The way I thought about it is that the trigonometric, hyperbolic, and exponential functions are all nontrivial solutions of the differential equation y'''' = y.

The way I thought about it is that the trigonometric, hyperbolic, and exponential functions are all nontrivial solutions of the differential equation y'''' = y. Sharing differential properties is a very powerful kinship, which is also why you can substitute any of them for any of the other: the solution space of a linear differential equation is a vector space, and these three families of functions are just a change of basis in this space. Well, I don't know, this is what makes sense to me.