Chapter 4.7, Problem 28E

Solution

To find: find the exact value.

Exact value of $\sin \left({\tan }^{-1}\left(-1\right)\right)$ is $-\frac{1}{\sqrt{2}}$ .

Given: $\sin \left({\tan }^{-1}\left(-1\right)\right)$

Concept Used: inverse trigonometry $\cos \left(-\theta \right)=\cos \theta$ and $\tan \left(-\frac{\pi }{4}\right)=-1$

Calculation:

  $\begin{array}{c}\sin \left({\tan }^{-1}\left(-1\right)\right)=\sin \left(-\frac{\pi }{4}\right)\left\{\because \tan \left(-\frac{\pi }{4}\right)=-1\right\}\\ =-\sin \left(\frac{\pi }{4}\right)\left\{\because \sin \left(-\theta \right)=-\sin \left(\theta \right)\right\}\\ =-\frac{1}{\sqrt{2}}\end{array}$

Conclusion:

Exact value of $\sin \left({\tan }^{-1}\left(-1\right)\right)$ is $-\frac{1}{\sqrt{2}}$ .