I apologise if this is trivial or well known to be impossible:

Can one find a finite set of integers $2\leq a_1<a_2<\ldots<a_m<\infty$ such that for the function defined as $$ f_{a_1,\ldots,a_m}(x)=\max\left\{\left|\frac{\sin (a_1 x)}{\sin x}\right| ,\left|\frac{\sin (a_2 x)}{\sin x}\right| , \cdots,\left|\frac{\sin (a_m x)}{\sin x}\right| \right\} $$ the minimum satisfies $$ \min\{f_{a_1,\ldots,a_m}(x):0\leq ~x~\leq 2\pi\}>1? $$

Edit I can restrict to $x$ not equal to an odd multiple of $\pi/2$ as well.

More generally can the lower bound be made larger? Say larger than a small positive integer? Or a slowly growing function of $m$?