 Solution

Write $z=e^{i\theta }$. The trigonometric sum formulas giveThus the imaginary part is zero either when $\theta =k\pi /3$, $k=0,\dots ,5$, or when $\theta =\pi /4+k\pi /2$, $k=0,\dots ,3$. These two lists are disjoint. At the latter four points $p=0$. At the former six points the real values, in increasing angular order, areEquivalently, $z=1$ maps to $2$, $z=-1$ to $-2$, $z=e^{\pm i\pi /3}$ to $1$, $z=e^{\pm 2i\pi /3}$ to $-1$, and the four roots of $z^4=-1$ map to zero. These are all the unit-circle points with real image.