Example

(a) Graph the polar equation $r=2 \ \cos ( 2\theta ) ,$ $0\leq \theta \leq 2\pi$ .

(b) Find parametric equations for $r=2 \ \cos ( 2\theta )$ .

Solution (a) The polar equation $r=2 \ \cos ( 2\theta )$ contains $\cos ( 2\theta )$ , which has the period $\pi$ . So, we construct Table 5 using common values of $\theta$ that range from $0$ to $2\pi$ , noting that the values for $\pi \leq \theta \leq 2\pi$ repeat the values for $0\leq \theta \leq \pi .$ Then we plot the points $( r,\theta)$ and trace out the graph. Figure 46(a) on page 674 shows the graph from the point $( 2,0)$ to the point $( 2, \ \pi).$ Figure 46(b) completes the graph from the point $( 2, \ \pi)$ to the point $( 2, \ 2\pi )$ .

TABLE 5

$\theta$	$r=2 \ \cos ( 2\theta )$	$( r,\theta )$
$0$	$2(1) =2$	$( 2,0)$
$\dfrac{\pi }{6}$	$2\left( \dfrac{1}{2}\right) =1$	$\left( 1,\dfrac{\pi }{6}\right)$
$\dfrac{\pi }{4}$	$2\left( 0\right) =0$	$\left( 0,\dfrac{\pi }{4}\right)$
$\dfrac{\pi }{3}$	$2\left( -\dfrac{1}{2}\right) =-1$	$\left( -1,\dfrac{\pi }{3}\right)$
$\dfrac{\pi }{2}$	$2\left( -1\right) =-2$	$\left(-2,\dfrac{\pi }{2}\right)$
$\dfrac{2\pi }{3}$	$2\left( -\dfrac{1}{2}\right) =-1$	$\left( -1,\dfrac{2\pi }{3}\right)$
$\dfrac{3\pi }{4}$	$2\left( 0\right) =0$	$\left( 0,\dfrac{3\pi }{4}\right)$
$\dfrac{5\pi }{6}$	$2\left( \dfrac{1}{2}\right) =1$	$\left( 1,\dfrac{5\pi }{6}\right)$
$\pi$	$2(1) =2$	$( 2, \ \pi)$
$\dfrac{7\pi }{6}$	$2\left( \dfrac{1}{2}\right) =1$	$\left( 1,\dfrac{7\pi }{6}\right)$
$\dfrac{5\pi }{4}$	$2(0) =0$	$\left( 0,\dfrac{5\pi }{4}\right)$
$\dfrac{4\pi }{3}$	$2\left( -\dfrac{1}{2}\right) =-1$	$\left( -1,\dfrac{4\pi }{3}\right)$
$\dfrac{3\pi }{2}$	$2\left( -1\right) =-2$	$\left( -2,\dfrac{3\pi }{2}\right)$
$\dfrac{5\pi }{3}$	$2\left( -\dfrac{1}{2}\right) =-1$	$\left( -1,\dfrac{5\pi }{3}\right)$
$\dfrac{7\pi }{4}$	$2\left(0\right) =0$	$\left( 0,\dfrac{7\pi }{4}\right)$
$\dfrac{11\pi }{6}$	$2\left( \dfrac{1}{2}\right) =1$	$\left( 1,\dfrac{11\pi }{6}\right)$
$2\pi$	$2(1) =2$	$( 2, \ 2\pi )$

Graphs of polar equations of the form $r=a \ \cos (n\theta )$ or $r =a \ \sin ( n\theta ),\;a>0,\;n$ an integer, are called roses. If $n$ is an even integer, the rose has $2n$ petals and passes through the pole $4n$ times. If $n$ is an odd integer, the rose has $n$ petals and passes through the pole $2n$ times.

(b) Parametric equations for $r=2 \ \cos ( 2\theta )$ : $\begin{equation*} x=r \ \cos \theta =2 \ \cos ( 2\theta ) \ \cos \theta \qquad y=r \ \sin \theta =2 \ \cos ( 2\theta ) \ \sin \theta \end{equation*}$

where $\theta$ is the parameter, and if $0\leq \theta \leq 2\pi ,$ then the graph is traced out exactly once in the counterclockwise direction.