# Real world uses of hyperbolic trigonometric functions

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?

• Lorentz transforms can be understood as hyperbolic rotations. The caternary curve (a dangling string/chain) is really just cosh – crasic Oct 30 '10 at 23:48

If you take a rope, fix the two ends, and let it hang under the force of gravity, it will naturally form a hyperbolic cosine curve.

• See Catenary for more analysis on the curve that is mentioned in this answer. – Justin L. Jul 20 '10 at 21:22
• In real life you use the catenary shape to know how much cable to place between two poles in high power transmission lines. Too much cable and it sags too much making it a hazzard. Too little cable and it breaks due to high tension as it stretches. Quite important to get it right when you have 768kV @ 6000 Amps through the cable. – ja72 Jan 8 '11 at 5:33

The catenary has been mentioned a number of times, but apparently not the corresponding surface of revolution, the catenoid. It and the plane are the only surfaces of revolution that have zero mean curvature (i.e. they are minimal surfaces). This surface is the form a soap bubble (approximately) takes when it is stretched across two rings:

(image from here)

On a map using the Mercator projection, the relationship between the latitude L of a point and its y coordinate on the map is given by $y = \operatorname{arctanh}(\sin(L))$, where $\operatorname{arctanh}$ is the inverse of the hyperbolic tangent function.

Velocity addition in (special) relativity is not linear, but becomes linear when expressed in terms of hyperbolic tangent functions.

More precisely, if you add two motions in the same direction, such as a man walking at velocity $v_1$ on a train that moves at $v_2$ relative to the ground, the velocity $v$ of the man relative to ground is not $v_1 + v_2$; velocities don't add (otherwise by adding enough of them you could exceed the speed of light). What does add is the inverse hyperbolic tangent of the velocities (in speed-of-light units, i.e., $v/c$).

$$\tanh^{-1}(v/c)=\tanh^{-1}(v_1/c) + \tanh^{-1}(v_2/c)$$

This is one way of deriving special relativity: assume that a velocity addition formula holds, respecting a maximum speed of light and some other assumptions, and show that it has to be the above.

I don't know if you consider General Relativity "outside acadamia"(and I don't care to argue the point!) but if you do,

the group of symmetries with respect to the Lorentzian Metric can be written as Matrices containing hyperbolic trig functions as elements.

Note Kenny's comment.

• (1) That's called Minkowski metric. Or do you mean Lorentz transforms? (2) It is just special relativity because no curved spacetime is involved. – kennytm Jul 24 '10 at 20:08
• I'll count it. Here's a bit more. Relativity is interested in the space-time interval. If time is the base of a triangle and distance the hypotenuse the interval is the other base. So space-time = d^2 - t^2. So in the d and t plane we have this weird distance metric. Now comes the hand waving. Take the geometric product of d and t = d * t + d ^ t = 0 + d ^ t. Squared it equals 1. Usually it equals -1. So when we expand e^(dt)x = 1 + dtx + x^2 + dtx^3 ... There is no alternating signs anymore. e^(dt)x = cosh(x) + dt*sinh(x). Explained here geocalc.clas.asu.edu/pdf/CompGeom-ch2.pdf – Jonathan Fischoff Jul 24 '10 at 20:28

Many kinds of nonlinear PDE have wave solutions explicitly expressed using hyperbolic tangents and secants: shock-wave profiles, solitons, reaction-diffusion fronts, and phase-transition fronts, for starters.

An equation for a catenary curve can be given in terms of hyperbolic cosine. Catenary curves appear in many places, such as the Gateway Arch in St. Louis, MO.

If a pendulum oscillates near its stable equilibrium, the equation of motion is $x''=-\omega^2 x$, and the solution is any linear combination of $\sin\omega t$ and $\cos\omega t$. If the pendulum has a stiff arm (rather than a string), then there is a second, unstable equilibrium, where it's straight up. This is like balancing a pencil on its tip. The equation of motion is $x''=\omega^2 x$, and the solution is any linear combination of $\sinh\omega t$ and $\cosh\omega t$.

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.
• is there any real benefit in knowing the expression of the logistic function in terms of tanh though? I appreciate that it's a nice example, I'm just wondering whether it's of much use to know ( I don't think that I've seen it expressed in terms of tanh before ) – baxx May 26 at 13:33
• @baxx There's no practical benefit, but the equivalence implies that the use-cases of one are also covered by the other. Normally, not many consider for example that population dynamics can be described by $\tanh (x)$ but that's just not painting the whole picture. The same can be said about $\sin(x)$, $\cos(x)$, $\sin^2(x)$, $\cos^2(x)$ which are all equivalent up to an appropriate transformation. – Jepsilon May 29 at 12:42

Most curves that look parabolic are actually Catenaries, which is based in the hyperbolic cosine function. A good example of a Catenary would be the Gateway Arch in Saint Louis, Missouri.

The tractrix, which is based in the hyperbolic secant, is also known as the pursuit curve, which models objects like cargo trucks turning corners, or a dog on a porch starting to chase a car in the street.