Chapter 9.2, Problem 75AYU

In Problems 73-82, graph each polar equation.

$r\text{ = }\theta ,\theta \ge \text{ 0}$ (spiral of Archimedes)

To determine

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

Answer to Problem 75AYU

Solution:

Graph of the polar equation $r=\theta$, $\theta \ge 0$.

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 $r=\left(-\theta \right)=-\theta$.

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

$r=\pi -\theta$

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 $2\pi$.

$\theta$	$r=\theta$
0	$0$
$\frac{\pi }{6}$	$\frac{\pi }{6}=0.52$
$\frac{\pi }{3}$	$\frac{\pi }{3}=1.05$
$\frac{\pi }{2}$	$\frac{\pi }{2}=1.57$
$\frac{2\pi }{3}$	$\frac{2\pi }{3}=2.09$
$\frac{5\pi }{6}$	$\frac{5\pi }{6}=2.62$
$\pi$	$\pi =3.14$
$\frac{7\pi }{6}$	$\frac{7\pi }{6}=3.66$
$\frac{4\pi }{3}$	$\frac{4\pi }{3}=4.19$
$\frac{3\pi }{2}$	$\frac{3\pi }{2}=4.71$
$\frac{5\pi }{3}$	$\frac{5\pi }{3}=5.24$
$\frac{11\pi }{6}$	$\frac{11\pi }{6}=5.76$
$2\pi$	$2\pi =6.28$

To sketch:

$r=\theta$