Chapter 9.2, Problem 77AYU

To determine

To find: The graph of the polar equation $r=\csc \theta -2$, $0<\theta <\frac{\pi }{2}$.

Answer to Problem 77AYU

Solution:

Graph of the polar equation $r=\csc \theta -2$, $0<\theta <\frac{\pi }{2}$.

Given:

It is asked to find the graph of the polar equation $r=\csc \theta -2$, $0<\theta <\frac{\pi }{2}$:

Explanation of Solution

Check for symmetry test:

Polar axis:

Replace $\theta$ by $-\theta$. The result is $r=\csc \left(-\theta \right)-2=-\csc \theta -2$.

The test fails. So the graph may or may not be symmetric with respect to the polar axis.

The line $\theta =\frac{\pi }{2}$:

Replace $\theta$ by $\pi -\theta$. The result is

$r=\csc \pi -\theta -2=\frac{1}{\text{sin}\pi -\theta }-2=\frac{1}{\text{sin}\pi \text{cos}\theta -\text{cos}\pi \text{sin}\theta }-2=-\frac{1}{\text{sin}\theta }-2, 0<\theta$

$<\frac{\pi }{2}$

The test fails, so the graph may or may not symmetric with respect to the line $\theta =\frac{\pi }{2}$.

The Pole:

Replace $r$ by $-r$. Then the result is $-r=\csc \theta -2$, so $r=-\csc \theta +2$. This test fails. Replace $\theta$ by $\theta +\pi$. The result is

$r=\csc \left(\theta +\pi \right)-2=\frac{1}{\text{sin}\theta +\pi }-2=\frac{1}{\text{sin}\theta \text{cos}\pi +\text{cos}\theta \text{sin}\pi }-2=-\frac{1}{\text{sin}\theta }-2.$

This test also fails, so the graph may or may not be symmetric with respect to the pole.

Let’s identify points on the graph by assigning values to the angle $\theta$ and calculating the The test for symmetry with respect to the pole, the polar axis, and the line $\theta =\frac{\pi }{2}$ fail. The graph pass through the pole at $\theta =0$. With the help of graphing utility, the values in the following table can be obtained.

As the polar equation fails the symmetric test, find the values from 0 to $\frac{\pi }{2}$.

$\theta$	$r=\csc \theta -2$
0	$undefined=\infty$
$\frac{\pi }{6}$	$2-2=0$
$\frac{\pi }{4}$	$\sqrt{2}-2$
$\frac{\pi }{3}$	$\frac{2}{\sqrt{3}}-2$
$\frac{\pi }{2}$	$1-2=-1$

To sketch:

$r=\csc \theta -2$