Chapter 14, Problem 13CR

To determine

To find:The correct option.

Answer to Problem 13CR

The correct answer is (c).

Explanation of Solution

Given:

  ${\csc }^{-1}\left(-\frac{2\sqrt{3}}{3}\right)$

Calculation:

Solve the given term,

  ${\csc }^{-1}\left(-\frac{2\sqrt{3}}{3}\right)={\sin }^{-1}\left(-\frac{3}{2\sqrt{3}}\right)$

Let

  $\begin{array}{l}y={\sin }^{-1}\left(-\frac{3}{2\sqrt{3}}\right)\\ y={\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\end{array}$

Then,

  $\sin y=-\frac{\sqrt{3}}{2}\text{ and }-\frac{\pi }{2}\le y\le \frac{\pi }{2}$

We know,

  $\sin \left(-\frac{\pi }{3}\right)=-\frac{\sqrt{3}}{2}$

So,

  ${\csc }^{-1}\left(-\frac{2\sqrt{3}}{3}\right)=-\frac{\pi }{3}$

Thus, the correct answer is (c).