A cyclotomic equation

Modulus. $|z|^5=1$, so $r=1$.

Argument. $5\theta\equiv0\pmod{2\pi}$, so $\theta=2k\pi/5$, $k=0,1,2,3,4$.

Solutions. $z_k=e^{2ik\pi/5}$ for $k=0,1,2,3,4$:
$$z=1,\quad e^{2i\pi/5},\quad e^{4i\pi/5},\quad e^{6i\pi/5},\quad e^{8i\pi/5}.$$

Conclusion. The fifth roots of unity are the vertices of a regular pentagon on the unit circle.