Trigonometric function

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.

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 $\pi$.

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.

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.

The half-angle identities include

The tangent addition formula iswhere both sides are defined.

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.

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 $\cos x\sim 1-x^2/2$.