How to find inverse of $\sin(x) + \sin(2x) = y$?

Question

I was wondering if there were any way to solve the equation $$\sin(x) + \sin(2x) = y$$ in terms of $x$. This of course would allow us to express the inverse for this function on $-\frac{\pi}{4}$ to $\frac{\pi}{4}$. Are there any techniques one can use to do this?

Answer 1

$$y=\sin x+\sin(2x)=\sin(x)+2\sin x \cos x=\sin x +2\sin x\sqrt{1-\sin^2 x}$$ $$\sin x =\cdots$$

Answer 2

What you could do is a series expansion $$y=3 x-\frac{3 x^3}{2}+\frac{11 x^5}{40}-\frac{43 x^7}{1680}+\frac{19 x^9}{13440}-\frac{683 x^{11}}{13305600}+\frac{2731 x^{13}}{2075673600}+O\left(x^{15}\right)$$ and then use series reversion to get $$x=\frac{y}{3}+\frac{y^3}{54}+\frac{79 y^5}{29160}+\frac{1151 y^7}{2204496}+\frac{2605 y^9}{22674816}+\frac{1104509 y^{11}}{40406522112}+\frac{35319907 y^{13}}{5157341549568}+O\left(y^{15}\right)$$