Chapter 6.5, Problem 30E

Solution

To analyze: The graph given by the polar equation $r=-3\cos 4\theta$ .

The equation represents a rose curve with $8$ petals. Domain of the graph is $\left(-\infty ,\infty \right)$ .

The range is $\left[-3,3\right]$ . The graph is continuous and symmetric about both the axes and the origin. Maximum value of $r$ is $3$ . There are no asymptotes to the graph.

Given information:

The given polar equation is $r=-3\cos 4\theta$ .

Calculation:

Plot the graph for the polar equation.

  

Figure (1)

From the figure (1), it can be observed that the equation represents a rose curve with $8$ petals.

The equation is defined for all real numbers of $\theta$ in the domain $\left(-\infty ,\infty \right)$ .

Range of $\cos 4\theta$ is, $-1\le \cos 4\theta \le 1$ .

The range of $r=2\sin 3\theta$ will be $\left[-3,3\right]$ considering the $1$ as maximum value and $-1$ as minimum value for $\cos 4\theta$ .

The graph is continuous. The rose curve is symmetric about both the axes and the origin.

Since $r$ values are bounded between $\left[-3,3\right]$ , it is a bounded graph.

The maximum value of $r$ is $\left|-3\right|=\left|3\right|=3$ .

There are no asymptotes to the graph.

Therefore, the polar equation $r=-3\cos 4\theta$ is analyzed with the help of figure (1).