Cartesian to polar confusion (simple)?

1. Mar 18, 2014

noahsdev

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. Mar 18, 2014

arildno

For which interval of angles is the standard tangent function defined?

3. Mar 18, 2014

HallsofIvy

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

You distinguish between them by noting that $-\pi/3$ is in the fourth quadrant, (+,-), while $2\pi/3$ is in the second quadrant, (-, +).

4. Mar 18, 2014

noahsdev

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. :)

5. Mar 18, 2014

SteamKing

Staff Emeritus
Are you sure that -2+2SQRT(3)i is in the first quadrant? Why don't you make a sketch?

6. Mar 18, 2014

noahsdev

Yes that makes sense. Thanks.
P.S I know the quadrants haha I misstyped :)