Find the principal value of

Let $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)=y$. Then, $\cos y=\frac{\sqrt{3}}{2}=\cos \left(\frac{\pi}{6}\right)$.

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

$[0, \pi]$ and $\cos \left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}$.

Therefore, the principal value of $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$ is $\frac{\pi}{6}$.