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We know that for any integer $n$, $\sin(n)$ can never equal to one. So what is the maximum value of $sin(n)$? How close can the sine of an integer get to unity?

I'll be really interested to know if such a value exists.

HPP_00
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    There is no max. You may find a sequence of integers $n_k$ for which $\lim_k \sin(n_k)=1$. – H. H. Rugh Aug 25 '17 at 12:24
  • @skyking Oops. Thank you. – HPP_00 Aug 25 '17 at 12:38
  • So it is, that given an integer $n$, I can make $sin(n)$ as close to 1 as I like? If so, can we plot distributions of how close $sin(n)$ approaches 1 for an arbitrary small precision value? – HPP_00 Aug 25 '17 at 12:42

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It doesn't exist. It can be proved that $\sup\{\sin n\,|\,n\in\mathbb{Z}\}=1$, but, as you wrote, $\sin(n)$ is never $1$, if $n\in\mathbb Z$.

José Carlos Santos
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