Chapter 13, Problem 13CT

(a)

To determine

To find: The value of expression $\tan 2\theta$ .

Answer to Problem 13CT

The value of the expression is $\tan 2\theta =-\frac{120}{119}$ .

Explanation of Solution

Given information:

The given expression is $\sin \theta =-\frac{5}{13}$ and $\cos \theta >0$ .

Calculation:

Consider the function.

  $\sin \theta =-\frac{5}{13}$ and $\cos \theta >0$ .

By apply Pythagorean theorem,

  $\begin{array}{c}\cos \theta =\sqrt{1-{\left(-\frac{5}{13}\right)}^2}\\ =\sqrt{1-\frac{25}{169}}\\ =\sqrt{\frac{144}{169}}\\ =\frac{12}{13}\end{array}$

Now,

  $\begin{array}{c}\tan \theta =\frac{\left(-\frac{5}{13}\right)}{\left(\frac{12}{13}\right)}\\ =\frac{-5}{12}\end{array}$

We have the double angle formula:

  $\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$

Substitute the value of $\tan \theta =\frac{-5}{12}$

  $\begin{array}{c}\tan 2\theta =\frac{2\cdot -\frac{5}{12}}{1-\left(-\frac{5}{12}\right)}\\ =\frac{\left(-\frac{5}{6}\right)}{\left(\frac{119}{144}\right)}\\ =-\frac{120}{119}\end{array}$

Therefore, the value of the expression is $\tan 2\theta =-\frac{120}{119}$ .

(b)

To determine

To find: The value of expression $\tan \frac{\theta }{2}$ .

Answer to Problem 13CT

The value of the expression is $\tan \frac{\theta }{2}=-\frac{1}{5}$ .

Explanation of Solution

Given information:

The given expression is $\sin \theta =-\frac{5}{13}$ and $\cos \theta >0$ .

Calculation:

Consider the function.

Substitute the values,

  $\begin{array}{c}\tan \frac{\theta }{2}=\frac{1-\frac{12}{13}}{-\frac{5}{13}}\\ =-\frac{1}{5}\end{array}$

Therefore, the value of the expression is $\tan \frac{\theta }{2}=-\frac{1}{5}$ .