Solving large trigonometric fucntions

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It seems to me that the hard work is already done properly by the solver. Finally, you got:

sage: var("theta phi M_1 M_2")
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)

Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$

Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$, recall that:

• if $\theta$ is an integer multiple of $\pi$, then ${0,\pm \pi}$ is the solution set.
• in general, there are exactly $2$ intersections (and the solutions differ by $\pi$).
• if the values of $M_1$ and $M_2$ allow it ($|M_1/M_2| \leq 1$), then there exists a value of $\theta$ for which the denominator vanishes, but the solution set is $\pm \pi/2$.

Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with SymPy solve or the new solveset.. but maybe I'm missing to pass some extra information.