Solving a seemingly-trivial trigonometric equation for a specific variable

Question

I'd like to express $y$ in $y = \sin (x*y)$. I can get $x = \frac{\arcsin y} y $ without any issues and then kind of don't know how to continue. WolframAlpha seems to have struggles with the equation as well — it doesn't return explicit results as usual. I wasn't able to find any similar problems online. Can anyone explain what's going on here or point me to some relevant literature please?

Answer 1

As said in comments, there is no analytical solution and, as you wrote, the problem is simple from a computational point of view.

To obtain quite good approximations, you could build the $[2n,2n]$ Padé approximant $P_n(y)$ of $\frac{\sin ^{-1}(y)}{y}$ and solve for $y$ the equation $$x=P_n(y)$$ which is a polynomial of degree $n$ in $y^2$.

The first decent one is $P_3(y)$.

You coul also expand $\frac{\sin ^{-1}(y)}{y}$ as a Taylor series around $y=0$ and use series reversion to get $$y=t-\frac{9 t^3}{40}+\frac{969 t^5}{22400}-\frac{21197 t^7}{2688000}+\frac{3092997 t^9}{2207744000}-\frac{1407584257 t^{11}}{5740134400000}+O\left(t^{13}\right)$$ where $t=\sqrt{6(x-1)}$