Graphing a Polar Equation (Limaçon Without an Inner Loop); Finding Parametric Equations

(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
\(\theta\) \(r=3+2 \ \cos \theta\) \(( r,\theta )\)
\(0\) \(3+2(1) =5\) \(( 5,0)\)
\(\dfrac{\pi }{3}\) \(3+2\left( \dfrac{1}{2}\right) =4\) \(\left( 4,\dfrac{\pi }{3}\right)\)
\(\dfrac{\pi }{2}\) \(3+2\left( 0\right) =3\) \(\left( 3,\dfrac{\pi }{2}\right)\)
\(\dfrac{2\pi }{3}\) \(3+2\left( -\dfrac{1}{2}\right) =2\) \(\left( 2,\dfrac{2\pi }{3}\right)\)
\(\pi\) \(3+2\left( -1\right) =1\) \(( 1, \ \pi )\)
\(\dfrac{4\pi }{3}\) \(3+2\left( -\dfrac{1}{2}\right) =2\) \(\left( 2,\dfrac{4\pi }{3}\right)\)
\(\dfrac{3\pi }{2}\) \(3+2\left( 0\right) =3\) \(\left( 3,\dfrac{3\pi }{2}\right)\)
\(\dfrac{5\pi }{3}\) \(3+2\left( \dfrac{1}{2}\right) =4\) \(\left( 4,\dfrac{5\pi }{3}\right)\)
\(2\pi\) \(3+2( 1) =5\) \(( 5, \ 2\pi )\)
Figure 44 The limaçon \(r=3+2 \ \cos \theta\).

Graphs of polar equations of the form \[\bbox[5px, border:1px solid black, #F9F7ED]{\bbox[#FAF8ED,5pt]{ \begin{array}{rcl@{\qquad}crcl} r&=&a+b \ \cos \theta & r=a+b \ \sin \theta\\ r&=&a-b \ \cos \theta & r=a-b \ \sin \theta \end{array} }} \]

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

(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.