What is Hyperbolic Cosine Used for?

[alech4466]

I just learned about hyperbolic functions in my calculus class, and though my professor attempted to explain the use of hyperbolic functions, he really did not go very far into it, just providing a weak example ("If two people are holding a chain, cosh(x) factors into how much the chain droops").

I was wondering what the uses of hyperbolic functions are. If you could provide any equations that would be used for an example


Thanks in advance

 

[SteamKing]

Cosh and other hyperbolics are used to analyze structural vibration, suspended chains and cables, and waves in shallow water.

 

[HallsofIvy]

Every function can be divided into "even" and "odd" parts. That is, if f(x) is the function, then there exist an even function $f_e(x)$ and an odd function $f_o(x)$ such that $f_e(x)+ f_o(x)= f(x)$. cosh(x) is the "even" part of $e^x$ and sinh(x) is the "odd" part.

More important in applications is that cosh(x) satisfies the differential equation y''= y with initial conditions y(0)= 1, y'(0)= 0 and sinh(x) satisfies the differential equation y''= y with initial conditons y(0)= 0, y'(0)= 1. Those are the "fundamental solutions" for that differential equation- The solution to the general problem y''= y with initial conditions y(0)= A, y'(0)= B is y(x)= Acosh(x)+ Bsinh(x). That is the basic reason for the applications SteamKing gives.

 

[alech4466]

Thank you. That is a much better explanation than my professor gave. What would be an example of an equation that uses cosh or other hyperbolics though?

 

[gomunkul51]

If you happen to have this expression: 0.5*(e^x + e^-x) it is nicer to write it as cosh(x) ;)
seriously, this is useful in finding eigenvalues in Sturm-Liouville problems when you use the known properties of k*cosh(ax) and do not mess with c*(e^ax + e^-ax). where a, c and k are some constants.

 

[klondike]

Thank you. That is a much better explanation than my professor gave. What would be an example of an equation that uses cosh or other hyperbolics though?

The shape of a suspended wire y(x) is governed by nonlinear ODE $$y''=\frac{\rho}{T}\sqrt{1+(y')^2}$$. When certain condition is met, i.e. y(0)=y(L), the solution of the ODE is a cosh.

 

[dextercioby]

Performing this integral

$$\int \frac{dx}{\sqrt{1+x^2}}$$

is trivial if you know your way with hyperbolic functions.

 

[micromass]

Hyperbolic cosines and sines are also useful in described geometric objects called hyperbola's (hence the name: hyperbolic cosine). In particular, the system

$$\left\{\begin{array}{l}x=a cosh(t)\\ y=b sinh(t)\end{array}\right.$$

describes a hyperbola. Now, why are hyperbola's useful? Well, because hyperbola's arise in a lot of place. See the link http://britton.disted.camosun.bc.ca/jbconics.htm for some occurences of conic sections (and thus of hyperbola's).

 

[Fredrik]

Check out this post for an example from special relativity. Skip the first few paragraphs and start reading at "Since you have...".

 

[ccamatthews]

One use is for determining Tractrix (pursuit curve)

As an example of having a shuttle craft link up with an orbiting space station and determining what its path should look like.

See http://www.csun.edu/~jb715473/math382/tractrix.pdf