Chapter 14.8, Problem 1ST

To determine

To find:the value of the given trigonometric function.

Answer to Problem 1ST

  $y=\frac{4\pi }{3}$

Explanation of Solution

Given:

The given trigonometric function is ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ .

Calculation:

As thegiven trigonometric function is ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ .

Now, to find the value of the given trigonometric function. Therefore,

  $\begin{array}{l}\text{let}y={\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\{\text{given equation}}\\ \therefore y=-{\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)\left\{{\sin }^{-1}\left(-x\right)=-{\sin }^{-1}x\right\}\\ \Rightarrow y=-{\sin }^{-1}\left(\sin \frac{\pi }{3}\right)\left\{\because \left(\sin \frac{\pi }{3}\right)=\frac{\sqrt{3}}{2}\right\}\\ \Rightarrow y=-\frac{\pi }{3}\left\{{\sin }^{-1}\left(-x\right)=-{\sin }^{-1}x\right\}\end{array}$

As $\sin x$ lies in quadrant II, therefore,

  $y=\pi -\left(-\frac{\pi }{3}\right)=\pi +\frac{\pi }{3}=\frac{4\pi }{3}$

Hence, the value of the given trigonometric function is $y=\frac{4\pi }{3}$ .