Answer
Graph:
Work Step by Step
Plotting points $(r,\theta)$ ,
$r$ is the directed distance of the point from the pole.
$\theta$ defines the angle of the ray on which the point lies,
- remains $\theta$ when $r$ is positive
- becomes $\theta\pm\pi$ when $r$ is negative
The points $(r,\theta)$ of the region are such that:
The angle $\pi/3$ terminates in the 1st quadrant, and
defines a line through the pole (the origin) with slope $\tan( \pi/3) =\sqrt{3}$ (in Cartesian coordinates, $y=\sqrt{3}x$ ).
The directed distance r is partly negative, from -1 to 0,
so some points in the opposite (3rd) quadrant are involved. These are points on the line that are at a distance 1 or less from the pole.
For the rest of the values of r, the points represented lie on the line segment from the pole to the point that is at a distance of 3 units from the pole.
