Question

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

Answer 1

Let $y={\sin }^{-1}(-\frac{1}{2})$
$\Rightarrow \sin y=-\frac{1}{2}$
We know that the range of the principal value branch of ${\sin }^{-1}x$ is $[-\frac{\pi }{2},\frac{\pi }{2}]$
$\Rightarrow \sin y=\sin (-\frac{\pi }{6})$
$\therefore y=-\frac{\pi }{6}$
Hence, the principal value ${\sin }^{-1}(-\frac{1}{2})$ is $-\frac{\pi }{6}$