If series $\sum a_n$ is convergent with positive terms does $\sum \sin a_n$ also converge?
Since $\lim\limits_{k \to \infty} a_k=0$, $$\tag 1\lim_{k \to \infty} \frac{\sin a_k}{a_k}=1$$
Thus $\sum_k |\sin a_k|$ converges $\iff \sum_k |a_k|=\sum_k a_k$ does. By your hypothesis and the above, $\sum_k \sin a_k$ will be absolutely convergent, so it will converge.
use that $|\sin(y)|\leq |y| $ (prove with the series) else you could prove it with mean value theorem.