does there exist such sequence?(Maybe Dirichlet Approximation Theory is needed)

Question

for any real number t between -1 and 1, does there exist a infinite sequence of positive integers $x_{n}$ such that $\lim_{n\to\infty}{\sin(x _{n})}=t$ ?

Answer 1

Yes. The set of points $\{e^{in}\}$ is dense on the unit circle in the complex plane, so you can find a subsequence converging to any point there.

See https://en.wikipedia.org/wiki/Equidistribution_theorem

Answer 2

Hints: as ,the set of subsequential limits of $\{\sin(n)\}_{n}$ is dense in $[-1,1]$ , so, for every $t\in [-1,1]$ , there is a subsequence $\{x_n\}$ of the sequence $\{n\}$ such that $\sin(x_n)=t$