Discussion Forum

The principal value of $\sin^{-1}\left[\sin\frac{5\pi}{3}\right]$ is

Answer:

Explanation:

Let $\theta = \sin^{-1}\left[\sin\frac{5\pi}{3}\right]$

$\Rightarrow\sin\theta = \sin\frac{5\pi}{3}=\sin\left[2\pi-\frac{\pi}{3}\right]$

$\Rightarrow\sin\theta =- \sin\frac{\pi}{3}=\sin\left(\frac{-\pi}{3}\right)$  $\because\sin\left(-\theta\right)$ = $- \sin\theta$

Therefore, principal value of $\sin^{-1}\left[\sin\frac{5\pi}{3}\right]$ is $\frac{-\pi}{3}$

as principal value of $\sin^{-1}$  x lies between $\frac{-\pi}{2}$ and $\frac{\pi}{2}$