Partial sums of $\sin(x)$

Is it true that
$$
\left|\sum_{k=1}^n \sin(k) \right|\leq M
$$
for every $n$?

I tried comparing this to the integral $\int_2^\infty \sin(x)dx$ but it is not monotone.

This is part of a problem that says if $A_n=a_1+a_2+\cdots+a_n$ and $|A_n|\leq M$ and $b_n$ is a decreasing sequence to zero, then $\sum a_n b_n$ converges. Then use this to show that
$$
\sum_{n=2}^\infty \frac{\sin(n)}{\log(n)}
$$

Solutions Collecting From Web of "Partial sums of $\sin(x)$"