Example

(a) Graph the polar equation $r=3+2 \ \cos \theta,\;0\leq \theta\leq 2\pi.$

(b) Find parametric equations for $r=3+2 \ \cos \theta$ .

Solution (a) The polar equation $r=3+2 \ \cos \theta$ contains $\cos \theta$ , which has the period $2\pi .$ We construct Table 3 using common values of $\theta$ that range from $0$ to $2\pi ,$ plot the points $(r,\theta )$ , and trace out the graph, beginning at the point $(5,0)$ and ending at the point $( 5, \ 2\pi )$ , as shown in Figure 44(a). Figure 44(b) shows the graph using technology.

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TABLE 3

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

(b) We obtain parametric equations for $r=3+2 \ \cos \theta$ by using the conversion formulas $x=r \ \cos \theta$ and $y=r \ \sin \theta$ : $x=r \ \cos \theta = ( 3+2 \ \cos \theta ) \ \cos \theta \qquad y=r \ \sin \theta =( 3+2 \ \cos \theta ) \ \sin \theta$

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

Graphs of polar equations of the form $\boxed{\bbox[#FAF8ED,5pt]{ \begin{array}{rcl@{\qquad}crcl} r&=&a+b \ \cos \theta & r=a+b \ \sin \theta\\ r&=&a-b \ \cos \theta & r=a-b \ \sin \theta \end{array} }}$

where $a> b > 0$ , are called limaçons without an inner loop. A lima\ciexn on without an inner loop does not pass through the pole.