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Cartesian to polar confusion (simple)?

  1. Mar 18, 2014 #1
    1. The problem statement, all variables and given/known data
    Convert -2+2√3i to polar coordinates.

    2. Relevant equations
    r = √x2+y2
    θ = tan-1(y/x)

    3. The attempt at a solution
    I am confused because θ = tan-1(2√3/2) = tan-1(√3) = -π/3 and r = 4, so that would make the polar form 4cis(-π/3), but the calculator gives: 4cis(2π/3).
    I think the calculator is right because when I convert my answer (4cis(-π/3)) back to cartesian it gives -2-2√3i, whereas the other (4cis(2π/3))gives the right answer, -2+2√3i.

    Can someone explain what I'm doing wrong?
    Thanks. :)
  2. jcsd
  3. Mar 18, 2014 #2


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    In which quadrant does your complex number lie?
    For which interval of angles is the standard tangent function defined?
  4. Mar 18, 2014 #3


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    Tangent is, of course, periodic and your calculator can give only one value- the "principal" value which, for tangent, is the value of [itex]\theta[/itex] with the smallest absolute value. Since tangent is periodic with period [itex]\pi[/itex], [itex]tan(-\pi/3)= tan(-\pi/3+ \pi)= tan(2\pi/3)[/itex].

    You distinguish between them by noting that [itex]-\pi/3[/itex] is in the fourth quadrant, (+,-), while [itex]2\pi/3[/itex] is in the second quadrant, (-, +).
  5. Mar 18, 2014 #4
    OK, I have found the angle using x and y (cos and sin) and they both confirm that the calculator is correct. And yes, it does make sense since the complex number lies in quadrant 1 but why is the tan function wrong? I'm guessing you were hinting at that part but I really don't know. :)
  6. Mar 18, 2014 #5


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    Are you sure that -2+2SQRT(3)i is in the first quadrant? Why don't you make a sketch?
  7. Mar 18, 2014 #6
    Yes that makes sense. Thanks.
    P.S I know the quadrants haha I misstyped :)
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