Consider the set $A=\{\sin k:k\in\mathbb Z \}$. I want to know whether this set is dense in $[-1,1]$.
I have a hunch that this problem can somehow be reduced to the approximation of $\pi$ using rationals, although I don't know how to make a connection.
Hint.
Consider the set $S=\{k+2\pi u | (k,u) \in \mathbb{Z}^2\}$. S is an additive subgroup of the reals. And you probably know that the additive subgroups of the reals are either closed (and have a least positive element) or dense.
$S$ cannot be closed as this would imply that $\pi$ would be rational. So $S$ is dense. As $\sin$ is continuous, $\sin(S)$ is dense in $[-1,1]$. As $\sin(A)=\sin(S)$, $\sin(A)$ is also dense in $[-1,1]$.
