Question

How do you change `10z^4` into polar coordinates where `z = 2 +4i`?

Answer 1

`4000(cos theta + i sin theta)`

Explanation:

Let's square it two times.

`z^2 = 4 + 16i - 16`

`z^4 = (-12 + 16i)^2 = 144 - 24*16i - 256 = -112 - 384i`

`10z^4 = -1120 - 3840i = w`

`|w| = sqrt{1120^2 + 3840^2} = 4000`

`tan (text{arg } w) = 3840/1120 = 24/7`

`theta = text{arg } w = pi + arctan frac{24}{7}`