Answer
(a) and (f) label the same point
(b) and (h) label the same point
(c) and (g) label the same point
(d) and (e) label the same point
Work Step by Step
$r$ represents the directed distance from the pole (the origin), and $\theta$ represents the angle with the initial ray ( the +x-axis).
Directed distance means:
- if $r$ is positive, then the point lies on the terminal side of $\theta$
- if $r$ is negative, then the point lies opposite the terminal side of $\theta$,
(it lies on the terminal side of $\theta\pm\pi$)
$ 2k\pi$ can be added to any angle to make terminal sides coincide.
$(a)$
Among other polar pairs representing $(-2,\pi/3):$is
$(2,\pi/3-\pi)=(2,-2\pi/3)$ ... choice ($f$)
$(b)$
Among other polar pairs representing $(2,-\pi/3):$is
$(-2,-\pi/3+\pi)=(-2,2\pi/3)$ ... choice ($h$)
$(c)$
Among other polar pairs representing $(r,\theta):$is
$(-r,\theta+\pi)$ ... choice ($g$)
$(d)$
Among other polar pairs representing $(+r,(\theta+\pi)):$ is
$(-r,(\theta+\pi)-\pi): =(-r,\theta)$ ... choice ($e$)
All choices have been paired:
(a) and (f) label the same point
(b) and (h) label the same point
(c) and (g) label the same point
(d) and (e) label the same point
