Question

Find the principal value of ${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right).$

Answer 1

Let $y={\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)$
$\Rightarrow \sin y=\frac{1}{\sqrt{2}}$
$\Rightarrow \sin y=\left(\frac{\pi }{4}\right)$
We know that the range of the principal value branch of ${\sin }^{-1}x$ is $[-\frac{\pi }{2},\frac{\pi }{2}]$

$\therefore y=\frac{\pi }{4}$
Hence, the principal value of ${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)$ is $\frac{\pi }{4}$