Question

Trig question?

Answer 1

True or False: When squaring a complex number using DeMoivre's Theorem, there is only one answer.
I think it's true

Answer 2

Ummm yah I would agree.

When you take the Nth root of a complex number,
you end up with N unique answers.

But when applying an integer exponent, you should one unique value.

Example:\[\Large\rm (1+\mathcal i)^2\]We have a magnitude of sqrt2,
factoring it out,\[\Large\rm \sqrt2 ^2 \left(\frac{\sqrt2}{2}+\frac{\sqrt2}{2}\mathcal i\right)^2\]So this is in the first quadrant, pi/4,
writing it in exponential form:\[\Large\rm 2 \left(e^{\mathcal i (\pi/4 + 2k \pi)}\right)^2\]Squaring gives us,\[\Large\rm 2 e^{\mathcal i( \pi/2 + 4k \pi)}\]But multiples of 4pi will just land in the same location when we spin around right?
So the +4k pi isn't giving us any new values.
It only does when we are looking for roots.

Answer 3

ok so based on that I think I'm going with true