Inverse Trig: Choosing One Angle — Cheat sheet

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The idea

Trig functions are not one-to-one, so each is restricted to a stretch where it is, and only that stretch is inverted. The restriction becomes the inverse's range.

The four windows

Function	Domain	Range
$\arcsin$	$[-1, 1]$	$\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$
$\arccos$	$[-1, 1]$	$[0, \pi]$
$\arctan$	$\mathbb{R}$	$\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$
$\text{arccot}$	$\mathbb{R}$	$(0, \pi)$

Round trips

Expression	Value
$\sin(\arcsin x)$	$x$ , for every $x$ in $[-1, 1]$
$\arcsin(\sin x)$	only $x$ when $x$ is already inside the range

Trap

Domains are forced by the original function's reach: sine never leaves $[-1, 1]$ , so $\arcsin 2$ does not exist. Tangent reaches everything, so $\arctan$ accepts everything.