Arcsin — Cheat sheet — Mr. Integral

Definition

$y = \arcsin x \iff \sin y = x$ with $y \in \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ . $D = [-1, 1]$ , $R = \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ .

Values worth knowing

$x$	$\arcsin x$
$-1$	$-\dfrac{\pi}{2}$
$-\dfrac{1}{2}$	$-\dfrac{\pi}{6}$
$0$	$0$
$\dfrac{1}{2}$	$\dfrac{\pi}{6}$
$\dfrac{\sqrt{2}}{2}$	$\dfrac{\pi}{4}$
$1$	$\dfrac{\pi}{2}$

Inverting

Trapped inside	Apply
$\sin(\ldots)$	$\arcsin$ to both sides
$\arcsin(\ldots)$	$\sin$ to both sides

Non-standard ratios

Let $\alpha$ be the inverse expression, draw the right triangle with that opposite side and hypotenuse, and use Pythagoras for the third side. Read any function of $\alpha$ off it.

The complementary sum

$x^2 + y^2 = 1 \Rightarrow \arcsin x + \arcsin y = \dfrac{\pi}{2}$ - the two legs of a unit-hypotenuse triangle, so the angles are complementary.

Properties

Property	Note
Odd	$\arcsin(-x) = -\arcsin x$
Increasing	across the whole domain
Closed range	$\pm\dfrac{\pi}{2}$ are reached
Outside $[-1,1]$	undefined