Rectangular to Polar Coordinates

I have recangular coordinates at (1,1) and am to convert to Polar coordinates

My Problem: I thought that (1,1) was a point on the cartesian plane that described which quadrant the vector appeared in. Right 1, up 1 Quadrant-1 yes?

My book's answers show 2 solutions (which you can prove easily enough), but I though that knowing the coordinate made finding alternate solutions superfluous?

Example
(1,1) to Polar
r= sqrt(2) and tanx = (pi/4) *since in QI, then tan=pi/4 is correct

Book ALSO shows
r=+-sqrt(2) and tanx=(pi/4) or (5pi/4)


Which gives
(sqrt(2),(pi/4)) OR (-sqrt(2),(5pi/4))

If you calculate the polar coordinates back to rectangular for either answer you will indeed get (1,1) - But... why the extra work? Am I misinterpreting the implications of rectangular coordinates?

Thank you for your time

Any coordinate only has one transformation from rectangular to polar and vice versa. So there's only one way to write in polars, and that's . You are correct.