If $A\subset \mathbb{N}$ is large, then does $\sum_{n\in A} \frac{\vert \sin n \vert }{n}$ diverge also?

Question

If $A\subset \mathbb{N}$ is large, that is, $\displaystyle\sum_{n\in A} \frac{1}{n}$ diverges, then does $\displaystyle\sum_{n\in A} \frac{\vert \sin n \vert }{n}$ diverge also?

I know that $\displaystyle\sum_{n\in \mathbb{N}} \frac{\vert \sin n \vert }{n}$ diverges. For example, see here. So this deals with the case $A=\mathbb{N}.$

I guess my question has to do with the irrationality measure of $\pi$ also.

Or is there some standard convergence test which can help here?