I'm trying to solve for all z values where $z^3 = 4\bar{z}$.
I tried using $z^3 = |z|(\cos(3\theta)+i\sin(3\theta)$ and that $|z| = \sqrt{x^2+y^2}$ so:
$$z^3 = \sqrt{x^2+y^2}(\cos(3\theta)+\sin(3\theta))$$ and $$4\bar z = 4x-4iy = 4r\cos(\theta)-i4r\sin(\theta)$$
but I have no idea where to go from there.
Best Answer
If $z^3=4\overline z$, then $z^4=4z\overline z=4|z|^2$. So, $|z|^4=|z^4|=4|z|^2$, and therefore $z=0$ or $|z|=2$. So, unless $z=0$, $z$ can be written as $2(\cos\theta+i\sin\theta)$, in which case$$z^3=4\overline z\iff8\bigl(\cos(3\theta)+i\sin(3\theta)\bigr)=8(\cos\theta-i\sin\theta).$$Can you take it from here?