Find the principal value of $\sin^{-1} \left(\dfrac{\sqrt{3}}{2}\right)$
Let $\;$ $\sin^{-1} \left(\dfrac{\sqrt{3}}{2}\right) = y$, $\;$ where $\;$ $- \dfrac{\pi}{2} \leq y \leq \dfrac{\pi}{2}$
$\implies$ $\sin \left(y\right) = \dfrac{\sqrt{3}}{2} = \sin \left(\dfrac{\pi}{3}\right)$
$\implies$ $y = \dfrac{\pi}{3}$
$\therefore \;$ The principal value of $\sin^{-1} \left(\dfrac{\sqrt{3}}{2}\right)$ is $\dfrac{\pi}{3}$