We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by finding out what is $\overline{A}$ of this given set. And if $\overline{A} = [-1,1]$, we are through with the proof, but i having trouble here!
Similarly can one do this with cosine function also, that is proving $B= \{ \cos{n} \ : \ n \in \mathbb{N}\}$ being dense in $[-1,1]$
Answer
The hard part is to show that for any $x$ such that $0 \le x \le 2\pi$, and any $\epsilon>0$ there exists a real number $y$ and two integers $m$ and $n$ such that $|y-x|<\epsilon$ and $n=2\pi m+y$. Hint: break up $[0,2\pi]$ into small subintervals, remember that $\pi$ is irrational and apply the pigeonhole principle.
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