Rational Approximations Of Pi?

Computer Science Level pending

We have \(\sin(\pi/2 + 2\pi k) = 1\) for all integers \(k.\)

Because \(\pi\) is irrational, one can use this to show that \(\sin(m)\) gets arbitrarily close to \(1\) as \(m\) ranges over all integers. Find the integer \(m\) such that \(|m|\leq 1000\) and \(\sin(m)\) is as close as possible to \(1.\)

Details and assumptions

In other words, the answer to this question is the unique integer \(m\) such that \(-1000\leq m\leq 1000\) and \(1-\sin(m) \leq 1-\sin(n)\) for all integers \(-1000\leq n\leq 1000\).


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