Chapter 7.4, Problem 9CFU

To determine

To find: the value of $\sin 2\theta ,\cos 2\theta$ and $\tan 2\theta$ by using given information.

Answer to Problem 9CFU

  $\begin{array}{c}\sin 2\theta =\frac{24}{25}\end{array}$

  $\begin{array}{c}\cos 2\theta =-\frac{7}{25}\end{array}$

  $\begin{array}{c}\tan 2\theta =-\frac{24}{7}\end{array}$

Explanation of Solution

Given information:

  $\tan \theta =\frac{4}{3},\pi <\theta <\frac{3\pi }{2}$

Calculation:

Evaluate $\cos \theta$ as shown below.

  $\begin{array}{c}\sec \theta =\sqrt{1+{\tan }^2\theta }\left[\text{Pythagorean identity}\right]\end{array}$

  $\begin{array}{c}\frac{1}{\cos \theta }=\sqrt{1+{\left(\frac{4}{3}\right)}^2}\left[\text{Put the value of tan}\theta \right]\end{array}$

  $\begin{array}{c}=\sqrt{1+\frac{16}{9}}\end{array}$

  $\begin{array}{c}=\sqrt{\frac{25}{9}}\end{array}$

  $\begin{array}{c}\frac{1}{\cos \theta }=\frac{5}{3}\end{array}$

  $\begin{array}{c}\cos \theta =\frac{3}{5}\end{array}$

Since $\pi <\theta <\frac{3\pi }{2}$ the value of cosine will be negative. Thus the value of $\cos \theta$ is $-\frac{3}{5}$ .

Evaluate $sin\theta$ by using the Pythagorean identity as shown below.

  $\begin{array}{c}\sin \theta =\sqrt{1-{\cos }^2\theta }\end{array}$

  $\begin{array}{c}=\sqrt{1-{\left(-\frac{3}{5}\right)}^2}\left[\text{Put the value of cos}\theta \right]\end{array}$

  $\begin{array}{c}=\sqrt{1-\frac{9}{25}}\end{array}$

  $\begin{array}{c}=\sqrt{\frac{16}{25}}\end{array}$

  $\begin{array}{c}=\frac{4}{5}\end{array}$

Since $\pi <\theta <\frac{3\pi }{2}$ the value of sine will be negative. Thus the value of $sin\theta$ is $-\frac{4}{5}$ .

Now evaluate $\sin 2\theta$ by using the double angle identity of sine as follows:

  $\begin{array}{c}\sin 2\theta =2\sin \theta \cos \theta \end{array}$

  $\begin{array}{c}=2\left(-\frac{4}{5}\right)\left(-\frac{3}{5}\right)\left[\text{Put the value of sin}\theta \text{ and cos}\theta \right]\end{array}$

  $\begin{array}{c}=\frac{24}{25}\end{array}$

Thus $\sin 2\theta$ is $\frac{24}{25}$ .

Now evaluate $\cos 2\theta$ by using the double angle identity of cosine as follows:

  $\begin{array}{c}\cos 2\theta =1-2{\sin }^2\theta \end{array}$

  $\begin{array}{c}=1-2{\left(-\frac{4}{5}\right)}^2\left[\text{Put the value of sin}\theta \right]\end{array}$

  $\begin{array}{c}=1-\frac{32}{25}\end{array}$

  $\begin{array}{c}=-\frac{7}{25}\end{array}$

Thus $\cos 2\theta$ is $-\frac{7}{25}$ .

Now evaluate $\tan 2\theta$ by using the quotient identity of tangent as follows:

  $\begin{array}{c}\tan 2\theta =\frac{\sin 2\theta }{\cos 2\theta }\end{array}$

  $\begin{array}{c}=\frac{\left(\frac{24}{25}\right)}{\left(-\frac{7}{25}\right)}\left[\text{Put the value of sin2}\theta \text{ and cos2}\theta \right]\end{array}$

  $\begin{array}{c}=-\frac{24}{7}\end{array}$

Thus $\tan 2\theta$ is $-\frac{24}{7}$ .