Inverse Trig: Choosing One Angle — Essentials

ESSENTIALS · FREE

The four windows

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

Trap	Rule
$\sin^{-1} x$	means $\arcsin x$ , never $\dfrac{1}{\sin x}$
$\arcsin 2$	undefined - sine never reaches $2$
$\arcsin(\sin x)$	equals $x$ only when $x$ is already inside the window
$\sin(\arcsin x)$	always $x$ , for $x$ in $[-1, 1]$