Chapter 13.1, Problem 59E

To determine

To calculate: The value of $\theta$ for the system of polar equations $r=2\text{ and }r=2\cos 2\theta$ . Graph the polar curves and compare the solutions.

Answer to Problem 59E

The value of $\theta$ for the system of polar equations $r=2\text{ and }r=2\cos 2\theta$ are $\left(2,180°\right)\text{ and }\left(2,0°\right)$ . Graph is provided below,

  

Explanation of Solution

Given information:

The system of polar equations $r=2\text{ and }r=2\cos 2\theta$ .

Formula used:

The initial horizontal velocity is $v_x=v\cos \theta$ and initial vertical velocity $v_y=v\sin \theta$ where v is the initial velocity with which object is thrown and $\theta$ is the angle it makes with the horizontal.

Calculation:

Consider the system of polar equations $r=2\text{ and }r=2\cos 2\theta$ .

Equate both the equations,

  $\begin{array}{c}2=2\cos 2\theta \\ \cos 2\theta =1\\ \cos 2\theta =\cos 0°\\ 2\theta =0°\end{array}$

Simplify it further as,

  $\theta =0°$

Therefore, when $\theta =0°$

  $\begin{array}{c}r=2\cos 2\left(0°\right)\\ r=2\end{array}$

Also, one more possible solution is,

  $\begin{array}{c}\cos 2\theta =\cos 360°\\ 2\theta =360°\end{array}$

Simplify it further as,

  $\theta =180°$

Therefore, when $\theta =180°$

  $\begin{array}{c}r=2\cos 2\left(180°\right)\\ r=2\end{array}$

Therefore, two solutions are $\left(2,180°\right)\text{ and }\left(2,0°\right)$ .

Sketch the polar curves $r=2\text{ and }r=2\cos 2\theta$ .

The curve $r=2$ is a circle of radius 2 units and the curve $r=2\cos 2\theta$ is a 4 petal rose.

  

The solution of system of polar equations are points of intersection of the two curves.

Thus, the value of $\theta$ for the system of polar equations $r=2\text{ and }r=2\cos 2\theta$ are $\left(2,180°\right)\text{ and }\left(2,0°\right)$ .