Answer
See graph
Work Step by Step
$r=3 \cos 3\theta$
The equation $r=3 \cos 3\theta$ is a rose curve. Rose curves have the general form:
$r=a\cos (n\theta)$ or $r=a\sin(n\theta)$.
If $𝑛$ is odd, the curve has $n$ petals.
If $n$ is even, the curve has $2n$ petals.
Here:
$a=3$
$n=3$ → So the curve has 3 petals
Make a table of values:
\[
\begin{array}{|c|c|c|c|}
\hline
\theta&3\theta &\cos(3\theta) & r=3\cos(3\theta) \\ \hline
0 & 0 &1 &3 \\ \hline
\frac{\pi}{6} & \frac{\pi}{2} & 0 & 0 \\ \hline
\frac{\pi}{3} & \pi & -1 & -3 \\ \hline
\frac{\pi}{2} & \frac{3\pi}{2} & 0 & 0 \\ \hline
\frac{2\pi}{3} & 2\pi & 1 & 3 \\ \hline
\end{array}
\]
Plotting $r$ vs $\theta$ gives a wave-like curve with period $\frac{2\pi}{3}$, oscillating between 3 and -3.
