Chapter 6.5, Problem 57E

Solution

To analyze:The graphs represented by the polar equations $r=a\cos n\theta$ and $r=a\sin n\theta$ where $n$ is a positive integer.

The polar equation represents a rose curve with $2n$ petals.

The domain of the graph is $\left(\infty ,-\infty \right)$ .

The range of the polar equation is $\left[-\left|a\right|,\left|a\right|\right]$ .

The graph is symmetric about both the axes and the origin.

The graph is continuous and bounded between the minimum and maximum $r-$ values.

The maximum value of $r=-\left|a\right|$ and minimum value of $r=\left|a\right|$ .

Given information:

The given polar equations are $r=a\cos n\theta$ and $r=a\sin n\theta$ where $n$ is a positive integer.

Calculation:

Plot the graph for $r=a\cos n\theta$ .

Consider the value of $a$ as $4$ and value of $n=2,4,6$ .

The graph of $r=4\cos 2\theta$ is as shown below:

  

The graph of $r=4\cos 4\theta$ is as shown below:

  

The graph of $r=4\cos 6\theta$ is as shown below:

  

From the graphs for $r=a\cos n\theta$ , it can be deduced that the polar equation represents a rose curve with $2n$ petals.

The domain of the graph is $\left(\infty ,-\infty \right)$ since it is valid for all real values of $\theta$ .

The range of $\cos n\theta$ is, $-1\le \cos n\theta \le 1$ .

So, the range of the polar equation is $\left[-\left|a\right|,\left|a\right|\right]$ taking $-1$ as minimum value of $\cos n\theta$ and $1$ as maximum value of $\cos n\theta$ .

The graph is symmetric about both the axes and the origin.

The graph is continuous and bounded between the minimum and maximum $r-$ values.

The maximum value of $r=-\left|a\right|$ and minimum value of $r=\left|a\right|$ .

Plot the graph for $r=a\sin n\theta$ .

Consider the value of $a$ as $4$ and value of $n=2,4,6$ .

The graph of $r=4\sin 2\theta$ is as shown below:

  

The graph of $r=4\sin 4\theta$ is as shown below:

  

The graph of $r=4\sin 6\theta$ is as shown below:

  

From the graphs for $r=a\sin n\theta$ , it can be deduced that the polar equation represents a rose curve with $2n$ petals.

The domain of the graph is $\left(\infty ,-\infty \right)$ since it is valid for all real values of $\theta$ .

The range of $\sin n\theta$ is, $-1\le \sin n\theta \le 1$ .

So, the range of the polar equation is $\left[-\left|a\right|,\left|a\right|\right]$ taking $-1$ as minimum value of $\sin n\theta$ and $1$ as maximum value of $\sin n\theta$ .

The graph is symmetric about both the axes and the origin.

The graph is continuous and bounded between the minimum and maximum $r-$ values.

The maximum value of $r=-\left|a\right|$ and minimum value of $r=\left|a\right|$ .