Mathematica finds

    Integrate[Sin[a*Cos[t]]*b^n*Cos[t]^n,{t, 0, 2*Pi},Assumptions->n \[Element] Integers&&n>0]

> $$-\frac{1}{2} \pi  a \left((-1)^n-1\right) b^n \Gamma \left(\frac{n}{2}+1\right) \, _1\tilde{F}_2\left(\frac{n+2}{2};\frac{3}{2},\frac{n+3}{2};-\frac{a^2}{4}\right) $$

Therefore, the integral under consideration can be expressed as a series.