System of two non-linear equations

Question

I have the following system of equations: \begin{align} \cos(\alpha)+\cos(\beta)&=1.6, \\ \cos(3\alpha)+\cos(3\beta)&=0. \end{align}

How can i use an iterative method to solve these equations?

Answer 1

Using the triple angle formula,

$$\begin{cases}a+b&=p,\\4a^3-3a+4b^3-3b&=q.\end{cases}$$

Eliminating $b$,

$$12pa^2-12p^2a+4p^3-3p-q=0$$

is a mere quadratic equation.

With the given numerical values,

$$\cos\alpha=a=\frac{24\pm\sqrt{33}}{30}.$$

By symmetry,

$$\cos\beta=b=\frac{24\mp\sqrt{33}}{30}.$$