Chapter 13.4, Problem 35PPS

a)

To determine

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

Answer to Problem 35PPS

  $\frac{-4\sqrt{21}}{25}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\cos ec\theta =\frac{-5}{2},\frac{3\pi }{2}<\theta <2\pi$

Formula used:

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

Calculation:

  $\begin{array}{l}\cos ec\theta =\frac{-5}{2}\\ \sin \theta =-\frac{2}{5}\\ \cos \theta =\frac{\sqrt{21}}{5}\\ \sin 2\theta =2\sin \theta \cos \theta \\ \sin 2\theta =2\times \left(\frac{-2}{5}\right)\times \left(\frac{\sqrt{21}}{5}\right)=\frac{-4\sqrt{21}}{25}\end{array}$

  

b)

To determine

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

Answer to Problem 35PPS

  $\frac{17}{25}$

Explanation of Solution

Given Information:

We know that, the given expression is the value of $\cos ec\theta =\frac{-5}{2},\frac{3\pi }{2}<\theta <2\pi$

Formula used:

  $\cos 2\theta =2{\cos }^2\theta -1$

Calculation:

  $\begin{array}{l}\cos 2\theta =2{\cos }^2\theta -1\\ \cos 2\theta =2\times {\left(\frac{\sqrt{21}}{5}\right)}^2-1\\ \cos 2\theta =\frac{17}{25}\end{array}$

c)

To determine

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

Answer to Problem 35PPS

  $-\frac{4\sqrt{21}}{17}$

Explanation of Solution

Given Information:

The given expression is the value of $\cos ec\theta =\frac{-5}{2},\frac{3\pi }{2}<\theta <2\pi$

Formula used:

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

Calculation:

  $\begin{array}{l}\tan 2\theta =\frac{\sin 2\theta }{\cos 2\theta }\\ \tan 2\theta =\frac{\frac{-4\sqrt{21}}{25}}{\frac{17}{25}}\\ \tan 2\theta =-\frac{4\sqrt{21}}{17}\end{array}$