What is $\sin^{-1}\left(\sin \frac{4 \pi}{5}\right)$?

Explanation

1. 01

   $\sin \frac{4 \pi}{5} = \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$.

2. 02

   The principal value with that sin is $\frac{\pi}{5}$.