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

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