# How do you evaluate sin^-1(-sqrt2/2) without a calculator?

Nov 30, 2016

${\sin}^{- 1} \left(- \frac{\sqrt{2}}{2}\right) = - \frac{\pi}{4}$

#### Explanation:

This is a known value for $\sin \left(x\right)$, you should remember that:

$\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

and that:

$\sin \left(- x\right) = - \sin \left(x\right)$

Although many other angles have the same value for sine, namely:

$\alpha = - \frac{\pi}{4} + 2 k \pi$
and
$\alpha = - \frac{3 \pi}{4} + 2 k \pi$

conventionally $\arcsin \left(x\right)$ is defined in the interval $\left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$
so you pick the value:

$x = - \frac{\pi}{4}$