Chapter 7.4, Problem 8CFU

To determine

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

Answer to Problem 8CFU

  $\begin{array}{l}\sin 2\theta =\frac{4\sqrt{21}}{25}\end{array}$

  $\begin{array}{c}\cos 2\theta =\frac{17}{25}\end{array}$

  $\begin{array}{c}\tan 2\theta =\frac{4\sqrt{21}}{17}\end{array}$

Explanation of Solution

Given information:

  $\sin \theta =\frac{2}{5},\text{ 0}°<\theta <90°$

Calculation:

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

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

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

  $\begin{array}{c}=\sqrt{1-\frac{4}{25}}\end{array}$

  $\begin{array}{c}=\sqrt{\frac{21}{25}}\end{array}$

  $\begin{array}{c}=\frac{\sqrt{21}}{5}\end{array}$

Thus the value of $\cos \theta$ is $\frac{\sqrt{21}}{5}$ .

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

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

  $\begin{array}{c}=2\left(\frac{2}{5}\right)\left(\frac{\sqrt{21}}{5}\right)\left[\text{Put the value of sin}\theta \text{ and cos}\theta \right]\end{array}$

  $\begin{array}{c}=\frac{4\sqrt{21}}{25}\end{array}$

Thus $\sin 2\theta$ is $\frac{4\sqrt{21}}{25}$ .

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

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

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

  $\begin{array}{c}=1-\frac{8}{25}\end{array}$

  $\begin{array}{c}=\frac{17}{25}\end{array}$

Thus $\cos 2\theta$ is $\frac{17}{25}$ .

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

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

  $\begin{array}{c}=\frac{\left(\frac{4\sqrt{21}}{25}\right)}{\left(\frac{17}{25}\right)}\left[\text{Put the value of sin2}\theta \text{ and cos2}\theta \right]\end{array}$

  $\begin{array}{c}=\frac{4\sqrt{21}}{17}\end{array}$

Thus $\tan 2\theta$ is $\frac{4\sqrt{21}}{17}$ .