The Bessel functions are particular functions defined as the solution of the differential equation:
$$x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2-\alpha^2)y = 0$$
for an arbitrary complex number $\alpha$ (I'll consider it real for what follows).
There are different kinds of Bessel functions, depending on the value of $\alpha$:
I only consider here the Bessel function of the first kind, which is $J_0(x)$. It really looks like to some kind of sinc function to me, or something similar. Hence I try to approximate the Bessel function (which has a tough expression) with "nice" trigonometrical functions.
Here is a graph comparing Bessel function of the first kind $J_0(x)$ to $A\frac{\sin(bx+\varphi)}{cx+\phi}$ (which is some kind of "generalised" sinc function since $bx + \varphi = cx + \phi = \pi x$ in a classical one) with $A = 1.1091$, $b = c= 1$ and $\varphi = \phi = 0.78$:
Isn't the ressemblance between those two functions stunning ? Is it just a coincidence or is there a specific reason to it ? Do you think that I could find the exact Bessel function by changing the parameters $a, b, c, \varphi$ and $\phi$ ?
EDIT: the result of the proposition of @Gary (see comments below):
