Is $x_k=\sin(k)$ eventually/periodic?

Question

A sequence is eventually periodic if we can drop a finite number of terms from the beginning and make it periodic.

$$x_k=\sin(k)$$

I think this is periodic since the function is periodic and it seems to converge to 0 by iterations. The thing that confused me is that there are infinite numbers in the domain in the period of $\sin(k)$ since it's continuous. So I can't really assume that if a function is periodic -> sequence is periodic?

Answer 1

If you have $\sin(k+n)=\sin k$, then¹ either $n$ or $2k+n$ is an integer multiple of $\pi$. Can this happen for integers $k,n\ge1$? (Knowing that $\pi$ is irrational² might be useful.)

You should arrive to the conclusion that the sequence is not periodic.

¹We know that $\sin x=\sin y$ holds if and only if $x=y+2z\pi$ or $x=\pi-y+2z\pi$ for some $z\in\mathbb Z$.

² Wikipedia: Proof that $\pi$ is irrational

Answer 2

It cannot be eventually periodic since it can be shown the set of values of $ \sin k\ (k\in \mathbf N)$ is dense in $[-1,1]$, and not equal to the whole of $[-1,1]$.