Question

In Problems 61–66, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph.

$$\begin{gathered} r=3; \\ r=2+2\cos \theta \end{gathered}$$

Short answer

The graph of the polar equations on the same polar grid is:

Step-by-step solution

Step 1. Given

The polar equations:

$$\begin{gathered} r=3; \\ r=2+2\cos \theta \end{gathered}$$

Step 2. Substitute the values and equate them.

Substitute the values of $r$and equate them.

$$\begin{gathered} 3=2+2\cos \theta \\ 2\cos \theta =1 \\ \cos \theta =\frac{1}{2} \\ \theta ={\cos }^{-1}\left(\frac{1}{2}\right) \\ =\frac{\mathrm{\pi }}{3},\frac{5\mathrm{\pi }}{3} \end{gathered}$$

The point of intersection is $(3,\frac{\mathrm{\pi }}{3}),(3,\frac{5\mathrm{\pi }}{3})$

Step 3. Sketch the graph of r=3

Sketch the graph of $r=3$

Step 4. Polar axis

Replace $\theta by-\theta$

$$\begin{gathered} r=2+2\cos (-\theta ) \\ =2+2\cos \theta \end{gathered}$$

The test is satisfied, so the graph is symmetric with respect to the polar axis.

Step 5. The line

Replace $\theta$by $\mathrm{\pi }-\mathrm{\theta }$

$$\begin{gathered} r=2+2\cos (\pi -\theta ) \\ =2+2(\cos \pi .\cos \theta +\sin \pi .\sin \theta ) \\ =2+2\left(\right(-1)\cos \theta +0) \\ =2-2\cos \theta \end{gathered}$$

The test fails, so the graph may or may not be symmetric with respect to the line $\frac{\mathrm{\pi }}{2}$

Step 6. The pole

Replace $rby-r$

$$\begin{gathered} -r=2+2\cos \theta \\ r=-2-2\cos \theta \end{gathered}$$

The test fails, so the graph may or may not be symmetric with respect to the pole.

Step 7. Sketch the graph r=2+2 cos θ

Graph the polar function $r=2+2\cos \theta$on the same polar grid.