91Ó°ÊÓ

The given equation is

$r=4\sin \left(5\mathrm{θ}\right)$

Substitute $-\mathrm{θ}$for $\mathrm{θ}$in the equation

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

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 $\mathrm{π}-\mathrm{θ}$for$\mathrm{θ}$ in the equation

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

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

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

Substitute $-r$for r in the equation

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

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 $\mathrm{θ}$in interval$\left(\frac{-\mathrm{π}}{2},\frac{\mathrm{π}}{2}\right)$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 $r=4\sin \left(5\mathrm{θ}\right)$using a graphing utility

Graph matches so our graph is correct.