Chapter 9.2, Problem 76AYU

To determine

To find: The graph of the polar equation $r=\frac{3}{\theta }$, $\theta \ge 0$.

Answer to Problem 76AYU

Solution:

Graph of the polar equation $r=\frac{3}{\theta }$, $\theta \ge 0$.

Given:

It is asked to find the graph of the polar equation $r=\theta$, $\theta \ge 0$:

Explanation of Solution

Check for symmetry test:

Polar axis:

Replace $\theta$ by $-\theta$. The result is $r=\frac{3}{\left(-\theta \right)}=-\frac{3}{\theta }$.

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

$r=\frac{3}{\pi -\theta }$

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

$r=\theta +\pi .$

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 $2\pi$.

$\theta$	$r=\frac{3}{\theta }$
0	$\infty$
$\frac{\pi }{6}$	$\frac{3}{\frac{\pi }{6}}=5.73$
$\frac{\pi }{3}$	$\frac{3}{\frac{\pi }{3}}=2.86$
$\frac{\pi }{2}$	$\frac{3}{\frac{\pi }{2}}=1.90$
$\frac{2\pi }{3}$	$\frac{3}{\frac{2\pi }{3}}=1.43$
$\frac{5\pi }{6}$	$\frac{3}{\frac{5\pi }{6}}=1.14$
$\pi$	$\frac{3}{\pi }=0.95$
$\frac{7\pi }{6}$	$\frac{3}{\frac{7\pi }{6}}=0.82$
$\frac{4\pi }{3}$	$\frac{3}{\frac{4\pi }{3}}=0.72$
$\frac{3\pi }{2}$	$\frac{3}{\frac{3\pi }{2}}=0.63$
$\frac{5\pi }{3}$	$\frac{3}{\frac{5\pi }{3}}=0.57$
$\frac{11\pi }{6}$	$\frac{3}{\frac{11\pi }{6}}=0.52$
$2\pi$	$\frac{3}{2\pi }=0.48$

To sketch:

$r=\frac{3}{\theta }$