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In view of a result by Khinchine (see e.g. Theorem 3.6(a), there exist infinitely many pairs $(p,q)$ of natural numbers such that $|2\pi q+\tfrac\pi2-p|<1/q$. For such $p$ and $q$, letting $q\to\infty$, we have $\sin p=1-O(1/q^2)=1-O(1/p^2)$ and hence $\sin^p p\to1$. So, $f(1)=1$.