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

The Correct Option is 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 $\frac{\pi }{2}.$