Question

If $\theta ={\sin }^{-1}\left[\sin \right(-600°\left)\right]$, then one of the possible value of $\theta$ is:

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{2}$
• C. $\frac{2\pi }{3}$
• D. $-\frac{2\pi }{3}$

Answer 1

The correct option is A

$\frac{\pi }{3}$

Explanation for the correct option.

$\begin{array}{rcl}\theta & =& {\sin }^{-1}\left[\sin \right(-600°\left)\right]\\ & \Rightarrow & \sin \theta =-\sin \left(600°\right)\\ & =& -\sin \left(360°+240°\right)\\ & =& -\sin 240°\left[by\sin \left(360°+\theta \right)=\sin \theta \right]\end{array}$

Now,

$\begin{array}{rcl}\sin \theta & =& -\sin 240°\\ & =& -\sin \left(180°+60°\right)\\ & =& \sin 60°\left[by\sin \left(180°+\theta \right)=-\sin \theta \right]\end{array}$

So,

$\begin{array}{rcl}\sin \theta & =& \sin 60°\\ & \Rightarrow & \sin \theta =\sin \frac{\mathrm{\pi }}{3}\\ \Rightarrow \theta & =& \frac{\mathrm{\pi }}{3}\end{array}$

Hence, option A is correct.