Both sin(x) and sinh(x) can be expanded as power series. The series for sin(x) is . The series for sinh(x) is the same except that all the terms have a + sign. So . For small x, this is close to .
I just learnt about the hyperbolic functions so I tried some things.
If I take sinh(x)-sin(x), it looks a lot like a x^3 function (at least for small x)
So I tried to transform x^3 to look more like that function.
As I see I get the closest if I use (x*ln(2))^3 (not sure if it's really ln(2) but a number around it)
I tried the same with cosh(x)-cos(x), and there x^2 is already a quite nice approximation.
Are these coincidences? Especially that number looking like ln(2) is confusing. Why 2?