Question

Identify and graph each polar equation. Verify your graph using a graphing utility.

$r^2=9\cos \left(2\theta \right)$

Short answer

The equation $r^2=9\cos \left(2\theta \right)$is lemniscate and its graph is

Step-by-step solution

Step 1. Given Information 

The given equation is

$r^2=9\cos \left(2\theta \right)$

Step 2. Symmetry with the polar axis  

Substitute $-\theta$for $\theta$in the equation

$$\begin{gathered} r^2=9\cos \left(2\left(-\theta \right)\right) \\ {r}^2=9\cos \left(2\theta \right) \end{gathered}$$

The equation matches with the original equation so the test is satisfied.

the graph is symmetric with respect to the polar axis

Step 3. Symmetry with line θ=π2

Substitute $\pi -\theta$for $\theta$in the equation

role="math" localid="1646689488157" $$\begin{gathered} r^2=9\cos \left(2\left(\mathrm{\pi }-\theta \right)\right) \\ {r}^2=9\cos \left(2\mathrm{\pi }-2\theta \right) \\ {r}^2=9\cos \left(2\theta \right) \end{gathered}$$

The equation matches the original equation so the test is satisfied.

the graph is symmetric with respect to the line$\theta =\frac{\mathrm{\pi }}{2}$

Step 4. Symmetry with the pole  

The equation is symmetric with respect to the polar axis and line$\theta =\frac{\mathrm{\pi }}{2}$

so the equation is symmetric with respect to the pole

Step 5. Points for the graph  

Consider different values for $\theta$in interval$\left(0,\frac{\mathrm{\pi }}{2}\right)$and determine coordinates of several points for graph

Step 6. Graph of the equation  

Locates points and use them to plot the graph of the equation

The graph state that the equation is a lemniscate.

Step 7. Verification  

Plot the graph of the equation$r^2=9\cos \left(2\theta \right)$using a graphing utility

Graph matches so our graph is correct.