Given Point (-2, 3pi) need to find r>0 , -2pi< or equal to theta < 0

[Jurrasic]

question is (-2, 3pi)
they want to find r>0 , -2pi< or equal to theta < 0

Isn't it just (-2 , pi)
because you subtract 2pi from 3pi don't you? But the book says the answer is (2, -2pi) What in the world?

The 2 in the actual correct answer is because of how polar coordinates are plotted it changed to a positive as a result. But the -2pi part instead of being pi, is not clear?

 

[Slats18]

No, as pi does not fall within the inequality. You can take a further 2pi away though, but this doesn't give a positive value for r. So, if you make r = 2 and not -2, how far has it rotated around the axis?

 

[QuarkCharmer]

Think of it like this. The original point is (-2, 3pi). If I were to lead you to that point from the origin, I would say, turn 3pi (1 and 1/2 rotations anticlockwise, so now you are facing the opposite direction), and then walk backwards -2 steps. (This takes you to the cartesian point (2,0)

They are expressing this as turning clockwise 2pi, so you make a full rotation but are still facing the same direction, and then walk 2 steps forward. This also takes you to the cartesian point (2,0)

The resulting location is the same.

You might find this selection of videos most helpful:
http://www.khanacademy.org/v/polar-coordinates-1?p=Trigonometry