Chapter 7, Problem 33SGA

To determine

To find: the value of $\sin 4\theta$ by using given information.

Answer to Problem 33SGA

  $-\frac{336}{625}$

Explanation of Solution

Given information:

  $\cos \theta =\frac{3}{5},\text{ 0}°<\theta <90°$

Calculation:

Evaluate $\sin \theta$ by using Pythagorean identity as follows:

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

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

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

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

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

Thus the value of $\sin \theta$ is $\frac{4}{5}$ .

Now evaluate $\sin 4\theta$ as follows:

  $\begin{array}{c}\sin 4\theta =2\left(\sin 2\theta \right)\left(\cos 2\theta \right)\left[\text{Double-angle identity of sine}\left(\text{let 4}\theta =2x\right)\right]\end{array}$

  $\begin{array}{c}=2\left(2\sin \theta \cos \theta \right)\left(2{\cos }^2\theta -1\right)\left[\text{Double-angle identity of sine and cosine}\right]\end{array}$

  $\begin{array}{c}=2\left[2\left(\frac{4}{5}\right)\left(\frac{3}{5}\right)\right]\left[2{\left(\frac{3}{5}\right)}^2-1\right]\left[\text{Put the value of sin}\theta \text{ and cos}\theta \right]\end{array}$

  $\begin{array}{c}=\frac{48}{25}\left[\frac{18}{25}-1\right]\end{array}$

  $\begin{array}{c}=\frac{48}{25}\left[\frac{-7}{25}\right]\end{array}$

  $\begin{array}{c}=-\frac{336}{625}\end{array}$

Thus the value of $\sin 4\theta$ is $-\frac{336}{625}$ .