Question

The principal value of $\sin^{-1} \left(\sin \frac{5\pi}{3}\right) $ is :

• A. $ - \frac{5 \pi}{3} $
• B. $ \frac{5 \pi}{3} $
• C. $ - \frac{\pi}{3} $
• D. $ \frac{4 \pi}{3} $

Answer 1

Answer: (C)

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