Identify and graph each polar equation. Verify your graph using a graphing utility.
$r=4\sin \left(5\theta \right)$

The given equation is

$r=4\sin \left(5\theta \right)$

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

$$\begin{gathered} r=4\sin \left(5\left(-\theta \right)\right) \\ r=-4\sin \left(5\theta \right) \end{gathered}$$

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

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

$$\begin{gathered} r=4\sin \left(5\left(\mathrm{\pi }-\theta \right)\right) \\ r=4\sin \left(5\mathrm{\pi }-5\theta \right) \\ r=4\sin \left(5\theta \right) \end{gathered}$$

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

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

Substitute $-r$for r in the equation

$$\begin{gathered} -r=4\sin \left(5\theta \right) \\ r=-4\sin \left(5\theta \right) \end{gathered}$$

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

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

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

The graph state that the equation is a rose with five petals

Plot the graph of the equation $r=4\sin \left(5\theta \right)$using a graphing utility

Graph matches so our graph is correct.