Chapter 14.4, Problem 12PPS

a)

To determine

ToFind:The value of $\sin 2\theta$

Answer to Problem 12PPS

  $\sin 2\theta =-\frac{4\sqrt{5}}{9}$

Explanation of Solution

Given information:

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

Calculation:

  

  $\begin{array}{l}\sin \theta =\frac{2}{3}\\ \cos \theta =\frac{\sqrt{5}}{3}\\ \sin 2\theta =2\sin \theta \cos \theta =2\times \frac{2}{3}\times \frac{-\sqrt{5}}{3}=\frac{-4\sqrt{5}}{3}\end{array}$

b)

To determine

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

Answer to Problem 12PPS

  $\cos 2\theta =\frac{1}{9}$

Explanation of Solution

Given information:

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

Calculation:

  

  $\cos 2\theta =2{\cos }^2\theta -1=2\times {(\frac{-\sqrt{5}}{2})}^2-1=\frac{1}{9}$

c)

To determine

To Find:The value of $\sin \frac{\theta }{2}$

Answer to Problem 12PPS

  $\sin \frac{\theta }{2}=\frac{2}{3}$

Explanation of Solution

Given information:

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

Calculation:

  

  $\sin \frac{\theta }{2}=\overline{)\frac{1-\cos 2\theta }{2}}=\overline{)\frac{1-\frac{1}{9}}{2}}=\frac{2}{3}$

d)

To determine

To Find:The value of $\cos \frac{\theta }{2}$

Answer to Problem 12PPS

  $\cos \frac{\theta }{2}=\frac{1}{2}$

Explanation of Solution

Given information:

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

Calculation:

  

  $\begin{array}{l}\sin \theta =2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\\ \cancel{\frac{2}{3}}=2\times \cancel{\frac{2}{3}}\times \cos \frac{\theta }{2}\\ \cos \frac{\theta }{2}=\frac{1}{2}\end{array}$