Chapter 10.3, Problem 9E

To determine

To graph: The set of points whose polar coordinates satisfy the given equation.

Answer to Problem 9E

All the points that satisfy the polar equation $\left|\theta \right|=\frac{\pi }{6}$ is shown in Figure (1).

Explanation of Solution

Given information: The given equation is $\left|\theta \right|=\frac{\pi }{6}$ .

Calculation:

Simplify the equation.

  $\theta =\pm \frac{\pi }{6}$

The angle $\theta =-\frac{\pi }{6}$ makes an angle $\frac{5\pi }{6}$ with the positive $x$ -axis.

Draw the graph of equations $\theta =-\frac{\pi }{6}$ and $\theta =\frac{\pi }{6}$ on the $xy$ -coordinate plane.

  

Figure (1)

The set of all points of positive and negative directed distance from the pole in the $-\frac{\pi }{6}$ and $\frac{\pi }{6}$ direction is a line through the origin with slope $\tan \left(-\frac{\pi }{6}\right)$ and $\tan \left(-\frac{\pi }{6}\right)$ .

Therefore, all the points that satisfy the polar equation $\left|\theta \right|=\frac{\pi }{6}$ is shown in Figure (1).