Suppose one has multiple $\sin/\cos$ wave curves with different periodicity. The question is to determine the unique count of intersections of the combined curves with the $x$-axis. For example, if we have two curves $y=\sin(\pi x)$ and $y=\sin(\pi x/3)$ it is clear that for the first curve, whenever $x$ is divisible by $2$ the first function yields $0$ and so the wave intersects the $x$-axis at all $x$ where $x$ is even. For the second curve, it will intersect the $x$-axis whenever $x$ is a multiple of $3$. In fact, one can say that for the function $y=\sin(\pi x/n)$ the curve will intersect the $x$-axis with a periodicity $n$.
So, if there are two curves, say our example above, each graphed on the same $x/y$-axis, the number of combined unique intersections with the $x$-axis is actually not the sum of the two curves since there will be combined intersections for numbers that are multiples of $6$. The question then is how to construct a counting function for the number of unique intersections that n curves will have on the $x$-axis? The compound function
$$C(x)= \sin(\pi x/a)\sin(\pi x/b)\sin(\pi x/c)\cdots\sin(\pi x/n)$$
creates a curve that intersects the x-axis exactly at the unique points of each individual curve. The reason that this works is that an $x$-axis intersection point for any of the individual curves gives $0$ and thus determines the value of $C(x)$ regardless of the values of the other elements of the function.
One can create a function $A(x)$ such that it is defined to be $1$ whenever $C(x)=0$, and $0$ otherwise. Then one can sum $C(x)$ for each integer value of $x$ over its range, 1 to R. This accomplishes the task, but it is awkward. It would be preferable to create a function, $C(x)$, such that it takes on the value of 1 at each x-axis intersection point.
Using cos, in theory, would provide an elegant solution to this problem, but unfortunately, it doesn’t work. If, for example, $x=2$, $|\cos(\pi x/2)| = 1$, but $|\cos(\pi x/3)| = 0.5$. Thus, the compound equation of the 2, even using absolute values, is 0.5. This becomes more difficult as one superimposes more curves, e.g.$|\cos(\pi x/5)| = 0.8$ and $|\cos(\pi x/7)| = 0.623$. This means that it doesn’t yield a count of the intersections. Unless there is a different radian expression that I am missing, cos cannot be made to work for this problem.
Another possible solution would be to use combinations to determine the count, basically counting the intersections as the sum of $1$/periodicities less the intersection points, but this also becomes cumbersome quickly as the number of curves increases.
While this count issue seems to be a problem in analytic geometry, I haven’t thought of or found an answer. Any suggestions?
A possible implementation for $A(x)$ would be:
\begin{equation} A(x) = \begin{cases} 1, & \text{if} \hspace{3mm} (x\equiv 0 \mod a) \land (x\equiv 0 \mod b) \land \ldots \land (x\equiv 0 \mod n) \\ 0, & \text{otherwise} \end{cases} \end{equation} as $C(x) = 0$ only when $x$ is a multiple of the denominator ($a, b, ...$).
If you have cosines as a function, I would suggest making a separate $C(x)$ for cosines and use the same idea as before.