# Math Help - Rectangular to Polar Coordinates

1. ## 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?

2. Any coordinate only has one transformation from rectangular to polar and vice versa. So there's only one way to write $\displaystyle (x, y) = (1, 1)$ in polars, and that's $\displaystyle (r,\theta) = \left(\sqrt{2}, \frac{\pi}{4}\right)$. You are correct.