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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?

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"trigonometric" (not "trigonomic" or "triganomic") –  Jason S Jul 20 '10 at 23:04
    
Outside academia — such as? –  KennyTM Jul 21 '10 at 12:23
    
@Kenny anything... –  Jacob Jul 21 '10 at 13:26
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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

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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.

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See Catenary for more analysis on the curve that is mentioned in this answer. –  Justin L. Jul 20 '10 at 21:22
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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

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 = atanh(sin(L)), where atanh is the inverse of the hyperbolic tangent function.

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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:

catenoid bubble

(image from here)

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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.

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(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
    
Thanks Kenny, it's been a while... –  BBischof Jul 24 '10 at 20:20
    
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.

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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.

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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.

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Apart from occuring naturally in the form of the catenary mentioned, hyperbolic trig functions are also fundamentally important in calculus. I hope you'll agree that calculus has many, many real world applications!

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Can you elaborate further? I am only aware of the use of their inverses in helping with integration. –  Justin L. Jul 24 '10 at 23:59

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.

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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 $\cosh\omega t$.

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