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

$-\frac{2\pi}{3}$ $-\frac{\pi}{3}$ $\frac{4\pi}{3}$  $\frac{5\pi}{3}$

$-\frac{\pi}{3}$

$ sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ is an angle $\theta ∈ \left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ such that $ sin \theta = - \frac{\sqrt{3}}{2}$. Clearly, such an angle is $-\frac{\pi}{3}$.