Find the value of $\cos ^{-1}\left(\frac{1}{2}\right)+2 \sin ^{-1}\left(\frac{1}{2}\right)$
Question:

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

Solution:

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

$\therefore \cos ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}$

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

$\therefore \sin ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{6}$

$\therefore \cos ^{-1}\left(\frac{1}{2}\right)+2 \sin ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}+\frac{2 \pi}{6}=\frac{\pi}{3}+\frac{\pi}{3}=\frac{2 \pi}{3}$