<s>[INST] solution for complex number equation with power i have the following equation $$z^3 = 2 + 2i$$
and I need to find the solutions of this equation in trigonometric form.
how do I find multiple solutions of such an equation?
 [/INST] $$z^3=2+2i$$
$$z^3=|2+2i|e^{\arg(2+2i)i}$$
$$z^3=2\sqrt{2}e^{\tan^{-1}\left(\frac{2}{2}\right)i}$$
$$z^3=2\sqrt{2}e^{\tan^{-1}(1)i}$$
$$z^3=2\sqrt{2}e^{\frac{\pi}{4}i}$$
$$z=\left(2\sqrt{2}e^{\left(\frac{\pi}{4}+2\pi k\right)i}\right)^{\frac{1}{3}}$$
$$z=\sqrt{2}e^{\left(\frac{1}{12}\pi  +\frac{2}{3}\pi k\right) i}$$
And k goes from 0-2 -> $k=0-2$ and k is the element of Z. So:
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