Chapter 9.2, Problem 79AYU

To determine

To find: The graph of the polar equation $r = \tan \theta$.

Answer to Problem 79AYU

Solution:

Graph of the polar equation $r = \tan \theta$.

Given:

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

Explanation of Solution

Check for symmetry test:

Polar axis:

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

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

$\begin{array}{}$ r = \tan \pi - \theta = \frac{\tan \pi - \tan \theta }{1 + \tan \pi \tan \theta } = \frac{0 - \tan \theta }{1 + \left(0\right)\tan \theta } = - \tan \theta $\end{array}$.

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

$r = \tan \left(\theta + \pi \right) = \frac{\tan \pi + \tan \theta }{1 - \tan \pi \tan \theta } = \frac{0 + \tan \theta }{1 - \left(0\right) \tan \theta } = \tan \theta$.

This test satisfied. So the graph is 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 and the polar axis fail. The graph is symmetric with respect to the polar axis. With the help of graphing utility, the values in the following table can be obtained.

As the polar equation is symmetric with respect to the pole, find the values from 0 to $\frac{\pi }{2}$.

$\theta$	$r = tan\mathit{\theta }$
$0$	$0$
$\frac{\mathit{\pi }}{6}$	$\frac{1}{\sqrt{3}}$
$\frac{\mathit{\pi }}{4}$	$1$
$\frac{\mathit{\pi }}{3}$	$\sqrt{3}$
$\frac{\mathit{\pi }}{2}$	undefined

To sketch:

$r = \tan \theta$.