Chapter 6.5, Problem 24E

Solution

To find: The points where the maximum value occurs for $r=4\sin 2\theta$ .

The maximum value $\left|r\right|=4$ occurs at $\theta =\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$ . $$

Given information:

The given polar equation is $r=4\sin 2\theta$ .

Calculation:

The range of the function is $-1\le \sin 2\theta \le 1$ .

Thus, then the range of the polar equation $r=4\sin 2\theta$ is:

  $\begin{array}{c}4\left(-1\right)\le r\le 4\left(1\right)\\ -4\le r\le 4\end{array}$

So, the maximum $\left|r\right|$ is $\left|-4\right|=\left|4\right|=4$ which occurs when $\sin 2\theta =\pm 1$ .

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

  $\begin{array}{c}2\theta ={\sin }^{-1}\left(\pm 1\right)\\ 2\theta =\frac{\pi }{2}+2\pi n,\frac{3\pi }{2}+2\pi n\\ \theta =\frac{\pi }{4}+\pi n,\frac{3\pi }{4}+\pi n\\ \theta =\frac{\left(2n+1\right)\pi }{4}\end{array}$

For the interval $0\le \theta \le 2\pi$ , the solution is: $\theta =\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$ .

Therefore, maximum value $\left|r\right|=4$ occurs at $\theta =\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$ .