(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

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

The limaçon \(r=3+2 \ \cos \theta\).

(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.