Chapter 6.5, Problem 54E

Solution

a)

To describe: The graph of the polar equation $r=3\cos 2\theta -\sin 3\theta$ .

The graph of the polar equation $r=3\cos 2\theta -\sin 3\theta$ shows graph of rose cure having one short petal of length $2\text{unit}$ and one long petal of length $4\text{unit}$ .

Given information:

The given polar equation is $r=3\cos 2\theta -\sin 3\theta$ .

Calculation:

Plot the graph for the polar equation $r=3\cos 2\theta -\sin 3\theta$ .

  

From the graph, it can be observed that the polar equation represents a rose curve with $4$ petals.

When the value of $\theta$ is $\frac{\pi }{2}$ , the value of $r$ is $\left|r\right|=\left|-2\right|=2$ .

When the value of $\theta$ is $\frac{3\pi }{2}$ , the value of $r$ is $\left|r\right|=\left|-4\right|=4$ .

Therefore, the maximum value of the petal is $4$ and minimum value of the petal is $2$ .

b)

To find: If there exist any symmetry in the graph.

The graph is symmetric about the $y-$ axis.

Given information:

The given polar equation is $r=2{\sin }^{2}2\theta +\sin 2\theta$ .

Calculation:

From part (a), the graph represents a rose curve symmetric about the $y-$ axis.

c)

To find: The maximum value of $\left|r\right|$ if it exists.

The maximum value of $r$ is $4$ .

Given information:

The given polar equation is $r=2{\sin }^{2}2\theta +\sin 2\theta$ .

Calculation:

From part (a), the maximum value of $\left|r\right|$ exists.

The maximum value of $r$ is $\left|r\right|=\left|-4\right|=4$ .