Among the first $n$ positive integers, approximately how many $m$ are there such that $|\sin{m}|<|\sin{}k|$ for every positive integer $k<m$?

Question

Among the first $n$ positive integers, approximately how many $m$ are there such that $|\sin{m}|<|\sin{k}|$ for every positive integer $k<m$? (Let's say $1$ counts as such an $m$.) Here is the sequence of such numbers.

Let's call this quantity $f(n)$.

For example, $f(10^5)=5$. The integers $1, 3, 22, 333, 355$ have "record breaking" small values of $|\sin{m}|$.

Is there an approximation formula for $f(n)$? For example, something like $\ln{n}$.

Possibly related: If everyone in the world chooses a random real number between $0$ and $1$ one by one, how many times will a world record for lowest number be set?