Find the principal value of

Question:

Find the principal value of $\cos ^{-1}\left(-\frac{1}{2}\right)$

Solution:

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

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

$[0, \pi]$ and $\cos \left(\frac{2 \pi}{3}\right)=-\frac{1}{2}$

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