Hyperbolic Trigonometric Functions | Brilliant Math & Science Wiki

The hyperbolic trigonometric functions extend the notion of the parametric equations for a unit circle \((x = \cos t\) and \(y = \sin t)\) to the parametric equations for a hyperbola, which yield the following two fundamental hyperbolic equations:

\[x = \cosh a = \dfrac{e^a + e^{-a}}{2},\quad y = \sinh a = \dfrac{e^a - e^{-a}}{2}.\]

A very important fact is that the hyperbolic trigonometric functions take area as their argument (called "the hyperbolic angle," but this is just a name and has nothing to do with angles), as depicted below.

Hyperbolic functions show up in many real-life situations. For example, they are related to the curve one traces out when chasing an object that is moving linearly. They also define the shape of a chain being held by its endpoints and are used to design arches that will provide stability to structures. This shape, defined as the graph of the function \(y=\lambda \cosh \frac{x}{\lambda}\), is also referred to as a catenary.

The shape of a dangling chain is a hyperbolic cosine.

The St. Louis Gateway Arch—the shape of an upside-down hyperbolic cosine

Hyperbolas, which are closely related to the hyperbolic functions, also define the shape of the path a spaceship takes when it uses the "gravitational slingshot" effect to alter its course via a planet's gravitational pull propelling it away from that planet at high velocity.

The spaceship traces out a hyperbola as it uses the "slingshot" effect.