How can we solve $\sin(ax) + \sin(bx) = y$ exactly for constants $a,b > 0$ and $y\in [-2,2]$

Question

How can we find a closed-form expression for $x$ for constants $a,b > 0$ and $y\in [-2,2]$, where $x$ solves the equation

$$y=\sin(ax) + \sin(bx)\text{?}$$

Answer 1

See maximum value returned by $y$ is $2$ , only possible when $sin(ax) = 1 ; sin(bx)=1$ So, $x=\frac{\pi}{2a}and\space x=\frac{\pi}{2b}$ , Similarly $-2$ can be obtained when both the values are $-1$ . $[-2,2]$ are the highest and lowest values $y$ can ever take, hence since there is no domain restriction $x \epsilon (-\infty,\infty)$