Question

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

Answer 1

Let ${\sin }^{-1}(-\frac{1}{2})=y$.

Then $\sin y=-\frac{1}{2}=-\sin \left(\frac{\pi }{6}\right)=\sin (-\frac{\pi }{6})$.
We know that the range of the principal value of ${\sin }^{-1}x$ is $[-\frac{\pi }{2},\frac{\pi }{2}]$
and $\sin (-\frac{\pi }{6})=-\frac{1}{2}$.
Therefore, the principal value of ${\sin }^{-1}(-\frac{1}{2})$ is $-\frac{\pi }{6}$