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

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.

• Nice question! I wish more people thought about their questions before posting. – mackycheese21 Apr 25 at 23:05

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

• Usually I don't do this when I get the degree, why is it that in this case I needed to do this? – user221435 Apr 25 at 16:35
• We don't need to do this, but we usually try to work with angles in the range $[0^\circ,360^\circ)$. – José Carlos Santos Apr 25 at 16:36
• Great, thank you – user221435 Apr 25 at 16:39
• @ThomasWeller this is not exactly right. In the case above, the value measured in degrees is an angle, which should always be in the range given. In modelers and animators, the value given is something like angular distance (ie the some of the absolute values of changes in angle over time) which happens to be measured in the same units as angles (degrees, radians), but is not the same type of value. Because of this, applying trigonometric functions to angular distances is not meaningful (without further assumptions) – DreamConspiracy Apr 25 at 19:22
• Being very fond of the geometrical plane of complex numbers, I feel that this is backwards (if not formally, then at least intuitively). In my opinion, $\sin$ and $\cos$ are unchanged after increasing or decreasing an angle by $360^\circ$ because turning something $360^\circ$ around the origin puts it back where you started. This sounds more like you're saying that turning something $360^\circ$ around the origin gives you what you started because $\sin$ and $\cos$ are unchanged. – Arthur Apr 25 at 19:34

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 $$+180^{\circ}$$

• Acomment above shows the range [0∘,360). Curious why the difference. – JoeTaxpayer Apr 25 at 22:33
• @JoeTaxpayer,The idea is getting the smallest angle possible, and it may differ from a book to another. Some ask for the smallest positive angle possible and that will be in $[0^{\circ},360^{\circ}[$, and others ask for the angle that assures the least rotation in the complex plane and that is what is called the principal value of an angle and ofcourse it lies in the interval $]-180^{\circ},180^{\circ}]$ (or the interval $[-180^{\circ},180^{\circ}[$, but it is the same idea). Also remark that all the above intervals cover all the possible angles that you might face in the complex plane – Fareed AF Apr 26 at 3:45
• Understood, and I really appreciate the detailed answer/comment. Thank-you. – JoeTaxpayer Apr 26 at 10:36

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.