The precise question is this: given a irrational $r\in \mathbb{R}$, is it true that $r\mathbb{Z}$ is dense (topologically) in $\mathbb{R}/\mathbb{Z}$?

The reason this came up was actually a teaching moment for Calc II; I wanted to use $sin(n)$ as an example of a bounded divergent sequence, but I was actually not sure (and certainly not sure how to explain to Calc II students) that it doesn't converge to some limit. I suspect the question above is true, which would imply that the example works (among other, more interesting, things).

Edit: The question was originally phrased in terms of transcendentals, but given the answer below, this is overkill and deceptive.