Chapter 4, Problem 170RE

a.

To determine

To find exact value of expression.

Answer to Problem 170RE

The exact value of ${\cos }^{-1}1$ is $0$

Explanation of Solution

Given:

The expression is ${\cos }^{-1}1$

Concept Used:

The value of ${\cos }^{-1}\left(\cos \theta \right)=\theta$

Calculation:

The expression is ${\cos }^{-1}1$

  ${\cos }^{-1}1={\cos }^{-1}\left(\cos 0\right)=0$

Range of ${\cos }^{-1}\theta$ is $\left[0,\pi \right]$

  $0$ lies in $\left[0,\pi \right]$

Therefore , ${\cos }^{-1}1=0$

Conclusion:

Hence, the exact value of ${\cos }^{-1}1$ is $0$

b.

To determine

To find the exact value of expression.

Answer to Problem 170RE

The exact value of ${\tan }^{-1}\left(-\sqrt{3}\right)$ is $-\frac{\pi }{3}$

Explanation of Solution

Given:

The expression is ${\tan }^{-1}\left(-\sqrt{3}\right)$

Concept Used:

The value of ${\tan }^{-1}\left(-\theta \right)=-{\tan }^{-1}\theta$

Calculation:

The expression is ${\tan }^{-1}\left(-\sqrt{3}\right)$

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

Range of ${\tan }^{-1}\theta$ is $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

  $-\frac{\pi }{3}$ lies in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

Therefore , ${\tan }^{-1}\left(-\sqrt{3}\right)=-\frac{\pi }{3}$

Conclusion:

Hence, the exact value of ${\tan }^{-1}\left(-\sqrt{3}\right)$ is $-\frac{\pi }{3}$