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The first time I knew about the hyperbolic function was when I was studying the derivatives. And I know that the derivative of $\sinh x= ( e^x - e^{-x} )/ 2$ , but i still confused with what are they really are? and how did we get them and for what we are using them?

Thanks very much, and i hope that wasn't a long question.

José Carlos Santos
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Alselvdor
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5 Answers5

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So you know how $\sin$ and $\cos$ relate the side lengths of right triangles on the unit circle to the interior angle? Well, $\sinh$ and $\cosh$ do the same but the instead of having right triangle which lie on the unit circle we have them lie on the unit hyperbola.

The equation for the unit circle is,

$$x^2+y^2 = 1$$

While the equation for the unit hyperbola is,

$$x^2 -y^2 = 1$$

This picture helps depict what I am describing, https://en.wikipedia.org/wiki/Hyperbolic_function#/media/File:Hyperbolic_functions-2.svg

Jürgen Sukumaran
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    Sure, but why do people use them? What's there real-life application – Andrew Li Mar 04 '18 at 16:55
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    The hanging chain is a good example, often explored in an introduction to calculus of variations course. Such a chain forms a "catenary", https://en.wikipedia.org/wiki/Catenary this shape uses hyperbolic trigonometric functions. – Jürgen Sukumaran Mar 04 '18 at 16:56
  • yeah,i was going to ask that, too? why we need them? – Alselvdor Mar 04 '18 at 16:57
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    Just like $\cos$ and $\sin$, a lot of things in nature/physics/life are described by these functions. We don't enforce their usefulness, it is an intrinsic quality. This thread https://math.stackexchange.com/questions/123/real-world-uses-of-hyperbolic-trigonometric-functions has more examples (a bubble between two circles, general relativity, etc) – Jürgen Sukumaran Mar 04 '18 at 16:59
  • I would be careful with the claim about the angle measure, since unlike the unit circle case, the angle measure and the sector area are not proportional for the hyperbola. The hyperbolic functions are tied to sector area (as your link shows), but if you wanted functions on the unit hyperbola related to angle measure, IIRC you'd need to write some integrals that are not elementary functions. – Mark S. Mar 04 '18 at 18:38
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The catenary ( or chaînette) is the shape of the curve assumed by a hanging chain or cable with the two ends fixed, under its own weight. It happens its equation is $$y=a\cosh \frac xa$$ where the constant $a$ depends on physical parameters (tension and mass per unit length). It is used in architecture and engineering for archs, bridges, &c.

You also find a derived curve in the shape of a skipping rope.

Bernard
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  • So, we can just deal with it as normal trig, but using them with arches and parabolas instead of triangles? – Alselvdor Mar 04 '18 at 18:26
  • @Alselvdor: I don't know what you have in mind saying ‘deal with as normal trig’, but there are formulæ similar to (but different from) trigonometric formulæ, such as $\cosh^2x-\sinh^2x=1$ or $(\cosh)' (x)=\sinh x$. – Bernard Mar 04 '18 at 18:58
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The hyperbolic functions $\sinh$ and $\cosh$ parameterize the hyperbola $x^2-y^2=1$ since $\cosh^2 t-\sinh^2 t=1$ for all $t\in\mathbb{R}$.

A. Goodier
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    It's actually $\cosh^2 t - \sinh^2 t = 1$. And they parameterize the right-hand branch of the hyperbola only since $\cosh t > 0$ for all $t$. – Daniel Schepler Mar 04 '18 at 18:26
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The imverse hyperbolic functions are particularly useful in integration, for example when dealing with positive quadratic functions inside square roots. Although such integrals can be done with trig functions, using hyperbolic functions makes them much easier.

Find $\int\sqrt{x^2-1}dx$ for example

David Quinn
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Every real functions can be uniquely represented by the sum of the even function and the odd function. $$f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$$ Now let's put $f(x)=e^x$.