For $a,b$ in some compact interval $I$, consider the integral function $K_{a,b}:\mathbb{R}\rightarrow\mathbb{R},$ defined by
$$K_{a,b}(x)=\displaystyle\int\limits_{0}^{\pi} \cos(ax\cos(t))\sin(bx\sin(t)) \, dt \quad (x\in\mathbb{R}).$$
Has anybody an idea about an asymptotic expansion of this integral for $x\rightarrow\infty$ (for fixed $a,b$)? Is it perhaps even possible to have some kind of an uniform estimate, i.e. $$\vert K_{a,b}(x)\vert\leq K(x)$$ for all $a,b\in I, x\in\mathbb{R}$ and some function $K:\mathbb{R}\rightarrow\mathbb{R}$ with $K(x)\rightarrow 0 \ (x\rightarrow\infty)$ of which we know an asymptotic expansion?
Three parameters are unecessary. You can let $ax\to x$ and $bx\to y$ to reduce to $$F(x,y)=\int_0^\pi \cos(x\cos t)\sin(y\sin t)\mathrm dt$$ Let $f(x,y,t)=\cos(x\cos t)\sin(y \sin t)$. The key observation is that $f$ satisfies the PDE (check this) $$\partial_x^2 f+\partial_y^2 f+f=0$$ And, by the Leibniz rule, $F$ satisfies this PDE as well, as well as having the boundary condition $$F(x,0)=0$$
Separation of variables should do the trick here. Unfortunately I don't have the time right now to go through this myself.