Arcsin — Essentials

ESSENTIALS · FREE

Definition

$y = \arcsin x \iff \sin y = x$ , with $D = [-1, 1]$ and $R = \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ (closed - the ends are reached).

$x$	$0$	$\dfrac{1}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{3}}{2}$	$1$
$\arcsin x$	$0$	$\dfrac{\pi}{6}$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{3}$	$\dfrac{\pi}{2}$

Negative inputs: $\arcsin(-x) = -\arcsin x$ - arcsin is odd.

Job	Move
Free $x$ from $\sin(\ldots) = k$	apply $\arcsin$ to both sides
Free $x$ from $\arcsin(\ldots) = k$	apply $\sin$ to both sides
$\cos\left(\arcsin\dfrac{3}{5}\right)$ and friends	draw the triangle, Pythagoras gives the third side

$x^2 + y^2 = 1 \Rightarrow \arcsin x + \arcsin y = \dfrac{\pi}{2}$ (check the condition first)