Chapter 7, Problem 21RE

To determine

To calculate: The exact value of the expression ${\sin }^{-1}\left[\cos \left(\frac{2\pi }{3}\right)\right]$ .

Answer to Problem 21RE

The exact value of the expression ${\sin }^{-1}\left[\cos \left(\frac{2\pi }{3}\right)\right]$ is $-\frac{\pi }{6}$ .

Explanation of Solution

Given information:

The expression is ${\sin }^{-1}\left[\cos \left(\frac{2\pi }{3}\right)\right]$ .

Formula used:

For the sine function and its inverse, the following holds:

  $\begin{array}{c}{f}^{-1}\left(f\left(x\right)\right)={\sin }^{-1}\left(sin\left(x\right)\right)\\ =x\end{array}$

Where, $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$

Sine and cosine function related as,

  $\sin \left(x\right)=\cos \left(\frac{\pi }{2}-x\right)$

Calculation:

Consider the expression ${\sin }^{-1}\left[\cos \left(\frac{2\pi }{3}\right)\right]$ .

Recall that sine and cosine function related as,

  $\sin \left(x\right)=\cos \left(\frac{\pi }{2}-x\right)$

Applied it,

  $\begin{array}{c}{\sin }^{-1}\left[\cos \left(\frac{2\pi }{3}\right)\right]={\sin }^{-1}\left[\cos \left(\frac{\pi }{2}-\left(-\frac{\pi }{6}\right)\right)\right]\\ ={\sin }^{-1}\left[\sin \left(-\frac{\pi }{6}\right)\right]\end{array}$

Observe that the angle $-\frac{\pi }{6}$ is lies between $-\frac{\pi }{2}$ and $\frac{\pi }{2}$ .

Recall that for the sine function and its inverse is related as,

  $\begin{array}{c}{f}^{-1}\left(f\left(x\right)\right)={\sin }^{-1}\left(sin\left(x\right)\right)\\ =x\end{array}$

Where, $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$

Simplify further,

  ${\sin }^{-1}\left[sin\left(-\frac{\pi }{6}\right)\right]=-\frac{\pi }{6}$

Thus, the exact value of the expression ${\sin }^{-1}\left[\cos \left(\frac{2\pi }{3}\right)\right]$ is $-\frac{\pi }{6}$ .