Chapter H.1, Problem 34E

To determine

To sketch: the curve with the given polar equation

Answer to Problem 34E

  

Explanation of Solution

Given:

  $r=3\cos 6\theta$

Calculation:

Consider the polar equation,

  $r=3\cos 6\theta$

The objective is to sketch the curve with the given polar equation.

First construct the table form the given equation by Finding the values of $r$ for some convenient values of $\theta$ and plot the corresponding points $\left(r,\theta \right)$ .

$\theta$	$r=3\cos 6\theta$
$0$	$3$
$\frac{\pi }{6}$	$-3$
$\frac{\pi }{4}$	$0$
$\frac{\pi }{3}$	$3$
$\frac{\pi }{2}$	$-3$

Sketch the graph of $r=3\cos 6\theta$ in Cartesian coordinates as follows:

  

AS $\theta$ increase from $0$ to $0.5\pi$ , the above figure shows that $r$ decreases from $3to0$ and so draw the corresponding portion of the polar curve in below figure. As $\theta$ increase from $0.5\pi$ to $\pi ,r$ decreases from $0to-3$ , draw the corresponding portion of the polar curve in below figure. Hence the finally the required polar graph is shown below:

  

Conclusion:

Thus, the given equation is drawn.