Chapter 6.5, Problem 23E

Solution

To find: The points where the maximum value occurs for $r=3\sin 3\theta$ .

The maximum value $\left|r\right|=3$ occurs at $\theta =0,\frac{\pi }{3},\frac{2\pi }{3},\pi ,\frac{4\pi }{3},\frac{5\pi }{3},2\pi$ . $$

Given information:

The given polar equation is $r=3\sin 3\theta$ .

Calculation:

The range of the function is $-1\le \cos 3\theta \le 1$ .

Thus, then the range of the polar equation $r=3\sin 3\theta$ is:

  $\begin{array}{c}3\left(-1\right)\le r\le 3\left(1\right)\\ -3\le r\le 3\end{array}$

So, the maximum $\left|r\right|$ is $\left|-3\right|=\left|3\right|=3$ which occurs when $\cos 3\theta =\pm 1$ .

For inverse of the function find the value of $\theta$ for integer $n$ .

  $\begin{array}{c}3\theta ={\cos }^{-1}\left(\pm 1\right)\\ 3\theta =2\pi n,\pi +2\pi n\\ \theta =\frac{2\pi n}{3},\frac{\pi }{3}+\frac{2\pi n}{3}\end{array}$

  $\theta =\frac{\pi n}{3}$

For the interval $0\le \theta \le 2\pi$ , the solution is: $\theta =0,\frac{\pi }{3},\frac{2\pi }{3},\pi ,\frac{4\pi }{3},\frac{5\pi }{3},2\pi$ .

Therefore, maximum value $\left|r\right|=3$ occurs at $\theta =0,\frac{\pi }{3},\frac{2\pi }{3},\pi ,\frac{4\pi }{3},\frac{5\pi }{3},2\pi$ .