# Math Help - Approximating sinh(x)-sin(x)

1. ## Approximating sinh(x)-sin(x)

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!

2. Both sin(x) and sinh(x) can be expanded as power series. The series for sin(x) is $\sin x = x - \tfrac{x^3}{3!} + \tfrac{x^5}{5!} - \tfrac{x^7}{7!} + \ldots$. The series for sinh(x) is the same except that all the terms have a + sign. So $\sinh x - \sin x = \tfrac2{3!}x^3 + \tfrac2{7!}x^7 + \ldots$. For small x, this is close to $\tfrac13x^3$.

3. Thanks!
(1/3)^(1/3) looks so much like ln(2)