Find the principal value of

Let $\sin ^{-1}\left(-\frac{1}{2}\right)=y$. Then $\sin y=-\frac{1}{2}=-\sin \left(\frac{\pi}{6}\right)=\sin \left(-\frac{\pi}{6}\right)$.

We know that the range of the principal value branch of $\sin ^{-1}$ is

$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $\sin \left(-\frac{\pi}{6}\right)=-\frac{1}{2}$

Therefore, the principal value of $\sin ^{-1}\left(-\frac{1}{2}\right)$ is $-\frac{\pi}{6}$.