<s>[INST] Trigonometric system with different coefficients I have a trigonometric system as below and I want to solve it for $x$.
\begin{gather}
A\cos(x)+B\cos(2x)+C\sin(2x)  = D \\
-C\cos(2x)-A\sin(x)+B\sin(2x) = F
\end{gather}
Can anyone suggest a solution?
Regards.
 [/INST] Rewrite the two equations as
$$\sqrt{B^2+C^2}\cos(2x-a) = D-A\cos x $$
$$\sqrt{B^2+C^2}\sin(2x-a) = F+A\sin x $$
with $a = \tan^{-1} \frac CB$. Square and add the two equations
$$B^2+C^2= D^2+F^2 +A^2 +2A(F \sin x - D\cos x)\\
= D^2+F^2 +A^2 +2A\sqrt{D^2+F^2} \sin (x-b)$$
with $b= \tan^{-1} \frac DF$. Now, it should be straightforward.
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