Solving trigonometric equation with tangent

Question

Please help me the following equation, we need to calculate X relative to the other: \begin{equation} \tan{(2X-\phi_0)} = \frac{\rho\sin{\phi-a\sin{X}}}{\rho\cos{\phi}-a\cos{X}} \end{equation} Thank you very much!

Answer 1

I hope that you are not expecting a closed form (remember that $x=\cos(x)$ does not show explicit solutions). So, the work is purely numerical (Newton method is a perfect candidate).

In order to avoid discontinuities, I cross-multiplied and the equation becomes $$\rho \sin (2 x-\phi-\phi_0)-a \sin (x-\phi_0 )=0$$ Do not forget that there will be an infinite number of roots.

If you look for the first root, assuming that the solution is small, use a Taylor series around $x=0$ up to $O(x^2)$ and solve the quadratic; this will give probably a decent estimate and, stating with it, polish the solution using Newton method.