TITLE: Real world uses of hyperbolic trigonometric functions
QUESTION [44 upvotes]: I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful.
Is there any good examples of their uses outside academia?

REPLY [9 votes]: The hyperbolic tangent is also related to what's called the Logistic function:
$L(x)=\frac{1}{1+e^{-x}}=\frac{1+\tanh(\frac{x}{2})}{2}$
Among many uses and applications of the logistic function/hyperbolic tangent there are:

Being an activation function for Neural Networks. These are universal function approximators that are pretty much becoming central to modern A.I.
The Fermi-Dirac Distribution and Ising Model in statistical mechanics
Being a sigmoid function ("S-shaped") means that it can be a candidate to a cumulative distribution function assuming that its derivative can be used to model some random variable
Modelling population growths/declines. Although this is more in the realm of Biology, it certainly has quite some appeal from purely the perspective of a dynamical system.
Considering $\tanh(kx)$, one can approximate the Heaviside step function (by setting $k$ to a sufficiently large number) in such a way that it is still continuous and infinitely differentiable. This can be used when solving DEs in physics to analyse the action of, for one example, turning on a switch.

Moving on to $\cosh (x)$, it also has some nice use-cases:

A hanging inelastic chain takes the shape of $\cosh (x)$. This shape is called a catenary.
A soap film joining two parallel, disjoint wireframe circles is the surface of revolution of $\cosh (x)$. This in general pops up a lot when studying minimal surfaces.
In the canonical formalism of Statistical Mechanics, the partition function of a 2-level system with state energies of $\pm\varepsilon$ system is given by $Z\propto\cosh(\varepsilon/k_B T)$. This then gives us that the average energy of the system is given by $\langle E\rangle \propto \varepsilon \tanh(\varepsilon/k_B T)$ taking us back to $\tanh (x)$
In architecture, if you have a free-standing (i.e. unloaded and unsupported) arch, the optimal shape to handle the lines of thrust produced by its own weight is $\cosh(x)$. The dome of Saint Paul's Cathedral in England has a $\cosh(x)$ cross-section. This type of arch was also favoured by architect Antoni Gaudí in his work.