Cardano's Formula/Trigonometric Form

Theorem
Let $P$ be the cubic equation:
 * $a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$

Let:

Let $a, b, c, d \in \R$.

Let the discriminant $D < 0$, where $D := Q^3 + R^2$.

Then the solutions of $P$ can be expressed as:

where:
 * $\cos \theta = \dfrac R {\sqrt {-Q^3} }$

Proof
From Cardano's Formula, the roots of $P$ are:

where:

Let $D = Q^3 + R^2 < 0$.

Then:
 * $S^3 = R + i \sqrt {\size {Q^3 + R^2} }$

We can express this in polar form:
 * $S^3 = r \paren {\cos \theta + i \sin \theta}$

where:
 * $r = \sqrt {R^2 + \paren {\sqrt {Q^3 + R^2} }^2} = \sqrt {R^2 - \paren {Q^3 + R^2} } = \sqrt {-Q^3}$
 * $\tan \theta = \dfrac {\sqrt {\size {Q^3 + R^2} } } R$

Then:
 * $\cos \theta = \dfrac R {\sqrt {-Q^3} }$

Similarly for $T^3$.

The result:

follows after some algebra.