Complex numbers $z=-3-4i$ polar form

Question

Let $z=-3-4i$ . In polar form this becomes $[5, 233° ]$. The question then asks for $z^2$, so the polar form becomes $[25,466]$ However in the solution they did $466° -360° $ and I am unsure why they did this. Is it a rule that needs to be applied.

Answer 1

It's because$$\cos(x-360^\circ)=\cos(x)\text{ and }\sin(x-360^\circ)=\sin(x).$$

Answer 2

It is just to get the principal value of the angle, since if you rotate by an angle $466^{\circ}$ you'll get to the same position as rotating $106^{\circ}$ so we usually take the smallest angle that is needed to arrive at the desired position.

The principal angle is an angle between $-180^{\circ}$ and $+180^{\circ}$

Answer 3

The polar form of a complex number is given by a distance from the origin and an angle against the positive real axis ("$x$-axis"). Increasing or decreasing this angle by $360^\circ$ will result in the same point. So adding or subtracting multiples of $360^\circ$ from the angle component of a set of polar coordinates will not change which point those coordinates represent.

By convention, we usually like this angle to be either in the range $[0^\circ, 360^\circ)$ or $(-180^\circ, 180^\circ]$. This is not a requirement by any means (unless explicitly stated in the exercise), but it's easier to tell by a glance what direction from the origin is represented by an angle of $270^\circ$ than by $2430^\circ$. So there is some merit to keeping the numbers small.