Sine and cosine

Sine and cosineGeneral informationGeneral definition sin ⁡ ( θ ) = opposite hypotenuse cos ⁡ ( θ ) = adjacent hypotenuse {displaystyle {begin{aligned}&sin(theta )={frac {textrm {opposite}}{textrm {hypotenuse}}}[8pt]&cos(theta )={frac {textrm {adjacent}}{textrm {hypotenuse}}}[8pt]end{aligned}}} Fields of applicationTrigonometry, Fourier series, Mathematical analysis.Domain, codomain and imageDomainreal number R {displaystyle mathbb {R} } Image [ − 1 , 1 ] {displaystyle [-1,1]}

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 θ {displaystyle theta } , the sine and cosine functions are denoted as sin ⁡ ( θ ) {displaystyle sin(theta )} and cos ⁡ ( θ ) {displaystyle cos(theta )} .

The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

The sine and cosine functions are commonly used to model 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 and koṭi-jyā functions used in Indian astronomy during the Gupta period.

To define the sine and cosine of an acute angle α {displaystyle alpha } , start with a right triangle that contains an angle of measure α {displaystyle alpha } ; in the accompanying figure, angle α {displaystyle alpha } in a right triangle A B C {displaystyle ABC} is the angle of interest. The three sides of the triangle are named as follows:[1]

• The opposite side is the side opposite to the angle of interest; in this case, it is a {displaystyle a} .
• The hypotenuse is the side opposite the right angle; in this case, it is h {displaystyle h} . The hypotenuse is always the longest side of a right-angled triangle.
• The adjacent side is the remaining side; in this case, it is b {displaystyle b} . 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:[1] sin ⁡ ( α ) = opposite hypotenuse , cos ⁡ ( α ) = adjacent hypotenuse . {displaystyle sin(alpha )={frac {text{opposite}}{text{hypotenuse}}},qquad cos(alpha )={frac {text{adjacent}}{text{hypotenuse}}}.}

The other trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and cosine functions. The 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:[1] tan ⁡ ( θ ) = sin ⁡ ( θ ) cos ⁡ ( θ ) = opposite adjacent , cot ⁡ ( θ ) = 1 tan ⁡ ( θ ) = adjacent opposite , csc ⁡ ( θ ) = 1 sin ⁡ ( θ ) = hypotenuse opposite , sec ⁡ ( θ ) = 1 cos ⁡ ( θ ) = hypotenuse adjacent . {displaystyle {begin{aligned}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{aligned}}}

As stated, the values sin ⁡ ( α ) {displaystyle sin(alpha )} and cos ⁡ ( α ) {displaystyle cos(alpha )} appear to depend on the choice of a right triangle containing an angle of measure α {displaystyle alpha } . However, this is not the case as all such triangles are similar, and so the ratios are the same for each of them. For example, each leg of the 45-45-90 right triangle is 1 unit, and its hypotenuse is 2 {displaystyle {sqrt {2}}} ; therefore, sin ⁡ 45 ∘ = cos ⁡ 45 ∘ = 2 2 {textstyle sin 45^{circ }=cos 45^{circ }={frac {sqrt {2}}{2}}} .[2] The following table shows the special value of each input for both sine and cosine with the domain between 0 < α < π 2 {textstyle 0<alpha <{frac {pi }{2}}} . 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.[3][4]

Angle, x sin(x) cos(x) Degrees Radians Gradians Turns Exact Decimal Exact Decimal 0° 0 0 g {displaystyle 0^{g}} 0 0 0 1 1 30° 1 6 π {displaystyle {frac {1}{6}}pi } 33 1 3 g {displaystyle 33{frac {1}{3}}^{g}} 1 12 {displaystyle {frac {1}{12}}} 1 2 {displaystyle {frac {1}{2}}} 0.5 3 2 {displaystyle {frac {sqrt {3}}{2}}} 0.866 45° 1 4 π {displaystyle {frac {1}{4}}pi } 50 g {displaystyle 50^{g}} 1 8 {displaystyle {frac {1}{8}}} 2 2 {displaystyle {frac {sqrt {2}}{2}}} 0.707 2 2 {displaystyle {frac {sqrt {2}}{2}}} 0.707 60° 1 3 π {displaystyle {frac {1}{3}}pi } 66 2 3 g {displaystyle 66{frac {2}{3}}^{g}} 1 6 {displaystyle {frac {1}{6}}} 3 2 {displaystyle {frac {sqrt {3}}{2}}} 0.866 1 2 {displaystyle {frac {1}{2}}} 0.5 90° 1 2 π {displaystyle {frac {1}{2}}pi } 100 g {displaystyle 100^{g}} 1 4 {displaystyle {frac {1}{4}}} 1 1 0 0

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.[5] Given a triangle A B C {displaystyle ABC} with sides a {displaystyle a} , b {displaystyle b} , and c {displaystyle c} , and angles opposite those sides α {displaystyle alpha } , β {displaystyle beta } , and γ {displaystyle gamma } , the law states, sin ⁡ α a = sin ⁡ β b = sin ⁡ γ c . {displaystyle {frac {sin alpha }{a}}={frac {sin beta }{b}}={frac {sin gamma }{c}}.} This is equivalent to the equality of the first three expressions below: a sin ⁡ α = b sin ⁡ β = c sin ⁡ γ = 2 R , {displaystyle {frac {a}{sin alpha }}={frac {b}{sin beta }}={frac {c}{sin gamma }}=2R,} where R {displaystyle R} is the triangle’s circumradius.

The law of cosines is useful for computing the length of an unknown side if two other sides and an angle are known.[5] The law states, a 2 + b 2 − 2 a b cos ⁡ ( γ ) = c 2 {displaystyle a^{2}+b^{2}-2abcos(gamma )=c^{2}} In the case where γ = π / 2 {displaystyle gamma =pi /2} from which cos ⁡ ( γ ) = 0 {displaystyle cos(gamma )=0} , the resulting equation becomes the Pythagorean theorem.[6]

The cross product and dot product are operations on two vectors in Euclidean vector space. The sine and cosine functions can be defined in terms of the cross product and dot product. If a {displaystyle mathbf {a} } and b {displaystyle mathbf {b} } are vectors, and θ {displaystyle theta } is the angle between a {displaystyle mathbf {a} } and b {displaystyle mathbf {b} } , then sine and cosine can be defined as:[7][8] sin ⁡ ( θ ) = | a × b | | a | | b | , cos ⁡ ( θ ) = a ⋅ b | a | | b | . {displaystyle {begin{aligned}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{aligned}}}

The sine and cosine functions may also be defined in a more general way by using unit circle, a circle of radius one centered at the origin ( 0 , 0 ) {displaystyle (0,0)} , formulated as the equation of x 2 + y 2 = 1 {displaystyle x^{2}+y^{2}=1} in the Cartesian coordinate system. A ray from the origin making an angle of θ {displaystyle theta } with the positive half of the x {displaystyle x} -axis intersects the unit circle at exactly one point. The x {displaystyle x} – and y {displaystyle y} -coordinates of this point of intersection are equal to cos ⁡ ( θ ) {displaystyle cos(theta )} and sin ⁡ ( θ ) {displaystyle sin(theta )} , respectively; that is,[9] sin ⁡ ( θ ) = y , cos ⁡ ( θ ) = x . {displaystyle sin(theta )=y,qquad cos(theta )=x.}

This definition is consistent with the right-angled triangle definition of sine and cosine when 0 < θ < π 2 {textstyle 0<theta <{frac {pi }{2}}} 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 y {displaystyle y} -coordinate. A similar argument can be made for the cosine function to show that the cosine of an angle when 0 < θ < π 2 {textstyle 0<theta <{frac {pi }{2}}} , even under the new definition using the unit circle.[10][11]

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 θ > 0 {displaystyle theta >0} . In a sine function, if the input is θ = π 2 {textstyle theta ={frac {pi }{2}}} , the point is rotated counterclockwise and stopped exactly on the y {displaystyle y} -axis. If θ = π {displaystyle theta =pi } , the point is at the circle’s halfway point. If θ = 2 π {displaystyle theta =2pi } , the point returns to its origin. This results in both sine and cosine functions having the range between − 1 ≤ y ≤ 1 {displaystyle -1leq yleq 1} .[12]

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 y {displaystyle y} -coordinate. In other words, both sine and cosine functions are periodic, meaning any angle added by the circle’s circumference is the angle itself. Mathematically,[13] sin ⁡ ( θ + 2 π ) = sin ⁡ ( θ ) , cos ⁡ ( θ + 2 π ) = cos ⁡ ( θ ) . {displaystyle sin(theta +2pi )=sin(theta ),qquad cos(theta +2pi )=cos(theta ).}

A function f {displaystyle f} is said to be odd if f ( − x ) = − f ( x ) {displaystyle f(-x)=-f(x)} , and is said to be even if f ( − x ) = f ( x ) {displaystyle f(-x)=f(x)} . The sine function is odd, whereas the cosine function is even.[14] Both sine and cosine functions are similar, with their difference being shifted by π 2 {textstyle {frac {pi }{2}}} . This phase shift can be expressed as cos ⁡ ( θ ) = sin ⁡ ( θ + π 2 ) {textstyle cos(theta )=sin left(theta +{frac {pi }{2}}right)} or sin ⁡ ( θ ) = cos ⁡ ( θ − π 2 ) {textstyle sin(theta )=cos left(theta -{frac {pi }{2}}right)} . This is distinct from the cofunction identities that follow below, which arise from right-triangle geometry and are not phase shifts: [15] sin ⁡ ( θ ) = cos ⁡ ( π 2 − θ ) , cos ⁡ ( θ ) = sin ⁡ ( π 2 − θ ) . {displaystyle {begin{aligned}sin(theta )&=cos left({frac {pi }{2}}-theta right),cos(theta )&=sin left({frac {pi }{2}}-theta right).end{aligned}}}

Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is sin ⁡ ( 0 ) = 0 {displaystyle sin(0)=0} . 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 cos ⁡ ( x ) = x {displaystyle cos(x)=x} . The decimal expansion of the Dottie number is approximately 0.739085.[16]

The sine and cosine functions are infinitely differentiable.[17] The derivative of sine is cosine, and the derivative of cosine is negative sine:[18] d d x sin ⁡ ( x ) = cos ⁡ ( x ) , d d x cos ⁡ ( x ) = − sin ⁡ ( x ) . {displaystyle {frac {d}{dx}}sin(x)=cos(x),qquad {frac {d}{dx}}cos(x)=-sin(x).} Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself.[17] These derivatives can be applied to the first derivative test, according to which the monotonicity of a function can be defined as the inequality of function’s first derivative greater or less than equal to zero.[19] It can also be applied to second derivative test, according to which the concavity of a function can be defined by applying the inequality of the function’s second derivative greater or less than equal to zero.[20] The following table shows that both sine and cosine functions have concavity and monotonicity—the positive sign ( + {displaystyle +} ) denotes a graph is increasing (going upward) and the negative sign ( − {displaystyle -} ) is decreasing (going downward)—in certain intervals.[3] This information can be represented as a Cartesian coordinates system divided into four quadrants.

Quadrant Angle Sine Cosine Degrees Radians Sign Monotony Convexity Sign Monotony Convexity 1st quadrant, I 0 ∘ < x < 90 ∘ {displaystyle 0^{circ }<x<90^{circ }} 0 < x < π 2 {displaystyle 0<x<{frac {pi }{2}}} + {displaystyle +} Increasing Concave + {displaystyle +} Decreasing Concave 2nd quadrant, II 90 ∘ < x < 180 ∘ {displaystyle 90^{circ }<x<180^{circ }} π 2 < x < π {displaystyle {frac {pi }{2}}<x<pi } + {displaystyle +} Decreasing Concave − {displaystyle -} Decreasing Convex 3rd quadrant, III 180 ∘ < x < 270 ∘ {displaystyle 180^{circ }<x<270^{circ }} π < x < 3 π 2 {displaystyle pi <x<{frac {3pi }{2}}} − {displaystyle -} Decreasing Convex − {displaystyle -} Increasing Convex 4th quadrant, IV 270 ∘ < x < 360 ∘ {displaystyle 270^{circ }<x<360^{circ }} 3 π 2 < x < 2 π {displaystyle {frac {3pi }{2}}<x<2pi } − {displaystyle -} Increasing Convex + {displaystyle +} Increasing Concave

Both sine and cosine functions can be defined by using differential equations. The pair of ( cos ⁡ θ , sin ⁡ θ ) {displaystyle (cos theta ,sin theta )} is the solution ( x ( θ ) , y ( θ ) ) {displaystyle (x(theta ),y(theta ))} to the two-dimensional system of differential equations y ′ ( θ ) = x ( θ ) {displaystyle y'(theta )=x(theta )} and x ′ ( θ ) = − y ( θ ) {displaystyle x'(theta )=-y(theta )} with the initial conditions y ( 0 ) = 0 {displaystyle y(0)=0} and x ( 0 ) = 1 {displaystyle x(0)=1} . 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 y ′ ( θ ) = x ( θ ) {displaystyle y'(theta )=x(theta )} and x ′ ( θ ) = − y ( θ ) {displaystyle x'(theta )=-y(theta )} starting from the initial conditions y ( 0 ) = 0 {displaystyle y(0)=0} and x ( 0 ) = 1 {displaystyle x(0)=1} .[citation needed]

Their area under a curve can be obtained by using the integral with a certain bounded interval. Their antiderivatives are: ∫ sin ⁡ ( x ) d x = − cos ⁡ ( x ) + C ∫ cos ⁡ ( x ) d x = sin ⁡ ( x ) + C , {displaystyle int sin(x),dx=-cos(x)+Cqquad int cos(x),dx=sin(x)+C,} where C {displaystyle C} denotes the constant of integration.[21] 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 0 {displaystyle 0} and t {displaystyle t} is ∫ 0 t 1 + cos 2 ⁡ ( x ) d x = 2 E ⁡ ( t , 1 2 ) , {displaystyle int _{0}^{t}!{sqrt {1+cos ^{2}(x)}},dx={sqrt {2}}operatorname {E} left(t,{frac {1}{sqrt {2}}}right),} where E ⁡ ( φ , k ) {displaystyle operatorname {E} (varphi ,k)} is the incomplete elliptic integral of the second kind with modulus k {displaystyle k} . It cannot be expressed using elementary functions.[22] In the case of a full period, its arc length is L = 4 2 π 3 Γ ( 1 / 4 ) 2 + Γ ( 1 / 4 ) 2 2 π = 2 π ϖ + 2 ϖ ≈ 7.6404 {displaystyle L={frac {4{sqrt {2pi ^{3}}}}{Gamma (1/4)^{2}}}+{frac {Gamma (1/4)^{2}}{sqrt {2pi }}}={frac {2pi }{varpi }}+2varpi approx 7.6404} where Γ {displaystyle Gamma } is the gamma function and ϖ {displaystyle varpi } is the lemniscate constant.[23][24]

The functions sin : R → R {textstyle sin :mathbb {R} to mathbb {R} } and cos : R → R {displaystyle cos :mathbb {R} to mathbb {R} } (as well as those functions with the same function rule and domain whose codomain is a subset of R {displaystyle mathbb {R} } containing the interval [ − 1 , 1 ] {displaystyle left[-1,1right]} ) are not bijective and therefore do not have inverse functions. For example, sin ⁡ ( 0 ) = 0 {displaystyle sin(0)=0} , but also sin ⁡ ( π ) = 0 {displaystyle sin(pi )=0} , sin ⁡ ( 2 π ) = 0 {displaystyle sin(2pi )=0} . Sine’s “inverse”, called arcsine, can then be described not as a function but a relation (for example, all integer multiples of π {displaystyle pi } would have an arcsine of zero). To define the inverse functions of sine and cosine, they must be restricted to their principal branches by restricting their domain and codomain; the standard functions used to define arcsine and arccosine are then sin : [ − π / 2 , π / 2 ] → [ − 1 , 1 ] {displaystyle sin :left[-pi /2,pi /2right]to left[-1,1right]} and cos : [ 0 , π ] → [ − 1 , 1 ] {displaystyle cos :left[0,pi right]to left[-1,1right]} .[25] These are bijective and have inverses: arcsin : [ − 1 , 1 ] → [ − π / 2 , π / 2 ] {displaystyle arcsin :left[-1,1right]to left[-pi /2,pi /2right]} and arccos : [ − 1 , 1 ] → [ 0 , π ] {displaystyle arccos :left[-1,1right]to left[0,pi right]} . Alternative notation is sin − 1 {displaystyle sin ^{-1}} for arcsine and cos − 1 {displaystyle cos ^{-1}} for arccosine. Using these definitions, one obtains the identity maps:

sin ∘ arcsin ( x ) = x x ∈ [ − 1 , 1 ] arcsin ∘ sin ( x ) = x x ∈ [ − π / 2 , π / 2 ] {displaystyle {begin{aligned}sin circ arcsin ,(x)&=xqquad xin left[-1,1right]arcsin circ sin ,(x)&=xqquad xin left[-pi /2,pi /2right]end{aligned}}} and

cos ∘ arccos ( x ) = x x ∈ [ − 1 , 1 ] arccos ∘ cos ( x ) = x x ∈ [ 0 , π ] {displaystyle {begin{aligned}cos circ arccos ,(x)&=xqquad xin left[-1,1right]arccos circ cos ,(x)&=xqquad xin left[0,pi right]end{aligned}}}

An acute angle θ {displaystyle theta } is given by: θ = arcsin ⁡ ( opposite hypotenuse ) = arccos ⁡ ( adjacent hypotenuse ) , {displaystyle theta =arcsin left({frac {text{opposite}}{text{hypotenuse}}}right)=arccos left({frac {text{adjacent}}{text{hypotenuse}}}right),} where for some integer k {displaystyle k} , sin ⁡ ( y ) = x ⟺ y = arcsin ⁡ ( x ) + 2 π k , or y = π − arcsin ⁡ ( x ) + 2 π k cos ⁡ ( y ) = x ⟺ y = arccos ⁡ ( x ) + 2 π k , or y = − arccos ⁡ ( x ) + 2 π k {displaystyle {begin{aligned}sin(y)=xiff &y=arcsin(x)+2pi k,{text{ or }}&y=pi -arcsin(x)+2pi kcos(y)=xiff &y=arccos(x)+2pi k,{text{ or }}&y=-arccos(x)+2pi kend{aligned}}} By definition, both functions satisfy the equations: sin ⁡ ( arcsin ⁡ ( x ) ) = x cos ⁡ ( arccos ⁡ ( x ) ) = x {displaystyle sin(arcsin(x))=xqquad cos(arccos(x))=x} and arcsin ⁡ ( sin ⁡ ( θ ) ) = θ for − π 2 ≤ θ ≤ π 2 arccos ⁡ ( cos ⁡ ( θ ) ) = θ for 0 ≤ θ ≤ π {displaystyle {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 0leq theta leq pi end{aligned}}}

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:[26][a] sin 2 ⁡ ( θ ) + cos 2 ⁡ ( θ ) = 1. {displaystyle sin ^{2}(theta )+cos ^{2}(theta )=1.}

Sine and cosine satisfy the following double-angle formulas:[27] sin ⁡ ( 2 θ ) = 2 sin ⁡ ( θ ) cos ⁡ ( θ ) , cos ⁡ ( 2 θ ) = cos 2 ⁡ ( θ ) − sin 2 ⁡ ( θ ) = 2 cos 2 ⁡ ( θ ) − 1 = 1 − 2 sin 2 ⁡ ( θ ) {displaystyle {begin{aligned}sin(2theta )&=2sin(theta )cos(theta ),cos(2theta )&=cos ^{2}(theta )-sin ^{2}(theta )&=2cos ^{2}(theta )-1&=1-2sin ^{2}(theta )end{aligned}}}

The cosine double angle formula implies that sin2 and cos2 are, themselves, shifted and scaled sine waves. Specifically,[28] sin 2 ⁡ ( θ ) = 1 − cos ⁡ ( 2 θ ) 2 cos 2 ⁡ ( θ ) = 1 + cos ⁡ ( 2 θ ) 2 {displaystyle sin ^{2}(theta )={frac {1-cos(2theta )}{2}}qquad cos ^{2}(theta )={frac {1+cos(2theta )}{2}}} 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]

Both sine and cosine functions can be defined by using a Taylor series, a power series involving the higher-order derivatives. As mentioned in § 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 sin ⁡ ( x ) {displaystyle sin(x)} are cos ⁡ ( x ) {displaystyle cos(x)} , − sin ⁡ ( x ) {displaystyle -sin(x)} , − cos ⁡ ( x ) {displaystyle -cos(x)} , sin ⁡ ( x ) {displaystyle sin(x)} , continuing to repeat those four functions. The ( 4 n + k ) {displaystyle (4n+k)} -th derivative, evaluated at the point 0: sin ( 4 n + k ) ⁡ ( 0 ) = { 0 when k = 0 1 when k = 1 0 when k = 2 − 1 when k = 3 {displaystyle sin ^{(4n+k)}(0)={begin{cases}0&{text{when }}k=01&{text{when }}k=1&{text{when }}k=2-1&{text{when }}k=3end{cases}}} where the superscript represents repeated differentiation. This implies the following Taylor series expansion at x = 0 {displaystyle x=0} . One can then use the theory of Taylor series to show that the following identities hold for all real numbers x {displaystyle x} —where x {displaystyle x} is the angle in radians.[29] More generally, for all complex numbers:[30] sin ⁡ ( x ) = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) ! x 2 n + 1 {displaystyle {begin{aligned}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{aligned}}} Taking the derivative of each term gives the Taylor series for cosine:[29][30] cos ⁡ ( x ) = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! x 2 n {displaystyle {begin{aligned}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{aligned}}}

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 its interpolation, and its extension of a periodic function known as the Fourier series. Let a n {displaystyle a_{n}} and b n {displaystyle b_{n}} be any coefficients, then the trigonometric polynomial of a degree N {displaystyle N} —denoted as T ( x ) {displaystyle T(x)} —is defined as:[31][32] T ( x ) = a 0 + ∑ n = 1 N a n cos ⁡ ( n x ) + ∑ n = 1 N b n sin ⁡ ( n x ) . {displaystyle T(x)=a_{0}+sum _{n=1}^{N}a_{n}cos(nx)+sum _{n=1}^{N}b_{n}sin(nx).}

The trigonometric series can be defined similarly analogous to the trigonometric polynomial, its infinite inversion. Let A n {displaystyle A_{n}} and B n {displaystyle B_{n}} be any coefficients, then the trigonometric series can be defined as:[33] 1 2 A 0 + ∑ n = 1 ∞ A n cos ⁡ ( n x ) + B n sin ⁡ ( n x ) . {displaystyle {frac {1}{2}}A_{0}+sum _{n=1}^{infty }A_{n}cos(nx)+B_{n}sin(nx).} In the case of a Fourier series with a given integrable function f {displaystyle f} , the coefficients of a trigonometric series are:[34] A n = 1 π ∫ 0 2 π f ( x ) cos ⁡ ( n x ) d x , B n = 1 π ∫ 0 2 π f ( x ) sin ⁡ ( n x ) d x . {displaystyle {begin{aligned}A_{n}&={frac {1}{pi }}int _{0}^{2pi }f(x)cos(nx),dx,B_{n}&={frac {1}{pi }}int _{0}^{2pi }f(x)sin(nx),dx.end{aligned}}}

Both sine and cosine can be extended further via complex number, a set of numbers composed of both real and imaginary numbers. For real number θ {displaystyle theta } , the definition of both sine and cosine functions can be extended in a complex plane in terms of an exponential function as follows:[35] sin ⁡ ( θ ) = e i θ − e − i θ 2 i , cos ⁡ ( θ ) = e i θ + e − i θ 2 , {displaystyle {begin{aligned}sin(theta )&={frac {e^{itheta }-e^{-itheta }}{2i}},cos(theta )&={frac {e^{itheta }+e^{-itheta }}{2}},end{aligned}}}

Alternatively, both functions can be defined in terms of Euler’s formula:[35] e i θ = cos ⁡ ( θ ) + i sin ⁡ ( θ ) , e − i θ = cos ⁡ ( θ ) − i sin ⁡ ( θ ) . {displaystyle {begin{aligned}e^{itheta }&=cos(theta )+isin(theta ),e^{-itheta }&=cos(theta )-isin(theta ).end{aligned}}}

When plotted on the complex plane, the function e i x {displaystyle e^{ix}} for real values of x {displaystyle x} traces out the unit circle in the complex plane. Both sine and cosine functions may be simplified to the imaginary and real parts of e i θ {displaystyle e^{itheta }} as:[36] sin ⁡ θ = Im ⁡ ( e i θ ) , cos ⁡ θ = Re ⁡ ( e i θ ) . {displaystyle {begin{aligned}sin theta &=operatorname {Im} (e^{itheta }),cos theta &=operatorname {Re} (e^{itheta }).end{aligned}}}

When z = x + i y {displaystyle z=x+iy} for real values x {displaystyle x} and y {displaystyle y} , where i = − 1 {displaystyle i={sqrt {-1}}} , both sine and cosine functions can be expressed in terms of real sines, cosines, and hyperbolic functions as:[37] sin ⁡ z = sin ⁡ x cosh ⁡ y + i cos ⁡ x sinh ⁡ y , cos ⁡ z = cos ⁡ x cosh ⁡ y − i sin ⁡ x sinh ⁡ y . {displaystyle {begin{aligned}sin z&=sin xcosh y+icos xsinh y,cos z&=cos xcosh y-isin xsinh y.end{aligned}}}

Sine and cosine are used to connect the real and imaginary parts of a complex number with its polar coordinates ( r , θ ) {displaystyle (r,theta )} : z = r ( cos ⁡ ( θ ) + i sin ⁡ ( θ ) ) = r cos ⁡ ( θ ) + i r sin ⁡ ( θ ) , {displaystyle z=r(cos(theta )+isin(theta ))=rcos(theta )+irsin(theta ),} and the real and imaginary parts are Re ⁡ ( z ) = r cos ⁡ ( θ ) , Im ⁡ ( z ) = r sin ⁡ ( θ ) , {displaystyle {begin{aligned}operatorname {Re} (z)&=rcos(theta ),operatorname {Im} (z)&=rsin(theta ),end{aligned}}} where r {displaystyle r} and θ {displaystyle theta } represent the magnitude and angle of the complex number z {displaystyle z} .[38]

For any real number θ {displaystyle theta } , Euler’s formula in terms of polar coordinates is stated as z = r e i θ {textstyle z=re^{itheta }} .[35]

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

sin ⁡ ( z ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) ! z 2 n + 1 = e i z − e − i z 2 i = sinh ⁡ ( i z ) i = − i sinh ⁡ ( i z ) cos ⁡ ( z ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! z 2 n = e i z + e − i z 2 = cosh ⁡ ( i z ) {displaystyle {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(izright)}{i}}&=-isinh left(izright)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 the hyperbolic sine and cosine. These are entire 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:

sin ⁡ ( x + i y ) = sin ⁡ ( x ) cos ⁡ ( i y ) + cos ⁡ ( x ) sin ⁡ ( i y ) = sin ⁡ ( x ) cosh ⁡ ( y ) + i cos ⁡ ( x ) sinh ⁡ ( y ) cos ⁡ ( x + i y ) = cos ⁡ ( x ) cos ⁡ ( i y ) − sin ⁡ ( x ) sin ⁡ ( i y ) = cos ⁡ ( x ) cosh ⁡ ( y ) − i sin ⁡ ( x ) sinh ⁡ ( y ) {displaystyle {begin{aligned}sin(x+iy)&=sin(x)cos(iy)+cos(x)sin(iy)&=sin(x)cosh(y)+icos(x)sinh(y)cos(x+iy)&=cos(x)cos(iy)-sin(x)sin(iy)&=cos(x)cosh(y)-isin(x)sinh(y)end{aligned}}}

Using the partial fraction expansion technique in complex analysis, one can find that the infinite series ∑ n = − ∞ ∞ ( − 1 ) n z − n = 1 z − 2 z ∑ n = 1 ∞ ( − 1 ) n n 2 − z 2 {displaystyle sum _{n=-infty }^{infty }{frac {(-1)^{n}}{z-n}}={frac {1}{z}}-2zsum _{n=1}^{infty }{frac {(-1)^{n}}{n^{2}-z^{2}}}} both converge and are equal to π sin ⁡ ( π z ) {textstyle {frac {pi }{sin(pi z)}}} . Similarly, one can show that π 2 sin 2 ⁡ ( π z ) = ∑ n = − ∞ ∞ 1 ( z − n ) 2 . {displaystyle {frac {pi ^{2}}{sin ^{2}(pi z)}}=sum _{n=-infty }^{infty }{frac {1}{(z-n)^{2}}}.}

Using product expansion technique, one can derive sin ⁡ ( π z ) = π z ∏ n = 1 ∞ ( 1 − z 2 n 2 ) . {displaystyle sin(pi z)=pi zprod _{n=1}^{infty }left(1-{frac {z^{2}}{n^{2}}}right).}

sin(z) is found in the functional equation for the Gamma function,

Γ ( s ) Γ ( 1 − s ) = π sin ⁡ ( π s ) , {displaystyle Gamma (s)Gamma (1-s)={pi over sin(pi s)},}

which in turn is found in the functional equation for the Riemann zeta-function,

ζ ( s ) = 2 ( 2 π ) s − 1 Γ ( 1 − s ) sin ⁡ ( π 2 s ) ζ ( 1 − s ) . {displaystyle zeta (s)=2(2pi )^{s-1}Gamma (1-s)sin left({frac {pi }{2}}sright)zeta (1-s).}

As a holomorphic function, sin z is a 2D solution of Laplace’s equation:

Δ u ( x 1 , x 2 ) = 0. {displaystyle Delta u(x_{1},x_{2})=0.}

The complex sine function is also related to the level curves of pendulums.[how?][39][better source needed]

Sine function in the complex plane Real component Imaginary component Magnitude Arcsine function in the complex plane Real component Imaginary component Magnitude

The word sine is derived, indirectly, from the Sanskrit word jyā ‘bow-string’ or more specifically its synonym jīvá (both adopted from Ancient Greek 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). This was transliterated in Arabic as jība, which is meaningless in that language and written as jb (جب). Since Arabic is written without short vowels, jb was interpreted as the homograph jayb (جيب), which means ‘bosom’, ‘pocket’, or ‘fold’.[40][41] When the Arabic texts of Al-Battani and al-Khwārizmī were translated into Medieval Latin in the 12th century by Gerard of Cremona, he used the Latin equivalent sinus (which also means ‘bay’ or ‘fold’, and more specifically ‘the hanging fold of a toga over the breast’).[42][43][44] 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.[45][46] The English form sine was introduced in Thomas Fale’s 1593 Horologiographia.[47]

The word cosine derives from an abbreviation of the Latin complementi sinus ‘sine of the complementary angle’ as cosinus in Edmund Gunter’s Canon triangulorum (1620), which also includes a similar definition of cotangens.[48]

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 function was discovered by Hipparchus of Nicaea (180-125 BCE) and Ptolemy of Roman Egypt (90-165 CE).[49]

The sine and cosine functions are closely related to the jyā and 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.[42][50]

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.[51] Al-Khwārizmī (c. 780-850) produced tables of sines, cosines and tangents.[52][53] 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°.[53]

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, Leibniz proved that sin x is not an algebraic function of x.[54] Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722).[55] 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.[42]

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.[56]

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, or 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]

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 sin and cos.

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

In programming languages, sin and cos 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 math.h: sin(double), sinf(float), and sinl(long double). 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 math.h, such as for cosine, arc sine, and hyperbolic sine (sinh). Similarly, Python defines math.sin(x) and math.cos(x) within the built-in math module. Complex sine and cosine functions are also available within the cmath module, e.g. cmath.sin(z). CPython’s math functions call the C math library, and use a double-precision floating-point format.

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 π {displaystyle pi } radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.[57][58] These functions are called sinpi and cospi in Fortran,[59] MATLAB,[57] OpenCL,[60] R,[58] Julia,[61] CUDA,[62] and ARM.[63] For example, sinpi(x) would evaluate to sin ⁡ ( π x ) , {displaystyle sin(pi x),} where x is expressed in half-turns, and consequently the final input to the function, πx can be interpreted in radians by sin. SciPy provides similar functions sindg and cosdg with input in degrees,[64] as do Fortran[65] but named sind and cosd.

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 π {displaystyle 2pi } , π {displaystyle pi } , and π 2 {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 π 2 {textstyle {frac {pi }{2}}} involves inaccuracies in representing π 2 {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.[66] 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 π 2048 {textstyle {frac {pi }{2048}}} would be incurred.

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