= Trigonometric function
{wiki}
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 is
$$
(\cos\theta,\sin\theta).
$$
The <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>.
\Image[/trigonometric-circle-projections.png]
{title=Sine and cosine as signed circle coordinates and periodic traces}
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 $\theta=\omega t+\phi$ gives $x=A\cos(\omega t+\phi)$ and $y=A\sin(\omega t+\phi)$. 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>:
$$
\cos z=\frac{e^{iz}+e^{-iz}}2,\qquad
\sin z=\frac{e^{iz}-e^{-iz}}{2i}.
$$
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. https://dlmf.nist.gov/4.14[NIST Digital Library of Mathematical Functions, §4.14: exponential definitions and periodicity]{ref}
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