What is the significance of hyperbolic trigonometric functions?

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qwertzuiop
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They aren't derived from their trig counterparts; how are they derived? Where do they appear in maths and science?
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TeeEm
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(Original post by qwertzuiop)
They aren't derived from their trig counterparts; how are they derived?
both trigonometric and hyperbolic functions are exponentials.

real exponentials are hyperbolic functions
complex exponentials are trigonometric functions

(Original post by qwertzuiop)
Where do they appear in maths and science?
everywhere
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qwertzuiop
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(Original post by TeeEm)
both trigonometric and hyperbolic functions are exponentials.

real exponentials are hyperbolic functions
complex exponentials are trigonometric functions



everywhere
Can you rewrite for me sin(x) in terms of e ? You purported that sin(x), which is a trigonometric function, is an "exponential".
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TeeEm
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(Original post by qwertzuiop)
Can you rewrite for me sin(x) in terms of e ? You purported that sin(x), which is a trigonometric function, is an "exponential".
sinx= (1/2i)[ eix - e-ix]= sinh(ix)
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qwertzuiop
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(Original post by TeeEm)
sinx= (1/2i)[ eix - e-ix]= sinh(ix)
Hence what is the limit as x-->infinity of sin(x), or as you put it, (1/2i)[ eix - e-ix].
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TeeEm
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(Original post by qwertzuiop)
Hence what is the limit as x-->infinity of sin(x), or as you put it, (1/2i)[ eix - e-ix].
there is no limit clearly
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username1599987
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(Original post by qwertzuiop)
They aren't derived from their trig counterparts; how are they derived? Where do they appear in maths and science?
Trig and hyperbolics unite when you start to look at complex numbers. They are actually extremely similar, and you would have seen that if you've done hyperbolic identities, a lot of them are exactly the same as the trig ones.
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qwertzuiop
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(Original post by TeeEm)
sinx= (1/2i)[ eix - e-ix]= sinh(ix)
Who derived that originally? I need to inquire; I am missing out.
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username1599987
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(Original post by qwertzuiop)
Who derived that originally? I need to inquire; I am missing out.
Euler? Not sure tho.
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atsruser
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(Original post by qwertzuiop)
They aren't derived from their trig counterparts; how are they derived? Where do they appear in maths and science?
Notwithstanding the result given by TeeEm, they essentially come from geometry. (\cos \theta, \sin \theta) are the coordinates of a point on the unit circle, and (\cosh \theta, \sinh \theta) are the coordinates of a point on the unit rectangular hyperbola.
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qwertzuiop
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(Original post by gagafacea1)
Euler? Not sure tho.
You'd think so, as e AND i are present
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TeeEm
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(Original post by qwertzuiop)
Who derived that originally? I need to inquire; I am missing out.
I am not up to date with history but I would imagine Euler and then Gauss are likely candidates for this. [Cauchy merely supplied formalism later in all things complex]
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qwertzuiop
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(Original post by atsruser)
Notwithstanding the result given by TeeEm, they essentially come from geometry. (\cos \theta, \sin \theta) are the coordinates of a point on the unit circle, and (\cosh \theta, \sinh \theta) are the coordinates of a point on the unit rectangular hyperbola.
Thanks!
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Riesel
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Haha. I remember disliking them at first. But you learn to enjoy them since they're basically the usual trig. With hyperbolic the same identities remain except you need to swap positive to negative, vice versa, when there's a sinh^2(x) (also applies to tanh^2(x)).

Who invented them I couldn't say. Maybe they were there to begin with and were just discovered? That sounds more like mathematical philosophy to me. Fun with physics... not maths.
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