In view of a [result by Scott][1], there exist infinitely many coprime natural numbers $p,q$ such that $q$ is odd and 
$|p-\tfrac\pi2\,q|<1/q$. 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, $f(1)=1$. 


  [1]: https://www.ams.org/journals/bull/1940-46-02/S0002-9904-1940-07152-6/?active=current