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 $\frac{\pi }{2}$