Complex numbers : How to decide what the 'correct' angle is when evaluating complex numbers in their polar form?

Question

$z = \cos Q + i \sin Q$

Find $z+1$ in polar form.

$R$, the magnitude, I've worked out to be $2 \cos \frac{Q}{2}$

In its polar form, $z+1 = 2 \cos \frac{Q}{2}$ $( \cos x + i\sin x)$

$\tan x = \frac{\sin Q}{1+\cos Q }$

I get:

$\tan x = \tan \frac{Q}{2}$

Now, $x= \pi + \frac{Q}{2}$, or $x=\frac{Q}{2}$.

How do I know what the correct answer is?

Answer 1

That's a good question. Here's a mediocre hint: You should be able to say something about the real part of $z$. It can't take on just any value. This will tell you enough about $Re(z+1)$ to decide in which quadrant(s) your $x$ can lie.

Answer 2

This problem is more widespread than just complex variables and just tangent. That's why most, if not all, programming languages distinguish between and ordinary arctangent, say atan(y/x) and one that accounts for the quadrant, as in atan2(y,x). A similar problem happened to me recently when I was using the law of sines; all my obtuse angles were coming out as acute. That was because $\sin(x)=\sin(\pi-x)$.