Question

Which of the following pairs of coordinates do not represent the same point in polar coordinates?
Check all that apply

A. (r,theta) and (r, theta + 2pi)
B. (r,theta) and (-r,theta + pi)
C. (r,theta) and (-r, -theta)
D. (r,theta) and (-r, theta+ 2pi)

Answer 1

There should be two answers, by the way.

Answer 2

I will explain each one:
A) theta + 2pi = theta, therefore (r,theta) = (r,theta+2pi)
B) Going from (r,theta) to (-r,theta) translates you 180 degrees around the unit circle, so you can correct this with a 180 degree, or pi radians rotation. Therefore, (r,theta) = (-r,theta + pi)
C) Again, we go from (r,theta) to (-r,theta), but instead of correcting this with a rotation of pi radians, they take the negative of the second theta. You can prove to yourself that -theta =/= theta + pi, so if B worked, then this does not work. (r,theta) =/= (-r,-theta). This is the first one.
D) Remember that theta + 2pi = theta. Therefore (-r,theta + 2pi) is equivalent to (-r,theta). So the question becomes, is (r,theta) equivalent to (-r,theta)? Looking back at B, you will see that going from (r,theta) to (-r,theta) translates you 180 degrees around the unit circle. So we can see that these are not equivalent.

Your answers should be C and D

Answer 3

Alright, thank you for your time!