The n-th roots of unity

The number $1$, as a point of the plane, has length $1$ and points along the positive real axis, so its angle is $0$, or equally $2\pi$, $4\pi$, and so on: $1 = \cos(2k\pi) + i \sin(2k\pi)$ for every whole number $k$. Here $i$ is the imaginary unit with $i^2 = -1$, and angles are in radians, where a full turn is $2\pi$.

Write an unknown solution as $z = \rho \left( \cos \varphi + i \sin \varphi \right)$ with length $\rho$ (rho) and angle $\varphi$ (phi). De Moivre's theorem says raising to the power $8$ raises the length to that power and multiplies the angle by $8$: $z^{8} = \rho^{8} \left( \cos 8\varphi + i \sin 8\varphi \right)$.