Chapter 7, Problem 22RE

To determine

To calculate: The exact value of the expression ${\cos }^{-1}\left[\tan \left(\frac{3\pi }{4}\right)\right]$ .

Answer to Problem 22RE

The exact value of the expression ${\cos }^{-1}\left[\tan \left(\frac{3\pi }{4}\right)\right]$ is $\pi$ .

Explanation of Solution

Given information:

The expression is ${\cos }^{-1}\left[\tan \left(\frac{3\pi }{4}\right)\right]$ .

Formula used:

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

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

Where, $0\le x\le \pi$

Values of the known angle of trigonometric function are:

  $\begin{array}{c}i)\tan \frac{3\pi }{4}=-1\\ ii)\cos \pi =-1\end{array}$

Calculation:

Consider the expression ${\cos }^{-1}\left[\tan \left(\frac{3\pi }{4}\right)\right]$ .

Recall that values of the known angle of trigonometric function are:

  $\begin{array}{c}i)\tan \frac{3\pi }{4}=-1\\ ii)\cos \pi =-1\end{array}$

Using it,

  $\begin{array}{c}{\cos }^{-1}\left[\tan \left(\frac{3\pi }{4}\right)\right]={\cos }^{-1}\left[-1\right]\\ ={\cos }^{-1}\left[\cos \left(\pi \right)\right]\end{array}$

Observe that the angle $\pi$ is lies between $0$ and $\pi$ .

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

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

Where, $0\le x\le \pi$

Simplify further,

  ${\cos }^{-1}\left[\cos \left(\pi \right)\right]=\pi$

Thus, the exact value of the expression ${\cos }^{-1}\left[\tan \left(\frac{3\pi }{4}\right)\right]$ is $\pi$ .