What is the significance of hyperbolic trigonometric functions?

They aren't derived from their trig counterparts; how are they derived? Where do they appear in maths and science?

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

Reply 2

qwertzuiop

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".

Reply 3

TeeEm

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)[ e^ix - e^-ix]= sinh(ix)

Reply 4

qwertzuiop

Original post by TeeEm

sinx= (1/2i)[ e^ix - e^-ix]= sinh(ix)

Hence what is the limit as x-->infinity of sin(x), or as you put it, (1/2i)[ e^ix - e^-ix].

Reply 5

TeeEm

Original post by qwertzuiop

Hence what is the limit as x-->infinity of sin(x), or as you put it, (1/2i)[ e^ix - e^-ix].

there is no limit clearly

Reply 6

username1599987

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.

Reply 7

qwertzuiop

Original post by TeeEm

sinx= (1/2i)[ e^ix - e^-ix]= sinh(ix)

Who derived that originally? I need to inquire; I am missing out.

Reply 8

username1599987

Original post by qwertzuiop

Who derived that originally? I need to inquire; I am missing out.

Euler? Not sure tho.

Reply 9

atsruser

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.

Reply 10

qwertzuiop

Original post by gagafacea1

Euler? Not sure tho.

You'd think so, as e AND i are present

Reply 11

TeeEm

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]

Reply 12

qwertzuiop

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!

Reply 13

Riesel

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.