Quick trivial trigonometry question for polar form complex number

Question

The question that I am about to ask is quite trivial, but I don’t understand it perfectly.

I am solving complex numbers in polar form.

$$w=-1+i$$ When I want to get the angle $\phi $. I do:

$$\phi=\arctan(\frac{1}{-1})+\pi$$

So the question is that the we have two solutions of arctan; $3\pi/4$ and $-\pi/4$ both of these solutions give me different outcome when I put the angle in $\cos$ and $\sin$.

Should I only do one solution or should I do both. Am I missing a trivial understanding of trigonometry?

Answer 1

We are looking for the angle between the positive x-axis and the point =−1+. Of course there are many solutions for this since we can continue rotating 2 around the axis, but we usually take to be between 0 and 2. So actually −/4 is the angle w makes with the negative x-axis so we add to find the angle it makes with the positive x-axis. Hence =3/4 is the correct answer. It is easier to understand if you do a simple sketch.

Answer 2

You only need one of the two solutions. To figure out which one, I would recommend plotting the complex plane, the point $w$ and the two angles.