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taylor series to show that the following identities hold for all real numbers x where x is the angle in radians 3 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 8pt sum _ n 0 infty frac 1 n 2n 1 x 2n 1 8pt end aligned taking the derivative of each term gives the taylor series for cosine 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 8pt sum _ n 0 infty frac 1 n 2n x 2n 8pt end aligned continued fraction definitions edit the sine function can also be represented as a generalized continued fraction sin x x 1 x 2 2 3 x 2 2 3 x 2 4 5 x 2 4 5 x 2 6 7 x 2 displaystyle 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 1 1 x 2 1 2 x 2 1 2 x 2 3 4 x 2 3 4 x 2 5 6 x 2 displaystyle 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 from euler s continued fraction formula and express the real number values both rational and irrational of the sine and cosine functions identities edit main article list of trigonometric identities exact identities using radians these apply for all values of θ displaystyle theta sin θ cos π 2 θ cos θ π 2 displaystyle sin theta cos left frac pi 2 theta right cos left theta frac pi 2 right cos θ sin π 2 θ sin θ π 2 displaystyle cos theta sin left frac pi 2 theta right sin left theta frac pi 2 right reciprocals edit the reciprocal of sine is cosecant i e the reciprocal of sin a is csc a or cosec a 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 a 1 sin a hypotenuse opposite displaystyle csc a frac 1 sin a frac textrm hypotenuse textrm opposite sec a 1 cos a hypotenuse adjacent displaystyle sec a frac 1 cos a frac textrm hypotenuse textrm adjacent inverses edit the usual principal values of the arcsin x and arccos x functions graphed on the cartesian plane the inverse function of sine is arcsine arcsin or asin or inverse sine sin 1 the inverse function of cosine is arccosine arccos acos or cos 1 the superscript of 1 in sin 1 and cos 1 denotes the inverse of a function not exponentiation as sine and cosine are not injective their inverses are not exact inverse functions but partial inverse functions for example sin 0 0 but also sin π 0 sin 2 π 0 etc it follows that the arcsine function is multivalued arcsin 0 0 but also arcsin 0 π arcsin 0 2 π 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 arcsin x will evaluate only to a single value called its principal value the standard range of principal values for arcsin is from π 2 to π and the standard range for arccos is from 0 to π 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 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 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 cos arccos x x displaystyle sin arcsin x x qquad 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 0 leq theta leq pi end aligned pythagorean trigonometric identity edit the basic relationship between the sine and the cosine is the pythagorean trigonometric identity 1 cos 2 θ sin 2 θ 1 displaystyle cos 2 theta sin 2 theta 1 where sin 2 x means sin x 2 double angle formulas edit sine and cosine satisfy the following double angle formulas sin 2 θ 2 sin θ cos θ displaystyle sin 2 theta 2 sin theta cos theta cos 2 θ cos 2 θ sin 2 θ 2 cos 2 θ 1 1 2 sin 2 θ displaystyle 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 sin 2 and cos 2 are themselves shifted and scaled sine waves specifically 4 sin 2 θ 1 cos 2 θ 2 cos 2 θ 1 cos 2 θ 2 displaystyle 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 the 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 derivative and integrals edit see also list of integrals of trigonometric functions and differentiation of trigonometric functions the derivatives of sine and cosine are 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 and their antiderivatives are sin x d x cos x c displaystyle int sin x dx cos x c cos x d x sin x c displaystyle int cos x dx sin x c where c denotes the constant of integration 1 properties relating to the quadrants edit 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 α 2 π sin α displaystyle 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 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 3 pi 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 3 pi 2 x 2 pi displaystyle increasing convex displaystyle increasing concave 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 degrees radians sin x displaystyle sin x cos x displaystyle cos x value point type value point type 0 displaystyle 0 circ 0 displaystyle 0 0 displaystyle 0 root inflection 1 displaystyle 1 maximum 90 displaystyle 90 circ π 2 displaystyle frac pi 2 1 displaystyle 1 maximum 0 displaystyle 0 root inflection 180 displaystyle 180 circ π displaystyle pi 0 displaystyle 0 root inflection 1 displaystyle 1 minimum 270 displaystyle 270 circ 3 π 2 displaystyle frac 3 pi 2 1 displaystyle 1 minimum 0 displaystyle 0 root inflection fixed points edit main article dottie number the fixed point iteration x n 1 cos x n with initial value x 0 1 converges to the dottie number 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 that is 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 0 739085 displaystyle 0 739085 ldots 5 arc length edit 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 t 1 sqrt 2 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 the arc length for a full period is 6 l 4 2 π 3 γ 1 4 2 γ 1 4 2 2 π 7 640395578 displaystyle l frac 4 sqrt 2 pi 3 gamma 1 4 2 frac gamma 1 4 2 sqrt 2 pi 7 640395578 ldots where γ displaystyle gamma is the gamma function this can also be written using π displaystyle pi and the lemniscate constant 6 7 law of sines edit main article 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 sin a a sin b b sin c c displaystyle frac sin a a frac sin b b frac sin c c this is equivalent to the equality of the first three expressions below a sin a b sin b c sin c 2 r displaystyle frac a sin a frac b sin b frac c sin c 2r where r is the triangle s circumradius 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 in triangulation a technique to determine unknown distances by measuring two angles and an accessible enclosed distance law of cosines edit main article law of cosines the law of cosines states that for an arbitrary triangle with sides a b and c and angles opposite those sides a b and c a 2 b 2 2 a b cos c c 2 displaystyle a 2 b 2 2ab cos c c 2 in the case where c π 2 displaystyle c pi 2 cos c 0 displaystyle cos c 0 and this becomes the pythagorean theorem for a right triangle a 2 b 2 c 2 displaystyle a 2 b 2 c 2 where c is the hypotenuse special values edit 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 values of sin x and cos x are particularly simple and can be expressed without nested square roots a table of these angles is given below for more complex angle expressions see exact trigonometric values common angles angle x sin x cos x degrees radians gradians turns exact decimal exact decimal 0 0 0 g 0 0 0 1 1 15 1 12 π 16 2 3 g 1 24 6 2 4 displaystyle frac sqrt 6 sqrt 2 4 0 2588 6 2 4 displaystyle frac sqrt 6 sqrt 2 4 0 9659 30 1 6 π 33 1 3 g 1 12 1 2 0 5 3 2 displaystyle frac sqrt 3 2 0 8660 45 1 4 π 50 g 1 8 2 2 displaystyle frac sqrt 2 2 0 7071 2 2 displaystyle frac sqrt 2 2 0 7071 60 1 3 π 66 2 3 g 1 6 3 2 displaystyle frac sqrt 3 2 0 8660 1 2 0 5 75 5 12 π 83 1 3 g 5 24 6 2 4 displaystyle frac sqrt 6 sqrt 2 4 0 9659 6 2 4 displaystyle frac sqrt 6 sqrt 2 4 0 2588 90 1 2 π 100 g 1 4 1 1 0 0 90 degree increments x in degrees 0 90 180 270 360 x in radians 0 π 2 π 3 π 2 2 π x in gons 0 100 g 200 g 300 g 400 g x in turns 0 1 4 1 2 3 4 1 sin x 0 1 0 1 0 cos x 1 0 1 0 1 relationship to complex numbers edit main article trigonometric functions relationship to exponential function euler s formula cos θ displaystyle cos theta and sin θ displaystyle sin theta are the real and imaginary parts of e i θ displaystyle e i theta sine and cosine are used to connect the real and imaginary parts of a complex number with its polar coordinates r φ z r cos φ i sin φ displaystyle z r cos varphi i sin varphi the real and imaginary parts are re z r cos φ displaystyle operatorname re z r cos varphi im z r sin φ displaystyle operatorname im z r sin varphi where r and φ represent the magnitude and angle of the complex number z for any real number θ euler s formula says that e i θ cos θ i sin θ displaystyle e i theta cos theta i sin theta therefore if the polar coordinates of z are r φ z r e i φ displaystyle z re i varphi complex arguments edit domain coloring of 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 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 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 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 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 edit 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 2z sum _ 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 z prod _ n 1 infty left 1 frac z 2 n 2 right alternatively the infinite product for the sine can be proved using complex fourier series proof of the infinite product for the sine using complex fourier series the function cos z x displaystyle cos zx can be decomposed as cos z x z sin π z π n 1 n e i n x z 2 n 2 z c z x π π displaystyle 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 π displaystyle x pi yields cos π z z sin π z π n 1 z 2 n 2 z sin π z π 1 z 2 2 n 1 1 z 2 n 2 displaystyle 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 π cot π z 1 z 2 n 1 z z 2 n 2 displaystyle pi cot pi z frac 1 z 2 displaystyle sum _ n 1 infty frac z z 2 n 2 the function π cot π z displaystyle pi cot pi z is the derivative of ln sin π z c 0 displaystyle ln sin pi z c_ 0 furthermore if d f d z z z 2 n 2 textstyle frac df dz frac z z 2 n 2 then the function f displaystyle f such that the emerged series converges on some open and connected subset of c displaystyle mathbb c is f 1 2 ln 1 z 2 n 2 c 1 textstyle f frac 1 2 ln left 1 frac z 2 n 2 right c_ 1 which can be proved using the weierstrass m test the interchange of the sum and derivative is justified by uniform convergence it follows that ln sin π z ln z n 1 ln 1 z 2 n 2 c displaystyle ln sin pi z ln z displaystyle sum _ n 1 infty ln left 1 frac z 2 n 2 ri...