Confused about complex number arguments.

To convert from regular form (a+bi) to exponential form re^iθ, θ=tan^-1(b/a) right?

For the complex number -1/2+i(sqrt3)/2, θ=tan^1(-sqrt3)=-pi/3. When inputting e^(-pi/3)i back into my calculator it gives an answer of 1/2-i(sqrt3)/2, the signs are wrong!

I worked out that it's the other solution of tan^1(-sqrt3), θ=2pi/3 that gives an answer of -1/2+i(sqrt3)/2 (the right one).

My question is, how can I tell which solution of tan^-1(b/a) to use?

Thanks.

Question

Confused about complex number arguments.

To convert from regular form (a+bi) to exponential form re^iθ, θ=tan^-1(b/a) right?

For the complex number -1/2+i(sqrt3)/2, θ=tan^1(-sqrt3)=-pi/3. When inputting e^(-pi/3)i back into my calculator it gives an answer of 1/2-i(sqrt3)/2, the signs are wrong!

I worked out that it's the other solution of tan^1(-sqrt3), θ=2pi/3 that gives an answer of -1/2+i(sqrt3)/2 (the right one).

My question is, how can I tell which solution of tan^-1(b/a) to use?

Thanks.

anonymous · Answer

Well you're talking about the complex number:
$$\Large - \frac{1}{2}+\frac{\sqrt{3}}{2}i$$
So if you compute an argument, you want to make sure that you get the precise one, this number lies in the 2nd quadrant, so for that quadrant you have:
$$\Large 90° \leq \varphi \leq 180° $$

anonymous · Answer

Or if you prefer working in radians:
$$\Large \frac{\pi}{2} \leq \theta \leq \pi  $$

anonymous · Answer

Ah yes, I forgot about taking the quadrants into account. Thanks!

anonymous · Answer

you're welcome