Question

What is the principal value of ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$?

Answer 1

Let $y={\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
Then,
$$\begin{gathered} \sin y=-\frac{\sqrt{3}}{2}=\sin \left(-\frac{\mathrm{\pi }}{3}\right) \\ y=-\frac{\mathrm{\pi }}{3}\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right] \end{gathered}$$

Here, $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$ is the range of the principal value branch of inverse sine function.

∴ ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)=-\frac{\mathrm{\pi }}{3}$