Complex argument

Hi, Am doing a question on complex numbers. The question ask you to show that z1 + z2 = 12e^2π/3 i, where zi = 6e^π/3 i and z2 = 6√ 3e^5π/6 i.

I converted z1 and z2 into cosθ/isinθ to get z1 + z2 = -6 + 6√ 3 i. Converting back into exponential form, I found r to be 12 and θ to be -π/3 but the MS states that it should be 2π/3. I had believed this to be the simpler part of complex numbers, but I can't see where I've gone wrong :")

A sketch helps and -6 + 6sqrt(3)i is clearly in quadrant 2, not 4. I guess you used atan() on your calculator which has a range -pi/2..pi/2, so it returns answers in quadrant 4, not 2. Smiilarly if both real and imaginary were negative, it would return answers in quadrant 1, not 3. Remember you +/-pi to get the other angle corresponding to the same gradient/extended line/value of tan and the sign of the real part tells you which one you want.

Note that the sides are in the ratio
Re : Im : Mod = 1 : sqrt(3) : 2
so youd expect a pi/6 - pi/3 - pi/2 right triangle in quadrant 2 without using your calculator. Similarly if you did tip to tail vector addition, you could have got there using (simple) geometry/triangles.