Question

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

$r^2=\sin \left(2\theta \right)$

Short answer

The equation $r^2=\sin \left(2\theta \right)$is lemniscate and the graph is

Step-by-step solution

Step 1. Given Information 

The given equation is

$r^2=\sin \left(2\theta \right)$

Step 2. Symmetry with the polar axis  

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

role="math" localid="1646690348938" $$\begin{gathered} r^2=\sin \left(2\left(-\theta \right)\right) \\ {r}^2=-\sin \left(2\theta \right) \end{gathered}$$

The equation does not match with the original equation so the test fails

the graph may or may not be symmetric with respect to the polar axis

Step 3. Symmetry with line θ=π2

Substitute$\mathrm{\pi }-\mathrm{\theta }$ for$\theta$ in the equation

$$\begin{gathered} r^2=\sin \left(2\left(\mathrm{\pi }-\theta \right)\right) \\ {r}^2=\sin \left(2\mathrm{\pi }-2\theta \right) \\ {r}^2=-\sin \left(2\theta \right) \end{gathered}$$

The equation does not match the original equation so the test fails.

the graph may be or may not be symmetric with respect to the line$\theta =\frac{\mathrm{\pi }}{2}$

Step 4. Symmetry with the pole  

Substitute$-r$ for r in the equation

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

The equation match with the original equation so the test satisfied

the graph is symmetric with respect to the pole

Step 5. Points for the graph  

Consider different values for $\theta$and determine coordinates of several points for the 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$r^2=\sin \left(2\theta \right)$using a graphing utility

Graph matches so our graph is correct.