Chapter 14.4, Problem 13PPS

a)

To determine

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

Answer to Problem 13PPS

  $\sin 2\theta =\frac{240}{289}$

Explanation of Solution

Given information:

We know that, the given expression is the value of $\sin \theta =\frac{-15}{17},\pi <\theta <\frac{3\pi }{2}$

Calculation:

  

  $\begin{array}{l}\sin \theta =\frac{-15}{17}\\ \cos \theta =\frac{-8}{17}\\ \sin 2\theta =2\sin \theta \cos \theta =2\times \frac{-15}{17}\times \frac{-8}{17}=\frac{240}{289}\end{array}$

b)

To determine

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

Answer to Problem 13PPS

  $\cos 2\theta =\frac{-161}{289}$

Explanation of Solution

Given information:

We know that, the given expression is the value of $\sin \theta =\frac{-15}{17},\pi <\theta <\frac{3\pi }{2}$

Calculation:

  

  $\cos 2\theta =2{\cos }^2\theta -1=2\times {(\frac{-8}{17})}^2-1=\frac{-161}{289}$

c)

To determine

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

Answer to Problem 13PPS

  $\sin \frac{\theta }{2}=\frac{15}{17}$

Explanation of Solution

Given information:

The given expression is the value of $\sin \theta =\frac{-15}{17},\pi <\theta <\frac{3\pi }{2}$

Calculation:

  

  $\sin \frac{\theta }{2}=\overline{)\frac{1-\cos 2\theta }{2}}=\overline{)\frac{1+\frac{161}{289}}{2}}=\overline{)\frac{225}{289}}=\frac{15}{17}$

d)

To determine

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

Answer to Problem 13PPS

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

Explanation of Solution

Given information:

The given expression is the value of $\sin \theta =\frac{-15}{17},\pi <\theta <\frac{3\pi }{2}$

Calculation:

  

  $\begin{array}{l}\sin \theta =2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\\ \frac{-15}{17}=2\times \frac{15}{17}\times \cos \frac{\theta }{2}\\ \cos \frac{\theta }{2}=-\frac{1}{2}\end{array}$