# Origin of hyperbolic functions

I read the wiki on hyperbolic functions but i don't really understand. I also plotted cosht,sinht on wolfram it made sideways v lying on the x axis. Can any one explain why people made hyperbolic functions.

I still don't really understand what it means apart from cosht being the x coordinate of the intercept of a ray passing through the origin intercepts a hyperbola, where the rays angle is 2A

Pengwuino
Gold Member
I can think off the top of the head they are solutions to various elementary forms of differential equations and the basis for some orthogonal function expansions.

Hurkyl
Staff Emeritus
Gold Member
The hyperbolic trig functions play the same role with hyperbolas that the ordinary (circular) trig functions play with circles and ellipses.

I found this page useful when I first met the hyperbolic functions in the context of special relativity. This is just one example of an application, rather than a comment how they originated, but you might find it interesting as it shows another parallel between circular and hyperbolic functions.

If you rotate an orthonormal coordinate system for Euclidean space, the way the coordinates of any point change can be expressed with a matrix of circular functions. For example, a rotation about the z axis:

$$\begin{bmatrix}\cos{\theta} & \sin{\theta} & 0\\ -\sin{\theta} & \cos{\theta} & 0\\ 0 & 0 & 1\\\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}$$

In special relativity, besides rotating an orthonormal coordinate system for spacetime in this way, you can make another kind of change of coordinates that also preserves (spacetime) distances between points. This switches to a coordinate system moving at some velocity relative to the one you started with, for example moving along the x axis:

$$\begin{bmatrix}\cosh{\phi} & -\sinh{\phi} & 0 & 0\\ -\sinh{\phi} & \cosh{\phi} & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}t\\x\\ y\\ z\end{bmatrix}$$

where $\phi = \text{tanh}^{-1}\left({\frac{v}{c}}\right)$, the inverse hyperbolic tangent, "artanh", of the speed of the new coordinate system as a fraction of the speed of light.

hotvette
Homework Helper
Based on a recent book I read on the history of the number e, y = 1/2(ex+e-x), which is cosh(x), is the solution to the 'hanging chain' problem posed by Jakob Bernoulli in 1690. One year later, three correct solutions were published (Leibniz, Johann Bernoulli, and Huygens). The analogy between the hyperbolic and trig functions appears to have been discovered in 1757 by Riccati.