Chapter 7.4, Problem 22E

To determine

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

Answer to Problem 22E

  $\begin{array}{l}\sin 2\theta =\frac{4\sqrt{2}}{9}\end{array}$

  $\begin{array}{c}\cos 2\theta =\frac{7}{9}\end{array}$

  $\begin{array}{c}\tan 2\theta =\frac{4\sqrt{2}}{7}\end{array}$

Explanation of Solution

Given information:

  $\sin \theta =\frac{1}{3},\text{ 0}<\theta <\frac{\pi }{2}$

Calculation:

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

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

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

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

  $\begin{array}{c}=\sqrt{\frac{8}{9}}\end{array}$

  $\begin{array}{c}=\frac{2\sqrt{2}}{3}\end{array}$

Thus the value of $\cos \theta$ is $\frac{2\sqrt{2}}{3}$ .

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

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

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

  $\begin{array}{c}=\frac{4\sqrt{2}}{9}\end{array}$

Thus $\sin 2\theta$ is $\frac{4\sqrt{2}}{9}$ .

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

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

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

  $\begin{array}{c}=1-\frac{2}{9}\end{array}$

  $\begin{array}{c}=\frac{7}{9}\end{array}$

Thus $\cos 2\theta$ is $\frac{7}{9}$ .

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

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

  $\begin{array}{c}=\frac{\left(\frac{4\sqrt{2}}{9}\right)}{\left(\frac{7}{9}\right)}\left[\text{Put the value of sin2}\theta \text{ and cos2}\theta \right]\end{array}$

  $\begin{array}{c}=\frac{4\sqrt{2}}{7}\end{array}$

Thus $\tan 2\theta$ is $\frac{4\sqrt{2}}{7}$ .