A Periodic But Not Fixed Point

Geometry Level 5

For every integer $k,$ we define a function $f_k$ by the formula

$f_k(x)=100x-k\sin x.$

What is the smallest positive integer value of $k$ such that, for some real $\alpha$ , we have $f_k\big(f_k(\alpha)\big)=\alpha,$ but $f_k(\alpha )\neq \alpha?$

Details and Assumptions: The function is evaluated in radians. There is no degree symbol in the problem.