Chapter 8.2, Problem 12E

To determine

To test the given polar equation for symmetry with respect to the polar axis, the pole and the line $\theta =\frac{\pi }{2}$ :

  $r=5cos\theta csc\theta$

Answer to Problem 12E

The graph of polar equation $r=5cos\theta csc\theta$ is symmetric with respect to the line $\theta =\frac{\pi }{2}$ .

Explanation of Solution

Given: Polar equation: $r=5cos\theta csc\theta$

Formula Used:

In the polar equation, if ( r ,  θ ) can be replaced by ( r , -  θ ) the graph is symmetric with respect to the polar axis. If in the polar equation, ( r ,  θ ) can be replaced by (-  r ,  θ ), the graph is symmetric with respect to the pole. If in the polar equation, ( r ,  θ ) can be replaced by (-  r , -  θ ), the graph is symmetric with respect to the line  $\theta =\frac{\pi }{2}$ .

  $\cos \left(-\theta \right)=cos\theta$

  $csc\left(-\theta \right)=-\csc \theta$

Calculation:

Given Polar equation is $r=5cos\theta csc\theta$

Replacing $\theta$ with $-\theta$ ; we have:

  $r=5cos\left(-\theta \right)csc\left(-\theta \right)$ ;

Solving above equation, we have:

  $r=-5cos\theta csc\theta$

If we replace $\theta$ with $-\theta$ ; the polar equation changes to $r=-5cos\theta csc\theta$ , so $r=5cos\theta csc\theta$ is not symmetric with respect to polar axis.

Replacing $r$ with $-r$ ; we have:

  $-r=-\left(5cos\theta csc\theta \right)$ ;

Solving above equation, we have:

  $r=-5cos\theta csc\theta$

If we replace $r$ with $-r$ ; the polar equation changes to $r=-5cos\theta csc\theta$ , so $r=5cos\theta csc\theta$ is not symmetric with respect to pole.

Replacing $r$ with $-r$ and $\theta$ with $-\theta$ ; we have:

  $r=-\left(5cos\left(-\theta \right)csc\left(-\theta \right)\right)$ ;

Solving above equation, we have:

  $r=5cos\theta csc\theta$

If we replace $r$ with $-r$ and $\theta$ with $-\theta$ ; the polar equation remains same, so $r=5cos\theta csc\theta$ is symmetric with respect to $\theta =\frac{\pi }{2}$ .

Conclusion:

Hence, the graph of polar equation $r=5cos\theta csc\theta$ is symmetric with respect to the line $\theta =\frac{\pi }{2}$ .