Chapter 5.2, Problem 58E

To determine

Write the first expression in terms of second if the terminal point determined by t.

Answer to Problem 58E

  $\tan t=\sqrt{\frac{1}{{\cos }^{2}t}-1}$

Explanation of Solution

Given information:

  $\tan t,\cos t;QuadrantIII$

Formula used:

  $\begin{array}{l}{\sin }^2\theta +{\cos }^2\theta =1\\ \\ \tan \theta =\frac{\sin \theta }{\cos \theta }\end{array}$

  $\tan \theta =\frac{\sin \theta }{\cos \theta }$

Now converting $\sin \theta$ to $\cos \theta$ by using ${\sin }^2\theta +{\cos }^2\theta =1$ and putting value of $\theta =t$

  $\tan t=\frac{\pm \sqrt{1-{\cos }^{2}t}}{\cos t}$

As we know that in $III$ quadrant tan is positive.

Hence the value of $\tan t=\sqrt{\frac{1}{{\cos }^{2}t}-1}$