Sin(n) and cos(n) dense in $[-1,1]

Question

We knows that $sin(x)$ and $cos(x)$ are two function with value in the closed set $[-1,1]$. How can I prove that $X=({sin(n)|n\in\mathbb{N}})$ and $Y=({cos(n)|n\in\mathbb{N}})$ are or not dense in $[-1,1]$.

Answer 1

Hint Use the irrationality of $\pi$ to show that $\{ m+2n \pi : m,n \in \mathbb Z \}$ is dense in $\mathbb R$.