So I am a bit surprised that there isn't a question on this. Recall the first time when you came across the cosine function $\cos (ax)$ as $a \to \infty$ graphically, it intersect the x axis more and more frequently, result in the limit to diverge
Now, more familiar with the different kinds of infinite sets, I am actually wondering whether the roots are dense as $a\to \infty$. It seems that plugging in any $a$ the position of the roots can be any real number, and the roots are regularly spaced by some nonzero intervals for any $a$, thus it seems as $a \to \infty$ there is never a case that the roots will be dense.
How to approach this problem?
The roots are dense, in the following interpretation of the word.
Given any $\epsilon>0$, there exists a real number $r=r_{\epsilon}$ such that if $a>r$ then given any real number $\beta$, $\cos ax$ has a zero within $\epsilon$ of $\beta$.