Cube Root of Unity if Modulus is 1 and Real Part is Minus Half

Theorem
Let $z \in \C$ be a complex number such that:


 * $\cmod z = 1$
 * $\Re \paren z = -\dfrac 1 2$

where:
 * $\cmod z$ denotes the complex modulus of $z$
 * $\Re \paren z$ denotes the real part of $z$.

Then:
 * $z^3 = 1$

Proof
Let $z = x + i y$.

From $\Re \paren z = -\dfrac 1 2$:
 * $x = -\dfrac 1 2$

by definition of the real part of $z$.

Then:

Thus:
 * $z = -\dfrac 1 2 \pm \dfrac {\sqrt 3} 2$

and the result follows from Cube Roots of Unity.