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?

Thank you!