Question

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

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

Answer 1

The correct option is C

$-\frac{\mathrm{\pi }}{3}$

Explanation for the correct option

Let $\theta ={\sin }^{-1}\left(\sin \frac{5\mathrm{\pi }}{3}\right)$

$\sin \theta =\sin \frac{5\mathrm{\pi }}{3}$

$$\begin{gathered} \sin \theta =\sin \left(2\mathrm{\pi }-\frac{\mathrm{\pi }}{3}\right)\mathbf{}\left[\because \mathrm{sin\theta }=\sin \left(2\pi -\theta \right)\right] \\ \sin \theta =-\sin \left(\frac{\mathrm{\pi }}{3}\right)=\sin \left(-\frac{\mathrm{\pi }}{3}\right) \end{gathered}$$

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

Hence the correct option is option(C)