Chapter 14.4, Problem 34PPS

a)

To determine

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

Answer to Problem 34PPS

  $\frac{-3\sqrt{7}}{8}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\sec \theta =\frac{-4}{3},{90}^{\circ }<x<{180}^{\circ }$

Formula used:

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

Calculation:

  $\begin{array}{l}\sin 2\theta =2\sin \theta \cos \theta \\ \sin 2\theta =2\times \left(\frac{\sqrt{7}}{4}\right)\times \left(\frac{-3}{4}\right)=\\ \sin 2\theta =\frac{-3\sqrt{7}}{8}\end{array}$

b)

To determine

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

Answer to Problem 34PPS

  $\frac{1}{8}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\sec \theta =\frac{-4}{3},{90}^{\circ }<x<{180}^{\circ }$

Formula used:

  $\cos 2\theta =2{\cos }^2\theta -1$

Calculation:

  $\begin{array}{l}\cos 2\theta =2{\cos }^2\theta -1\\ \cos 2\theta =2\times {\left(\frac{-3}{4}\right)}^2-1\\ \boxed{\cos 2\theta =\frac{1}{8}}\end{array}$

c)

To determine

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

Answer to Problem 34PPS

  $-3\sqrt{7}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\sec \theta =\frac{-4}{3},{90}^{\circ }<x<{180}^{\circ }$

Formula used:

  $\tan 2\theta =\frac{\sin 2\theta }{\cos 2\theta }$

Calculation:

  $\begin{array}{l}\tan 2\theta =\frac{\sin 2\theta }{\cos 2\theta }\\ \tan 2\theta =\frac{\frac{-3\sqrt{7}}{\cancel{8}}}{\frac{1}{\cancel{8}}}\\ \tan 2\theta =-3\sqrt{7}\end{array}$