Sketch set of complex numbers which satisfy $ℜ(z^4)\geq 0$

Question

I tried to figure it out but I get inequality

$x^4 - 6xy + y^4 \geq 0$

and its alternative

$(x^2-2xy-y^2)(x^2+2xy-y^2) \geq 0$

I wonder if there's solution which does not use mathematical analysis.

Answer 1

Using polar coordinates $z=re^{i\theta}$ we get $z^4=r^4e^{i4\theta}$ and so $\Re(z^4)=r^4\cos(4\theta)$.

The angles $\theta$ for which $\cos(4\theta)\ge0$ are those in the figure, that is, those in the intervals $$[-\pi/8,\pi/8],\quad[3\pi/8,5\pi/8],\quad[7\pi/8,9\pi/8]\quad\text{and}\quad[11\pi/8,13\pi/8].$$