Chapter 14.4, Problem 16PPS

a)

To determine

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

Answer to Problem 16PPS

  $\sin 2\theta =\frac{24}{25}$

Explanation of Solution

Given information:

We know that, the given expression is the value of $\tan \theta =\frac{4}{3},{180}^{\circ }<\theta <{270}^{\circ }$

Calculation:

  

  $\begin{array}{l}\tan \theta =\frac{4}{3}\\ \sin \theta =-\frac{4}{5}\\ \sin 2\theta =2\sin \theta \cos \theta =2\times \frac{-4}{5}\times \frac{-3}{5}=\frac{24}{25}\end{array}$

b)

To determine

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

Answer to Problem 16PPS

  $\cos 2\theta =\frac{-7}{25}$

Explanation of Solution

Given information:

We know that, the given expression is the value of $\tan \theta =\frac{4}{3},{180}^{\circ }<\theta <{270}^{\circ }$

Calculation:

  

  $\cos 2\theta =2{\cos }^2\theta -1=2\times \frac{3}{25}-1=\frac{-7}{25}$

c)

To determine

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

Answer to Problem 16PPS

  $\sin \frac{\theta }{2}=\frac{4}{5}$

Explanation of Solution

Given information:

We know that, the given expression is the value of $\tan \theta =\frac{4}{3},{180}^{\circ }<\theta <{270}^{\circ }$

Calculation:

  

  $\sin \frac{\theta }{2}=\overline{)\frac{1-\cos 2\theta }{2}}=\overline{)\frac{1+\frac{7}{25}}{2}}=\frac{4}{5}$

d)

To determine

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

Answer to Problem 16PPS

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

Explanation of Solution

Given information:

We know that, the given expression is the value of $\tan \theta =\frac{4}{3},{180}^{\circ }<\theta <{270}^{\circ }$

Calculation:

  

  $\begin{array}{l}\sin \theta =2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\\ \frac{-4}{5}=2\times \frac{4}{5}\times \cos \frac{\theta }{2}\\ \cos \frac{\theta }{2}=-\frac{1}{2}\end{array}$