Chapter 14.4, Problem 2CYU

a)

To determine

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

Answer to Problem 2CYU

  $\frac{-24}{25}$

Explanation of Solution

Given Information: The given expression is the value of $\sin \theta =\frac{4}{5},{90}^{\circ }<\theta <{180}^{\circ }$

Formula used: $\sin 2\theta =2\sin \theta \cos \theta$

Calculation:

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

  

b)

To determine

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

Answer to Problem 2CYU

  $\frac{-7}{25}$

Explanation of Solution

Given Information: The given expression is the value of $\sin \theta =\frac{4}{5},{90}^{\circ }<\theta <{180}^{\circ }$

Formula used: $\cos 2\theta =2{\cos }^2\theta -1$

Calculation:

  $\begin{array}{l}\cos 2\theta =2{\cos }^2\theta -1\\ \cos 2\theta =2\times {\left(\frac{-3}{5}\right)}^2-1\\ \cos 2\theta =\frac{18}{25}-1\\ \cos 2\theta =\frac{-7}{25}\end{array}$

c)

To determine

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

Answer to Problem 2CYU

  $\frac{2}{\sqrt{5}}$

Explanation of Solution

Given Information: The given expression is the value of $\sin \theta =\frac{4}{5},{90}^{\circ }<\theta <{180}^{\circ }$

Formula used: $\sin \frac{\theta }{2}=\sqrt{\frac{1-\cos \theta }{2}}$

Calculation:

  $\begin{array}{l}\sin \frac{\theta }{2}=\sqrt{\frac{1-\cos \theta }{2}}\\ \sin \frac{\theta }{2}=\sqrt{\frac{1+\frac{3}{5}}{2}}\\ \sin \frac{\theta }{2}=\frac{2}{\sqrt{5}}\end{array}$

d)

To determine

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

Answer to Problem 2CYU

  $\frac{1}{\sqrt{5}}$

Explanation of Solution

Given Information: The given expression is the value of $\sin \theta =\frac{4}{5},{90}^{\circ }<\theta <{180}^{\circ }$

Formula used: $\sin \theta =2\sin \frac{\theta }{2}\cos \frac{\theta }{2}$

Calculation:

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