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Math Help - Rectangular to Polar Coordinates

  1. #1
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    Question 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
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  2. #2
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    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.
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