Chapter 14.4, Problem 32PPS

a)

To determine

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

Answer to Problem 32PPS

  $\frac{4\sqrt{2}}{9}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\sin \theta =\frac{1}{3},0<\theta <\frac{\pi }{2}$

Formula used:

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

Calculation:

  $\begin{array}{l}\sin \theta =\frac{1}{3}\\ \cos \theta =\frac{2\sqrt{3}}{3}\\ \sin 2\theta =2\sin \theta \cos \theta \\ \sin 2\theta =2\times \left(\frac{1}{3}\right)\times \left(\frac{2\sqrt{2}}{3}\right)=\frac{4\sqrt{2}}{9}\end{array}$

  

b)

To determine

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

Answer to Problem 32PPS

  $\frac{7}{9}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\sin \theta =\frac{1}{3},0<\theta <\frac{\pi }{2}$

Formula used:

  $\cos 2\theta =2{\cos }^2\theta -1$

Calculation:

  $\begin{array}{l}\cos 2\theta =2{\cos }^2\theta -1\\ \cos 2\theta =2\times {\left(\frac{2\sqrt{2}}{3}\right)}^2-1\\ \boxed{\cos 2\theta =\frac{7}{9}}\end{array}$

c)

To determine

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

Answer to Problem 32PPS

  $\frac{4\sqrt{2}}{7}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\sin \theta =\frac{1}{3},0<\theta <\frac{\pi }{2}$

Formula used:

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

Calculation:

  $\begin{array}{l}\tan 2\theta =\frac{\sin 2\theta }{\cos 2\theta }\\ \tan 2\theta =\frac{\frac{4\sqrt{2}}{\cancel{9}}}{\frac{7}{\cancel{9}}}\\ \tan 2\theta =\frac{4\sqrt{2}}{7}\end{array}$