Trigonometric function

Trigonometric functions turn an angle into a coordinate or a ratio of lengths. Start at $(1,0)$ on the unit circle and rotate counterclockwise through $\theta$ radians: the resulting point isThe cosine records horizontal position and the sine records vertical position. This definition continues around the whole circle, including negative angles, so the signs encode direction rather than requiring a separate rule for each right triangle.

The unit circle picture explains why these functions are useful beyond measuring triangles. The two coordinates of a point rotating at constant angular speed are periodic functions. Scaling the radius to $A$ and using $\theta =\omega t+\varphi$ gives $x=A\cos (\omega t+\varphi )$ and $y=A\sin (\omega t+\varphi )$. A circular motion becomes two simple repeating traces, connecting Euclidean geometry, simple harmonic motion, and Fourier series.

For an acute angle in a right triangle, rescaling its hypotenuse to one recovers the familiar ratios: sine is opposite side divided by hypotenuse, and cosine is adjacent side divided by hypotenuse. The tangent function is their ratio, $\tan \theta =\sin \theta /\cos \theta$, wherever the denominator is nonzero; the secant function, cosecant, and cotangent give the other reciprocal ratios.

The same names extend to complex numbers through the complex exponential:These definitions agree with the circle interpretation for real angles. For complex arguments, the values need not lie between $-1$ and $1$; the literal circle-coordinate picture is restricted to real arguments. ^[ref]

An inequality involving trigonometric functions. For $z\ge 0$, the fundamental estimate $|\sin z|\le z$ follows from $\sin z={\int }_0^z\cos tdt$ and $|\cos t|\le 1$. Thus $1-\cos z=2{\sin }^2(z/2)\le z^2/2$, and integrating gives $0\le z-\sin z\le z^3/6$. For $0\le z\le \pi$, this implies $0\le z^2-{\sin }^{2}z\le z^4/3$. These bounds give explicit errors for central finite differences applied to sine modes.

The angle-addition identities give $\sin 2x=2\sin x\cos x$, $\cos 2x={\cos }^{2}x-{\sin }^{2}x=1-2{\sin }^{2}x=2{\cos }^{2}x-1$, and $\tan 2x=2\tan x/(1-{\tan }^{2}x)$ wherever the tangent expressions are defined. The sine formula converts ${\sin }^{2}x{\cos }^{2}x$ into ${\sin }^2(2x)/4$, which is useful for trigonometric inequalities and integration.

A trigonometric identity is an equality involving trigonometric functions that holds for every input in its stated domain.

A trigonometric function defined where $\cos x\ne 0$. It has real simple poles at $(n+1/2)\pi$, and $\tan ((n+1/2)\pi +\delta )=-\cot \delta$. This local pole identity generates the large roots near tangent poles.

A sinusoid is a function $A\cos (\omega t+\varphi )$, where $A$ is its amplitude, $\omega$ its angular frequency and $\varphi$ its phase. For $\omega \ne 0$ its period is $2\pi /|\omega |$. A uniformly randomized phase can produce a strictly stationary process even though individual paths look periodic.

Product-to-sum formulas turn products of trigonometric functions into sums at combined arguments. The identity $2\sin A\sin B=\cos (A-B)-\cos (A+B)$ is especially useful for verifying orthogonality of sine Fourier series modes on an interval.

The cosecant is $\csc z=1/\sin z$, wherever the sine denominator is nonzero. The square of the cosecant occurs in the derivative $d(\cot z)/dz=-{\csc }^{2}z$ of the cotangent.

Factor the sine expansion as $\sin z=z[1-z^2/6+z^4/120+O(z^6)]$. To invert the bracket, put $1+A{z}^2+B{z}^4+O(z^6)$ and equate coefficients after multiplication. The $z^2$ coefficient gives $A=1/6$, while the $z^4$ coefficient gives $B=A/6-1/120=7/360$. Dividing by $z$ proves the displayed Laurent series truncation for $z\to 0$, $z\ne 0$. This reciprocal-series calculation is useful when a small sine argument is itself an inverse power of a large variable; multiply the remainder by any outside prefactor when assigning its final order.

The cotangent is the quotient $\cot z=\cos z/\sin z$, wherever the sine denominator is nonzero. Its meromorphic continuation has simple poles at integer multiples of $\pi$.

Away from integer poles,The series is locally uniformly convergent. Integrate $\pi \cot (\pi \zeta )/({\zeta }^2-z^2)$ around the square with sides $\mathrm{Re}\zeta ,\mathrm{Im}\zeta =\pm (N+1/2)$. The cotangent is uniformly bounded there, so the integral tends to zero. The residue theorem then gives $\pi \cot (\pi z)/z+{\sum }_{n\in \mathbb{Z}}(n^2-z^2{)}^{-1}=0$, which is the expansion. Differentiation generates higher integer-lattice pole sums. Setting $w=e^{2\pi iz}$ gives $\pi \cot \pi z=\pi i(w+1)/(w-1)$. On $0<|z|<1$, expansion of each paired summand gives $\pi \cot \pi z=z^{-1}-2{\sum }_{m\ge 0}\zeta (2m+2)z^{2m+1}$, relating this trigonometric function to the Riemann zeta function.

For $\mathrm{Im}z>0$, writing cotangent as $i(e^{2\pi iz}+1)/(e^{2\pi iz}-1)$ and expanding a geometric series gives this Fourier series. Differentiating the cotangent partial-fraction expansion and this Fourier series $k-1$ times yields

For a real angle $\theta$, $\sin \theta$ is the vertical coordinate on the unit circle reached by a counterclockwise rotation through $\theta$ radians. For an acute angle in a right triangle, it is the opposite side divided by the hypotenuse. In particular,The first-quarter rise and second-quarter fall are visible in the circle projection; the lower semicircle gives negative values.

Reflecting the unit circle across the horizontal axis gives $\sin (-\theta )=-\sin \theta$, so sine is an odd function. A full turn gives $\sin (\theta +2\pi )=\sin \theta$, making it a periodic function. For real arguments, its range is $[-1,1]$ and its zeros of a function occur at $\theta =k\pi$ for integers $k$. The cosine describes the perpendicular coordinate of the same point, and the Pythagorean trigonometric identity keeps their two values tied together.

The sine function extends to an entire function on the complex plane by the displayed exponential formula. Substituting $z=x+iy$ and grouping the real and imaginary terms gives $\sin (x+iy)=\sin x\cosh y+i\cos x\sinh y$, for real $x,y$. Thus its values grow exponentially in imaginary directions, even though the real sine is bounded.

The paired zero factors converge locally uniformly because $\sum n^{-2}$ converges. Hadamard factorization for the order-one entire sine function gives a possible exponential multiplier; oddness removes its linear exponent and the derivative at zero fixes its constant. This product turns a Dirichlet oscillator determinant ratio into an elementary sine ratio.

For a real angle $\theta$, $\cos \theta$ is the horizontal coordinate on the unit circle reached by a counterclockwise rotation through $\theta$ radians. For an acute angle in a right triangle, it is the adjacent side divided by the hypotenuse. Basic values includeAt $\theta =2\pi /3$, the horizontal coordinate is $-1/2$ even though the vertical sine coordinate remains positive. A positive angle need not produce a positive cosine.

Reflection across the horizontal axis leaves this coordinate unchanged, giving $\cos (-\theta )=\cos \theta$: cosine is an even function. A full turn gives $\cos (\theta +2\pi )=\cos \theta$. For real arguments, its range is $[-1,1]$ and its zeros of a function occur at $\theta =\pi /2+k\pi$. A quarter-turn shifts between the two coordinate traces, with $\cos \theta =\sin (\theta +\pi /2)$.

The extension of the cosine to the complex plane is $\cos z=(e^{iz}+e^{-iz})/2$. Splitting the complex exponential function into its real part and imaginary part gives the displayed formula in terms of hyperbolic cosine and hyperbolic sine.

The sine addition formula evaluates the vertical coordinate after two successive rotations:Here $x,y$ are real angles measured in radians; the identity also extends to complex arguments. It is useful when a new angle can be assembled from familiar ones. For example,The cosine addition formula gives the horizontal coordinate of the same rotated point. Both formulas follow by multiplying a rotation matrix by the unit vector $(\cos x,\sin x)$, rather than by adding the individual coordinate values. ^[ref]

The cosine addition formula gives the horizontal coordinate after two successive rotations:The minus sign comes from the way a counterclockwise rotation mixes horizontal and vertical coordinates:The two rows recover the cosine addition formula and sine addition formula together.

With $x=\pi /6$ and $y=\pi /4$, the resulting angle is $5\pi /12$, andAlthough the horizontal coordinate becomes small, the vertical one approaches one; their squared sum still obeys the Pythagorean trigonometric identity. Taking $y=-x$ in the cosine addition formula, together with the even function property of cosine and odd function property of sine, also recovers that identity.

For every real angle $\theta$,It is the equation of the unit circle written in terms of its sine and cosine coordinates. For an acute angle, dividing the Pythagorean theorem by the square of the hypotenuse gives the same relation in a right triangle. The circle interpretation explains why it remains true outside that first-quadrant triangle.

For example, if $\sin \theta =3/5$, the identity gives ${\cos }^2\theta =16/25$, so $\cos \theta$ is either $4/5$ or $-4/5$. The sign needs the angle's location: it is positive in the first quadrant and negative in the second. Thus the identity determines the magnitude of the missing coordinate without determining the direction.

The relation is useful for eliminating one squared trigonometric function, checking that a coordinate pair has unit length, and checking that the columns of a rotation matrix are unit vectors. It also holds for complex arguments under the complex exponential definitions, although then the squares are algebraic squares rather than squared moduli.

The tangent is the trigonometric function $\tan x=\sin x/\cos x$ where $\cos x\ne 0$.

For a point $B$ of a curve, let its ordinate meet the chosen coordinate axis at $C$, and its tangent meet that axis at $E$. The segment $CE$ is the subtangent. It depends on the chosen axis. For the standard parabola, $CE=2CD$, where $D$ is its vertex of a parabola.

The secant function is the reciprocal of the cosine, $\sec x=1/\cos x$, where the denominator is nonzero. Its derivative is $\sec x\tan x$, and ${\sec }^{2}x=1+{\tan }^{2}x$ follows from the Pythagorean trigonometric identity.

These identities supply the formulas used in the tangent half-angle substitution. The half-angle identities include

The tangent addition formula iswhere both sides are defined.

For real $-1\le x\le 1$, the principal arccosine is the unique angle $\theta \in [0,\pi ]$ satisfying $\cos \theta =x$. Restricting to this interval makes the cosine one-to-one. Differentiating the identity on $-1<x<1$ gives $d(\arccos x)/dx=-1/\sqrt{1-x^2}$. Other solutions of $\cos \theta =x$ are $\theta =\pm \arccos x+2\pi n$, with integer $n$; an initial condition can select one of these branches.

Set $u=e^{iw}$ in $\cos w=z$. The resulting quadratic equation is $u^2-2zu+1=0$, with roots $z\pm i\sqrt{1-z^2}$. Their product is one, so taking a complex logarithm never encounters zero. Solving the exponential equation gives the displayed full inverse relation, with integer $n$. The principal complex logarithm and principal square root do not make both signs identical to the principal inverse cosine: at $z=0$ they yield $\pm \pi /2$, while principal $\arccos 0=\pi /2$. A single analytic branch requires compatible choices of sign and branches.

The inverse of the tangent restricted to $(-\pi /2,\pi /2)$, taking real values in that interval. Its derivative is $(1+x^2{)}^{-1}$. A formula involving an inverse tangent of a ratio may need a continuous branch of a multivalued function to describe a smooth field through a pole of that ratio.

For real $A,B$ with $AB<1$, the principal inverse tangent satisfiesTo prove it, multiply the complex numbers $1+iA$ and $1+iB$. Their product is $(1-AB)+i(A+B)$, whose positive real part places its complex argument in $(-\pi /2,\pi /2)$. The individual complex arguments are $\arctan A$ and $\arctan B$, and their sum is in $(-\pi ,\pi )$. Multiplication adds these complex arguments, so the positive real part forces the sum into the same principal range and gives the displayed formula.

If $AB>1$, both $A,B$ have the same sign. Add $\pi$ to the right-hand side when $A,B>0$, and subtract $\pi$ when $A,B<0$. If $AB=1$, the sum is $\pi /2$ or $-\pi /2$ according to the sign of $A+B$. These cases record the branch information that can be lost by applying the tangent addition formula alone.

The inverse sine is one member of the inverse trigonometric functions.

As $x\to 0$, the limit $\sin x/x\to 1$ gives $\sin x\sim x$, while the Taylor series gives $\arcsin x\sim x$ and $\cos x\sim 1-x^2/2$.

The tangent half-angle substitution $t=\tan (x/2)$ converts rational trigonometric expressions into rational functions: $\sin x=2t/(1+t^2)$, $\cos x=(1-t^2)/(1+t^2)$ and $dx=2dt/(1+t^2)$.