# Eliminate variable in trigonometry equations

Say you have the equations:

\begin{align} -S_1\sin\left(2\psi+\theta\right)+S_2\cos\left(\psi\right)&=S_3\\ S_1\cos\left(2\psi+\theta\right)+S_2\sin\left(\psi\right)&=S_4 \end{align}

or switching around the phase, which is perfectly ok for the system I am studying:

\begin{align} -S_1\sin\left(2\psi\right)+S_2\cos\left(\psi+\theta\right)&=S_3\\ S_1\cos\left(2\psi\right)+S_2\sin\left(\psi+\theta\right)&=S_4 \end{align}

and wish to eliminate $\psi$. How to do that? I have tried using 'square and add', rewriting using double-angle terms, and using Maple - without any luck so far.

• Can we start with $$2\psi+\theta=\psi+(\psi+\theta)$$ and solve for $\sin(\psi+\theta),\cos(\psi+\theta)$ – lab bhattacharjee Jul 29 '14 at 9:42
• Could you elaborate on that? – Ole Jul 29 '14 at 17:43
• Anybody who has an idea to solve this generic problem? – Ole Oct 30 '14 at 9:59