Question

Identify and graph each polar equation. Verify your graph using a graphing utility.
$r=2\sin \left(3\theta \right)$

Short answer

The equation$r=2\sin \left(3\theta \right)$is a rose with three petals and the graph is

Step-by-step solution

Step 1. Given Information 

The given equation is

$r=2\sin \left(3\theta \right)$

Step 2. Symmetry with the polar axis 

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

$$\begin{gathered} r=2\sin \left(3\left(-\theta \right)\right) \\ r=-2\sin \left(3\theta \right) \end{gathered}$$

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

$$\begin{gathered} r=2\sin \left(3\left(\mathrm{\pi }-\theta \right)\right) \\ r=2\sin \left(3\mathrm{\pi }-3\theta \right) \\ r=2\sin \left(3\theta \right) \end{gathered}$$

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 

Substitute$-r$ for r in the equation

$$\begin{gathered} -r=2\sin \left(3\theta \right) \\ r=-2\sin \left(3\theta \right) \end{gathered}$$

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 pole

Step 5. Points for the graph 

Consider different values for$\theta$in interval$\left(\frac{-\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$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 rose with three petals

Step 7. Verification 

Plot the graph of the equation$r=2\sin \left(3\theta \right)$using a graphing utility

Graph matches so our graph is correct.