Case 1: $(0 < a < b)$
To proceed, let $X$ have pdf $f(x)$:

where I am initially assuming parameter $a$ is positive. I will consider the alternative cases later. Second, let $W = sin(Y)$ with pdf $\phi(w)$:

The latter is relatively straightforwards to derive.
Next, we seek the pdf of the product of two random variables, namely $Z = X * W$. This can be cumbersome to do by hand, but can be derived easily using the TransformProduct
function in the mathStatica
suite. In particular, the pdf of $Z$, say $g(z)$ is:

All done. (I should add I am one of the authors of the software.)
Here is a plot of the theoretical pdf derived $g(z)$ (in red) when $a = 2$ and $b = 5$, and superimposed on top a Monte Carlo check (in blue).

The other 2 cases (with parameter $a< 0$) are derived in identical fashion ... just change the assumptions on parameters $a$ and $b$ in the first input. Doing so yields:
Case 2: $(a < 0, b > 0)$
The solution pdf is $g(z)$:

Here is a plot of the solution pdf $g(z)$ when $a = -3, b = 2$:

Case 3: $(a < b < 0)$
The solution pdf is $g(z)$:

Here is a plot of the solution pdf $g(z)$ when $a = -4, b = -2$:
