Chapter 14.4, Problem 31PPS

a)

To determine

To Find: The value of $\sin 2\theta$

Answer to Problem 31PPS

  $\frac{24}{25}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\cos \theta =\frac{4}{5},0^{\circ }<\theta <{90}^{\circ }$

Formula used:

  $\sin 2\theta =2\sin \theta \cos \theta$

Calculation:

  $\begin{array}{l}\cos \theta =\frac{4}{5}\\ \sin \theta =\frac{3}{5}\\ \sin 2\theta =2\sin \theta \cos \theta \\ \sin 2\theta =2\times \left(\frac{3}{5}\right)\times \left(\frac{4}{5}\right)=\frac{24}{25}\end{array}$

  

b)

To determine

To Find: The value of $\cos 2\theta$

Answer to Problem 31PPS

  $\frac{7}{25}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\cos \theta =\frac{4}{5},0^{\circ }<\theta <{90}^{\circ }$

Formula used:

  $\cos 2\theta =2{\cos }^2\theta -1$

Calculation:

  $\begin{array}{l}\cos 2\theta =2{\cos }^2\theta -1\\ \cos 2\theta =2\times {\left(\frac{4}{5}\right)}^2-1\\ \boxed{\cos 2\theta =\frac{7}{25}}\end{array}$

c)

To determine

To Find:The value of $\tan 2\theta$

Answer to Problem 31PPS

  $\frac{24}{7}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\cos \theta =\frac{4}{5},0^{\circ }<\theta <{90}^{\circ }$

Formula used:

  $\tan 2\theta =\frac{\sin 2\theta }{\cos 2\theta }$

Calculation:

  $\begin{array}{l}\tan 2\theta =\frac{\sin 2\theta }{\cos 2\theta }\\ \tan 2\theta =\frac{\frac{24}{\cancel{25}}}{\frac{7}{\cancel{25}}}\\ \tan 2\theta =\frac{24}{7}\end{array}$