Chapter 13.4, Problem 17PPS

a)

To determine

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

Answer to Problem 17PPS

  $\sin 2\theta =-\frac{4}{5}$

Explanation of Solution

Given information:

We know that, the given expression is the value of $\tan \theta =-2,{90}^{\circ }<\theta <{180}^{\circ }$

Calculation:

  

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

b)

To determine

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

Answer to Problem 17PPS

  $\cos 2\theta =-\frac{3}{5}$

Explanation of Solution

Given information:

We know that, the given expression is the value of $\tan \theta =-2,{90}^{\circ }<\theta <{180}^{\circ }$

Calculation:

  

  $\cos 2\theta =2{\cos }^2\theta -1=2\times \frac{1}{5}-1=-\frac{3}{5}$

c)

To determine

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

Answer to Problem 17PPS

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

Explanation of Solution

Given information:

We know that, the given expression is the value of $\tan \theta =-2,{90}^{\circ }<\theta <{180}^{\circ }$

Calculation:

  

  $\sin \frac{\theta }{2}=\overline{)\frac{1-\cos 2\theta }{2}}=\overline{)\frac{1+\frac{3}{5}}{2}}=\frac{2}{\sqrt{5}}$

d)

To determine

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

Answer to Problem 17PPS

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

Explanation of Solution

Given information:

We know that, the given expression is the value of $\tan \theta =-2,{90}^{\circ }<\theta <{180}^{\circ }$

Calculation:

  

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