic function.

Complex exponential function definitions

Main article:Euler's formula

Theexponential function$e^{z}$is defined on the entire domain of thecomplex numbers.The definition of sine and cosine can be extended to all complex numbers via

$\sin z={\frac {e^{iz}-e^{-iz}}{2i}}$

$\cos z={\frac {e^{iz}+e^{-iz}}{2}}$

These can be reversed to giveEuler's formula

$e^{iz}=\cos z+i\sin z$

$e^{-iz}=\cos z-i\sin z$

When plotted on thecomplex plane,the function$e^{ix}$for real values of$x$traces out theunit circlein the complex plane.

When$x$is a real number, sine and cosine simplify to the imaginary and real parts of$e^{ix}$or$e^{-ix}$,as:

$\sin x=\operatorname {Im} (e^{ix})=-\operatorname {Im} (e^{-ix})$

$\cos x=\operatorname {Re} (e^{ix})=\operatorname {Re} (e^{-ix})$

When$z=x+iy$for real values$x$and$y$,sine and cosine can be expressed in terms of real sines, cosines, andhyperbolic functionsas

${\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}$

Differential equation definition

$(\cos \theta,\sin \theta )$is the solution$(x(\theta ),y(\theta ))$to the two-dimensional system ofdifferential equations$y'(\theta )=x(\theta )$and$x'(\theta )=-y(\theta )$with theinitial conditions$y(0)=0$and$x(0)=1$.One could interpret the unit circle in the above definitions as defining thephase space trajectoryof the differential equation with the given initial conditions.

• 
  Animation demonstrating how the sine function (in red) is graphed from they-coordinate (red dot) of a point on theunit circle(in green), at an angle ofθ.The cosine (in blue) is thex-coordinate.

It can be interpreted as a phase space trajectory of the system of differential equations$y'(\theta )=x(\theta )$and$x'(\theta )=-y(\theta )$starting from the initial conditions$y(0)=0$and$x(0)=1$.

Series definitions

The sine function (blue) is closely approximated by itsTaylor polynomialof degree 7 (pink) for a full cycle centered on the origin.

This animation shows how including more and more terms in the partial sum of its Taylor series approaches a sine curve.

The successive derivatives of sine, evaluated at zero, can be used to determine its Taylor series. Using only geometry and properties oflimits,it can be shown that thederivativeof sine is cosine, and that the derivative of cosine is the negative of sine. This means the successive derivatives of sin(x) are cos(x), -sin(x), -cos(x), sin(x), continuing to repeat those four functions. The (4n+k)-th derivative, evaluated at the point 0:

$\sin ^{(4n+k)}(0)={\begin{cases}0&{\text{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\-1&{\text{when }}k=3\end{cases}}$

where the superscript represents repeated differentiation. This implies the following Taylor series expansion at x = 0. One can then use the theory ofTaylor seriesto show that the following identities hold for allreal numbersx(where x is the angle in radians), and more generally for allcomplex numbers:

${\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[5mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\end{aligned}}$

Taking the derivative of each term gives the Taylor series for cosine:

${\begin{aligned}\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[5mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\end{aligned}}$

These Taylor series provide explicit,computable,definitions of sine and cosine in terms ofelementary arithmeticoperations.

Continued fraction definitions

The sine function can also be represented as ageneralized continued fraction:

$\sin(x)={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}.$

$\cos(x)={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}.$

The continued fraction representations can be derived fromEuler's continued fraction formulaand express thereal numbervalues, bothrationalandirrational,of the sine and cosine functions.

Identities

Main article:List of trigonometric identities

Exact identities (usingradians):

These apply for all values of$\theta$.

$\sin(\theta )=\cos \left({\frac {\pi }{2}}-\theta \right)=\cos \left(\theta -{\frac {\pi }{2}}\right)$

$\cos(\theta )=\sin \left({\frac {\pi }{2}}-\theta \right)=\sin \left(\theta +{\frac {\pi }{2}}\right)$

Reciprocals

Thereciprocalof sine is cosecant, i.e., the reciprocal of$\sin {\theta }$is$\csc {\theta }$.Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the length of the hypotenuse to that of the adjacent side.

$\csc {\theta }={\frac {1}{\sin {\theta }}}={\frac {\textrm {hypotenuse}}{\textrm {opposite}}}$

$\sec {\theta }={\frac {1}{\cos {\theta }}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}$

Inverses

The usual principal values of thearcsin(x)andarccos(x)functions graphed on the Cartesian plane

Theinverse functionof sine is arcsine (arcsin or asin) or inverse sine (sin⁻¹). The inverse function of cosine is arccosine (arccos, acos, orcos⁻¹). (The superscript of −1 insin⁻¹andcos⁻¹denotes the inverse of a function, notexponentiation.) As sine and cosine are notinjective,their inverses are not exact inverse functions, but partial inverse functions. For example,sin(0) = 0,but alsosin(π) = 0,sin(2π) = 0etc. It follows that the arcsine function is multivalued:arcsin(0) = 0,but alsoarcsin(0) =π,arcsin(0) = 2π,etc. When only one value is desired, the function may be restricted to itsprincipal branch.With this restriction, for eachxin the domain, the expressionarcsin(x)will evaluate only to a single value, called itsprincipal value.The standard range of principal values for arcsin is from−π/2toπ/2and the standard range for arccos is from0toπ.

$\theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right).$

where (for some integerk):

${\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ or }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ or }}\\&y=-\arccos(x)+2\pi k\end{aligned}}$

By definition, arcsin and arccos satisfy the equations:

$\sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x$

and

${\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{for}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{for}}\quad 0\leq \theta \leq \pi \end{aligned}}$

Pythagorean trigonometric identity

The basic relationship between the sine and the cosine is thePythagorean trigonometric identity:

$\cos ^{2}(\theta )+\sin ^{2}(\theta )=1$

where sin²(x) means (sin(x))².

Double angle formulas

Sine and cosine satisfy the following double angle formulas:

$\sin(2\theta )=2\sin(\theta )\cos(\theta )$

$\cos(2\theta )=\cos ^{2}(\theta )-\sin ^{2}(\theta )=2\cos ^{2}(\theta )-1=1-2\sin ^{2}(\theta )$

Sine function in blue and sine squared function in red. The X axis is in radians.

The cosine double angle formula implies that sin²and cos²are, themselves, shifted and scaled sine waves. Specifically,

$\sin ^{2}(\theta )={\frac {1-\cos(2\theta )}{2}}\qquad \cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}$

The graph shows both the sine function and thesine squaredfunction, 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.

Derivative and integrals

See also:List of integrals of trigonometric functionsandDifferentiation of trigonometric functions

The derivatives of sine and cosine are:

${\frac {d}{dx}}\sin(x)=\cos(x)\qquad {\frac {d}{dx}}\cos(x)=-\sin(x)$

and their antiderivatives are:

$\int \sin(x)\,dx=-\cos(x)+C$

$\int \cos(x)\,dx=\sin(x)+C$

whereCdenotes theconstant of integration.

Properties relating to the quadrants

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$\sin(\ Alpha +2\pi )=\sin(\ Alpha )$of the sine function.

Quadrant	Angle		Sine			Cosine
	Degrees	Radians	Sign	Monotony	Convexity	Sign	Monotony	Convexity
1st quadrant, I	$0^{\circ }<x<90^{\circ }$	$0<x<{\frac {\pi }{2}}$	$+$	increasing	concave	$+$	decreasing	concave
2nd quadrant, II	$90^{\circ }<x<180^{\circ }$	${\frac {\pi }{2}}<x<\pi$	$+$	decreasing	concave	$-$	decreasing	convex
3rd quadrant, III	$180^{\circ }<x<270^{\circ }$	$\pi <x<{\frac {3\pi }{2}}$	$-$	decreasing	convex	$-$	increasing	convex
4th quadrant, IV	$270^{\circ }<x<360^{\circ }$	${\frac {3\pi }{2}}<x<2\pi$	$-$	increasing	convex	$+$	increasing	concave

The quadrants of the unit circle and of sin(x), using theCartesian coordinate system

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

Degrees	Radians	$\sin(x)$		$\cos(x)$
		Value	Point type	Value	Point type
$0^{\circ }$	$0$	$0$	Root,inflection	$1$	Maximum
$90^{\circ }$	${\frac {\pi }{2}}$	$1$	Maximum	$0$	Root,inflection
$180^{\circ }$	$\pi$	$0$	Root,inflection	$-1$	Minimum
$270^{\circ }$	${\frac {3\pi }{2}}$	$-1$	Minimum	$0$	Root,inflection

Fixed points

Main article:Dottie number

The fixed point iterationx_n+1= cos(x_n) with initial valuex₀= −1 converges to the Dottie number.

Zero is the only realfixed pointof the sine function; in other words the only intersection of the sine function and theidentity functionis$\sin(0)=0$.The only real fixed point of the cosine function is called theDottie number.That is, the Dottie number is the unique real root of the equation$\cos(x)=x.$The decimal expansion of the Dottie number is$0.739085\ldots$.

Arc length

Thearc lengthof the sine curve between$0$and$t$is

$\int _{0}^{t}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}\operatorname {E} (t,1/{\sqrt {2}}),$

where$\operatorname {E} (\varphi,k)$is theincomplete elliptic integral of the second kindwith modulus$k$.It cannot be expressed usingelementary functions.

The arc length for a full period is

$L={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}={\frac {2\pi }{\varpi }}+2\varpi =7.640395578\ldots$

where$\Gamma$is thegamma functionand$\varpi$is thelemniscate constant.

Laws

Main articles:Law of sinesandLaw of cosines

Thelaw of sinesstates that for an arbitrary triangle with sidesa,b,andcand angles opposite those sidesA,BandC:

${\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}.$

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

${\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,$

whereRis the triangle'scircumradius.

It can be proved 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 intriangulation,a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

Thelaw of cosinesstates that for an arbitrary triangle with sidesa,b,andcand angles opposite those sidesA,BandC:

$a^{2}+b^{2}-2ab\cos(C)=c^{2}$

In the case where$C=\pi /2$,$\cos(C)=0$and this becomes thePythagorean theorem:for a right triangle,$a^{2}+b^{2}=c^{2},$wherecis the hypotenuse.

Special values

Some common angles (θ) shown on theunit circle.The angles are given in degrees and radians, together with the corresponding intersection point on the unit circle, (cos(θ), sin(θ)).

Forinteger multiplesof 15° (that is,$\textstyle {\frac {\pi }{12}}$radians), the values of sin(x) and cos(x) are particularly simple and can be expressed in terms of${\sqrt {2}},{\sqrt {3}},{\sqrt {6}}$only. A table of these angles is given below. For more complex angle expressions seeExact trigonometric values § Common angles.

Angle,x				sin(x)		cos(x)
Degrees	Radians	Gradians	Turns	Exact	Decimal	Exact	Decimal
0°	0	0g	0	0	0	1	1
15°	⁠1/12⁠π	⁠16+2/3⁠g	⁠1/24⁠	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	0.2588	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	0.9659
30°	⁠1/6⁠π	⁠33+1/3⁠g	⁠1/12⁠	⁠1/2⁠	0.5	${\frac {\sqrt {3}}{2}}$	0.8660
45°	⁠1/4⁠π	50g	⁠1/8⁠	${\frac {\sqrt {2}}{2}}$	0.7071	${\frac {\sqrt {2}}{2}}$	0.7071
60°	⁠1/3⁠π	⁠66+2/3⁠g	⁠1/6⁠	${\frac {\sqrt {3}}{2}}$	0.8660	⁠1/2⁠	0.5
75°	⁠5/12⁠π	⁠83+1/3⁠g	⁠5/24⁠	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	0.9659	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	0.2588
90°	⁠1/2⁠π	100g	⁠1/4⁠	1	1	0	0

90 degree increments:

xin degrees	0°	90°	180°	270°	360°
xin radians	0	π/2	π	3π/2	2π
xingons	0	100g	200g	300g	400g
xin turns	0	1/4	1/2	3/4	1
sinx	0	1	0	−1	0
cosx	1	0	−1	0	1

Relationship to complex numbers

Main article:Trigonometric functions § Relationship to exponential function (Euler's formula)

$\cos(\theta )$and$\sin(\theta )$are the real and imaginary parts of$e^{i\theta }$.

Sine and cosine are used to connect the real and imaginary parts of acomplex numberwith itspolar coordinates(r,φ):

$z=r(\cos(\varphi )+i\sin(\varphi ))$

The real and imaginary parts are:

$\operatorname {Re} (z)=r\cos(\varphi )$

$\operatorname {Im} (z)=r\sin(\varphi )$

whererandφrepresent the magnitude and angle of the complex numberz.

For any real numberθ,Euler's formulasays that:

$e^{i\theta }=\cos(\theta )+i\sin(\theta )$

Therefore, if the polar coordinates ofzare (r,φ),$z=re^{i\varphi }.$

Complex arguments

Domain coloringof sin(z) in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.

sin(z) as a vector field

Applying the series definition of the sine and cosine to a complex argument,z,gives:

${\begin{aligned}\sin(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\\&={\frac {e^{iz}-e^{-iz}}{2i}}\\&={\frac {\sinh \left(iz\right)}{i}}\\&=-i\sinh \left(iz\right)\\\cos(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\\&={\frac {e^{iz}+e^{-iz}}{2}}\\&=\cosh(iz)\\\end{aligned}}$

where sinh and cosh are thehyperbolic sine and cosine.These areentire functions.

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

${\begin{aligned}\sin(x+iy)&=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\&=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\\cos(x+iy)&=\cos(x)\cos(iy)-\sin(x)\sin(iy)\\&=\cos(x)\cosh(y)-i\sin(x)\sinh(y)\\\end{aligned}}$

Partial fraction and product expansions of complex sine

Using the partial fraction expansion technique incomplex analysis,one can find that the infinite series

$\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{z-n}}={\frac {1}{z}}-2z\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}-z^{2}}}$

both converge and are equal to${\textstyle {\frac {\pi }{\sin(\pi z)}}}$.Similarly, one can show that

${\frac {\pi ^{2}}{\sin ^{2}(\pi z)}}=\sum _{n=-\infty }^{\infty }{\frac {1}{(z-n)^{2}}}.$

Using product expansion technique, one can derive

$\sin(\pi z)=\pi z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).$

Alternatively, the infinite product for the sine can be proved usingcomplex Fourier series.

Proof of the infinite product for the sine

Using complex Fourier series, the function$\cos(zx)$can be decomposed as $\cos(zx)={\frac {z\sin(\pi z)}{\pi }}\displaystyle \sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}\,e^{inx}}{z^{2}-n^{2}}},\,z\in \mathbb {C} \setminus \mathbb {Z},\,x\in [-\pi,\pi ].$ Setting$x=\pi$yields $\cos(\pi z)={\frac {z\sin(\pi z)}{\pi }}\displaystyle \sum _{n=-\infty }^{\infty }{\frac {1}{z^{2}-n^{2}}}={\frac {z\sin(\pi z)}{\pi }}\left({\frac {1}{z^{2}}}+2\displaystyle \sum _{n=1}^{\infty }{\frac {1}{z^{2}-n^{2}}}\right).$ Therefore, we get $\pi \cot(\pi z)={\frac {1}{z}}+2\displaystyle \sum _{n=1}^{\infty }{\frac {z}{z^{2}-n^{2}}}.$ The function$\pi \cot(\pi z)$is the derivative of$\ln(\sin(\pi z))+C_{0}$.Furthermore, if${\textstyle {\frac {df}{dz}}={\frac {z}{z^{2}-n^{2}}}}$,then the function$f$such that the emerged series converges on some open and connected subset of$\mathbb {C}$is${\textstyle f={\frac {1}{2}}\ln \left(1-{\frac {z^{2}}{n^{2}}}\right)+C_{1}}$,which can be proved using theWeierstrass M-test.The interchange of the sum and derivative is justified byuniform convergence.It follows that $\ln(\sin(\pi z))=\ln(z)+\displaystyle \sum _{n=1}^{\infty }\ln \left(1-{\frac {z^{2}}{n^{2}}}\right)+C.$ Exponentiating gives $\sin(\pi z)=ze^{C}\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).$ Since${\textstyle \lim _{z\to 0}{\frac {\sin(\pi z)}{z}}=\pi }$and${\textstyle \lim _{z\to 0}\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right)=1}$,we have$e^{C}=\pi$.Hence $\sin(\pi z)=\pi z\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right)$ for some open and connected subset of$\mathbb {C}$.Let${\textstyle a_{n}(z)=-{\frac {z^{2}}{n^{2}}}}$.Since${\textstyle \sum _{n=1}^{\infty }|a_{n}(z)|}$converges uniformly on any closed disk,${\textstyle \prod _{n=1}^{\infty }(1+a_{n}(z))}$converges uniformly on any closed disk as well.It follows that the infinite product is holomorphic on$\mathbb {C}$.By theidentity theorem,the infinite product for the sine is valid for all$z\in \mathbb {C}$,which completes the proof.$\blacksquare$

Usage of complex sine

sin(z) is found in thefunctional equationfor theGamma function,

$\Gamma (s)\Gamma (1-s)={\pi \over \sin(\pi s)},$

which in turn is found in thefunctional equationfor theRiemann zeta-function,

$\zeta (s)=2(2\pi )^{s-1}\Gamma (1-s)\sin \left({\frac {\pi }{2}}s\right)\zeta (1-s).$

As aholomorphic function,sinzis a 2D solution ofLaplace's equation:

$\Delta u(x_{1},x_{2})=0.$

The complex sine function is also related to the level curves ofpendulums.

Complex graphs

Sine function in the complex plane

real component	imaginary component	magnitude

Arcsine function in the complex plane

real component	imaginary component	magnitude

History

Main article:History of trigonometry

Quadrant from 1840sOttoman Turkeywith axes for looking up the sine andversineof angles

While the early study of trigonometry can be traced to antiquity, thetrigonometric functionsas they are in use today were developed in the medieval period. Thechordfunction was discovered byHipparchusofNicaea(180–125 BCE) andPtolemyofRoman Egypt(90–165 CE).

The sine and cosine functions can be traced to thejyāandkoṭi-jyāfunctions used inIndian astronomyduring theGupta period(AryabhatiyaandSurya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.

All six trigonometric functions in current use were known inIslamic mathematicsby the 9th century, as was thelaw of sines,used insolving triangles.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.Al-Khwārizmī(c. 780–850) produced tables of sines, cosines and tangents.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°.

The first published use of the abbreviationssin,cos,andtanis by the 16th-century French mathematicianAlbert Girard;these were further promulgated by Euler (see below). TheOpus palatinum de triangulisofGeorg Joachim Rheticus,a student ofCopernicus,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,Leibnizproved that sinxis not analgebraic functionofx.Roger Cotescomputed the derivative of sine in hisHarmonia Mensurarum(1722).Leonhard Euler'sIntroductio 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 abbreviationssin.,cos.,tang.,cot.,sec.,andcosec.

Etymology

Main article:History of trigonometry § Etymology

The wordsineis derived, indirectly, from theSanskritwordjyā'bow-string' or more specifically its synonymjīvá(both adopted fromAncient Greekχορδή'string'), due to visual similarity between the arc of a circle with its corresponding chord and a bow with its string (seejyā, koti-jyā and utkrama-jyā). This wastransliteratedinArabicasjība,which is meaningless in that language and written asjb(جب). Since Arabic is written without short vowels,jbwas interpreted as thehomographjayb(جيب), which means 'bosom', 'pocket', or 'fold'. When the Arabic texts ofAl-Battaniandal-Khwārizmīwere translated intoMedieval Latinin the 12th century byGerard of Cremona,he used the Latin equivalentsinus(which also means 'bay' or 'fold', and more specifically 'the hanging fold of atogaover the breast').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.The English formsinewas introduced in the 1590s.

The wordcosinederives from an abbreviation of the Latincomplementi sinus'sine of thecomplementary angle' ascosinusinEdmund Gunter'sCanon triangulorum(1620), which also includes a similar definition ofcotangens.

Software implementations

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.

Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g.sin(1022).

A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest pre-calculated value, orlinearly interpolatebetween 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.

TheCORDICalgorithm 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 tosinandcos.

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

In programming languages,sinandcosare typically either a built-in function or found within the language's standard math library.

For example, theC standard librarydefines sine functions withinmath.h:sin(double),sinf(float),andsinl(long double).The parameter of each is afloating pointvalue, specifying the angle in radians. Each function returns the samedata typeas it accepts. Many other trigonometric functions are also defined inmath.h,such as for cosine, arc sine, and hyperbolic sine (sinh).

Similarly,Pythondefinesmath.sin(x)andmath.cos(x)within the built-inmathmodule. Complex sine and cosine functions are also available within thecmathmodule, e.g.cmath.sin(z).CPython's math functions call theCmathlibrary, and use adouble-precision floating-point format.

Turns based implementations

Some software libraries provide implementations of sine and cosine using the input angle in half-turns,a half-turn being an angle of 180 degrees or$\pi$radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.In MATLAB, OpenCL, R, Julia, CUDA, and ARM, these functions are calledsinpiandcospi.For example,sinpi(x)would evaluate to$\sin(\pi x),$wherexis expressed in half-turns, and consequently the final input to the function,πxcan be interpreted in radians bysin.

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${\textstyle {\frac {\pi }{2}}}$in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits.

Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing modulo${\textstyle {\frac {\pi }{2}}}$involves inaccuracies in representing${\textstyle {\frac {\pi }{2}}}$.

For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to 4096 over one complete revolution.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${\textstyle {\frac {\pi }{2048}}}$would be incurred.

See also

• Āryabhaṭa's sine table
• Bhaskara I's sine approximation formula
• Discrete sine transform
• Dixon elliptic functions
• Euler's formula
• Generalized trigonometry
• Hyperbolic function
• Lemniscate elliptic functions
• Law of sines
• List of periodic functions
• List of trigonometric identities
• Madhava series
• Madhava's sine table
• Optical sine theorem
• Polar sine—a generalization to vertex angles
• Proofs of trigonometric identities
• Sinc function
• Sine and cosine transforms
• Sine integral
• Sine quadrant
• Sine wave
• Sine–Gordon equation
• Sinusoidal model
• SOH-CAH-TOA
• Trigonometric functions
• Trigonometric integral

References

• Traupman, Ph.D., John C. (1966),The New College Latin & English Dictionary,Toronto: Bantam,ISBN0-553-27619-0
• Webster's Seventh New Collegiate Dictionary,Springfield: G. & C. Merriam Company, 1969

External links

• Media related toSine functionat Wikimedia Commons