Chapter 7, Problem 23RE

To determine

To calculate: The exact value of the expression ${\tan }^{-1}\left[\tan \left(\frac{7\pi }{4}\right)\right]$ .

Answer to Problem 23RE

The exact value of the expression ${\tan }^{-1}\left[\tan \left(\frac{7\pi }{4}\right)\right]$ is $-\frac{\pi }{4}$ .

Explanation of Solution

Given information:

The expression is ${\tan }^{-1}\left[\tan \left(\frac{7\pi }{4}\right)\right]$ .

Formula used:

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

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

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

Addition of integer multiple of $\pi$ to $\theta$ ,the value of the tangent function remain unchanged.

That is, for all $\theta$

  $\tan \left(k\pi +\theta \right)=\tan \theta$

Where $k$ is any integer.

Calculation:

Consider the expression ${\tan }^{-1}\left[\tan \left(\frac{7\pi }{4}\right)\right]$ .

Recall that addition of integer multiple of $\pi$ to $\theta$ , the value of the tangent function remain

unchanged. That is, for all $\theta$

  $\tan \left(k\pi +\theta \right)=\tan \theta$

Where $k$ is any integer .

Applied it,

  $\begin{array}{c}{\tan }^{-1}\left[\tan \left(\frac{7\pi }{4}\right)\right]={\tan }^{-1}\left[\tan \left(2\pi +\left(-\frac{\pi }{4}\right)\right)\right]\\ ={\tan }^{-1}\left[\tan \left(-\frac{\pi }{4}\right)\right]\end{array}$

Observe that the angle $-\frac{\pi }{4}$ is lies between $-\frac{\pi }{2}$ and $\frac{\pi }{2}$ .

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

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

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

Simplify further,

  ${\tan }^{-1}\left[\tan \left(-\frac{\pi }{4}\right)\right]=-\frac{\pi }{4}$

Thus, the exact value of the expression ${\tan }^{-1}\left[\tan \left(\frac{7\pi }{4}\right)\right]$ is $-\frac{\pi }{4}$ .