In view of a result by Khinchine (see e.g. [Theorem 3.6(a)][1], 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, $f(1)=1$. 


  [1]: https://www-ams-org.services.lib.mtu.edu/journals/tran/1961-099-01/S0002-9947-1961-0121357-5/