Math Problem Solver

The solutions are given by the formula: $$a = \cos{\left(\frac{2 \pi k}{3} \right)} + \sin{\left(\frac{2 \pi k}{3} \right)} i$$ where $k$ is an integer in $0 \le k < 3$.

There are $2$ solutions with nonzero imaginary part.

Real solutions: $$\begin{aligned}a &= 1\end{aligned}$$

Solutions with nonzero imaginary part: $$\begin{aligned}a &= - \frac{1}{2} + \frac{\sqrt{3} i}{2} \\&\approx -0.5 + 0.8660254 i\\a &= - \frac{1}{2} - \frac{\sqrt{3} i}{2} \\&\approx -0.5 -0.8660254 i\end{aligned}$$

$i$ is the imaginary unit, defined as $i^2 = -1$.