You want to exand $\sin x$ as a cosine series on $(0,\pi)$. Extend $\sin x$ to $(-\pi, \pi)$ in an even manner. So $f(-x) = \sin x$ for $x\in (0,\pi)$. Then this will be a cosine series if exteneded periodically by $2\pi$. The coefficients are:
$\pi a_0 = 2\int_0^{\pi}\sin x dx, \ \pi a_n = 2\int_0^{\pi}\sin x \cos nx dx, \ b_n = 0 \mbox{ for }n\geq 1$.