Question

Find the principal values of ${\sin }^{-1}(-\frac{1}{\sqrt{2}})$

Answer 1

We know that ${\sin }^{-1}x$ denotes an angle in an interval $[\frac{-\pi }{2},\frac{\pi }{2}]$ whose sine is $x$ for $x\in [-1,1]$

$\therefore {\sin }^{-1}\left(\frac{-1}{\sqrt{2}}\right)=$ an angle in

$[\frac{-\pi }{2},\frac{\pi }{2}]$ whose sine is $\frac{-1}{\sqrt{2}}$

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