Chapter 13.4, Problem 3CYU

a)

To determine

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

Answer to Problem 3CYU

  $\frac{-120}{169}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\cos \theta =-\frac{5}{13},\frac{\pi }{2}<\theta <\pi$

Formula used:

  $\sin 2\theta =2\sin \theta \cos \theta$

Calculation:

  $\begin{array}{l}\cos \theta =-\frac{5}{13}\\ \sin \theta =\frac{12}{13}\\ \sin 2\theta =2\sin \theta \cos \theta \\ \sin 2\theta =2\times \frac{12}{13}\times \frac{-5}{13}=\frac{-120}{169}\end{array}$

  

b)

To determine

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

Answer to Problem 3CYU

  $\frac{-119}{169}$

Explanation of Solution

Given Information:

The given expression is the value of $\cos \theta =-\frac{5}{13},\frac{\pi }{2}<\theta <\pi$

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

Calculation:

  $\begin{array}{l}\cos 2\theta =2{\cos }^2\theta -1\\ \cos 2\theta =2\times {\left(\frac{-5}{13}\right)}^2-1\\ =\frac{-119}{169}\end{array}$

c)

To determine

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

Answer to Problem 3CYU

  $\frac{3}{\sqrt{13}}$

Explanation of Solution

Given Information:

The given expression is the value of $\cos \theta =-\frac{5}{13},\frac{\pi }{2}<\theta <\pi$

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

Calculation:

  $\begin{array}{l}\sin \frac{\theta }{2}=\sqrt{\frac{1-\cos \theta }{2}}\\ \sin \frac{\theta }{2}=\sqrt{\frac{1+\frac{5}{13}}{2}}\\ \sin \frac{\theta }{2}=\frac{3}{\sqrt{13}}\end{array}$

d)

To determine

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

Answer to Problem 3CYU

  $\frac{2}{\sqrt{13}}$

Explanation of Solution

Given Information: The given expression is the value of $\cos \theta =-\frac{5}{13},\frac{\pi }{2}<\theta <\pi$

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

Calculation:

  $\begin{array}{l}\sin \theta =2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\\ \frac{12}{13}=2\times \frac{3}{\sqrt{13}}\times \cos \frac{\theta }{2}\\ \cos \frac{\theta }{2}=\frac{2}{\sqrt{13}}\end{array}$