Convergent's numerators of the continued fraction for $\pi$

Question

Call $C_{\pi} = \{ 1,3,22,333,355,… \}$, it´s the sequence of the numerators of convergents of the continued fraction for $\pi$, its OEIS' A002485, http://oeis.org/A002485.

Let $n \in C_{\pi}$, such that $n \geq 208341$.

If $k$ is the digit's number of $n$, then $|sin(n)| < 10^{-(k-1)}$.

I tried to show it, but the only thing that I can do is check the affirmation.

Answer 1

If $p_k/q_k$ is the $k$'th convergent of the simple continued fraction for an irrational number $x$, we have $|\sin(p_k - q_k x)| < |p_k - q_k x| < 1/q_k$.