# Hyperbolic Functions Defined? **URL:** \ **Category:** Factual Questions\ **Created:** [November 27, 2002, 5:25pm UTC](https://boards.straightdope.com/t/hyperbolic-functions-defined/139144 "2002-11-27T17:25:27Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [November 27, 2002, 5:25pm UTC](https://boards.straightdope.com/t/hyperbolic-functions-defined/139144/1 "2002-11-27T17:25:27Z")
Ok, if trig functions are based on what’s referred to as the “Unit Circle”, then where do hyperbolic functions get their basis? Is there a Unit Hyperbola, perhaps? And, what are the definitions of the hyperbolic trig functions of sinh, cosh, and tanh? I mean, like sin = opposite/hypotenuse…what is the equivalent for sinh? Going hyper, - Jinx ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [November 27, 2002, 5:34pm UTC](https://boards.straightdope.com/t/hyperbolic-functions-defined/139144/2 "2002-11-27T17:34:32Z")
The hyperbolic functions are a parameterization of the curve x[sup]2[/sup] - y[sup]2[/sup] = 1, a hyperbola. They are defined in terms of exponentials, cosh x = (e[sup]x[/sup] + e[sup]-x[/sup])/2 sinh x = (e[sup]x[/sup] - e[sup]-x[/sup])/2 I do not immediately recall an opp/hyp type definition for the hyperbolic functions. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [November 27, 2002, 5:55pm UTC](https://boards.straightdope.com/t/hyperbolic-functions-defined/139144/3 "2002-11-27T17:55:23Z")
_Checks index of Calculus by Howard Anton._ unit hyperbola, p 430 _Turns to page 430._ “…the curve x[sup]2[/sup] - y[sup]2[/sup] = 1, called the **unit hyperbola**.” _Previews._ Dammit **Jabba**! ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [November 27, 2002, 6:09pm UTC](https://boards.straightdope.com/t/hyperbolic-functions-defined/139144/4 "2002-11-27T18:09:01Z")
Aside from the parameterization analogy, there is a rather contrived similarity that runs between circular and hyperbolic trig functions that involves the area defined by a line from the origin to a certain point on the curve, the curve itself, and one of the axes. [Check out pages 1-2 of this PDF file](http://www.etsu.edu/math/gardner/1920/c6s7bw.pdf). However, the _real_ way to see the similarities between sine and hyperbolic sine is to look at their Taylor series, or their definitions in the complex domain. ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [November 27, 2002, 10:54pm UTC](https://boards.straightdope.com/t/hyperbolic-functions-defined/139144/5 "2002-11-27T22:54:49Z")
Is there a branch of trig functions based on a conic section besides the circle or hyperbola? ---
**Author:** ![bonzer](https://avatars.discourse-cdn.com/v4/letter/b/45deac/32.png) [@bonzer](https://boards.straightdope.com/u/bonzer)\ **Post date:** [November 28, 2002, 12:22am UTC](https://boards.straightdope.com/t/hyperbolic-functions-defined/139144/6 "2002-11-28T00:22:58Z")
Well, the most useful generalisation of the trig functions that I’ve been able to use anywhere are the Jacobi elliptic functions. You have _sn_ and _cn_ that are _sin_ and _cos_ in a particular limit, but you also have _dn_. These generalise the standard trig function identities, but, I have to confess, I’ve never thought of these geometrically and have no idea offhand what they might correspond to. ---
**Author:** ![barking\_frog](https://avatars.discourse-cdn.com/v4/letter/b/96bed5/32.png) [@barking\_frog](https://boards.straightdope.com/u/barking_frog)\ **Post date:** [November 29, 2002, 5:18am UTC](https://boards.straightdope.com/t/hyperbolic-functions-defined/139144/7 "2002-11-29T05:18:14Z")
Damnit. On preview the letters didn’t work out right (I used the Windows character map). Anyways, just substitute è for theta and ö for phi. ------------ Heres my original post: There’s also another neat thing with the hyperbolic trigs. For the unit circle, if you draw a line from the origin to the circle, the area of the sector of that circle is given by: x/(2\*Pi) \* Pi = x/2 where x = è For the unit hyperbola, if you draw a line from the origin to a point (cosh(è), sinh(è)), the area between the line and hyperbola/x-axis is also è/2. To show that the area is è/2: The slope of the line from the origin to the point (cosh(è),sinh(è)) is m = tanh(è) and the equation is y = x_tanh(è) ==\> x = y/tanh(è) = y_coth(è). Let x2 be a point on the hyperbola and x1 be a point on the line. Use a horizontal area element for integration, which is: dA = (x2 - x1)dy = (sqrt(1+y^2) - y\*coth(è))dy Integrate and then, A = Integral[(sqrt(1 + y^2) - y\*coth(è))dy, {0, sinhè}] Read: A is the integral of (sqrt(1 + y^2) - y\*coth(è))dy from 0 to sinhè (where the hyperbola and line meet). Note: it’s sinhè and not coshè because a horizontal strip is being used. First, split up the integral: A = INT[sqrt(1+y^2)dy] - cothè_INT[ydy] (both on the same interval) = INT[sqrt(1 + y^2)dy] - cothè_(y^2/2) = … - cothè\*(sinh^2(è) - 0)/2 = INT[sqrt(1 + y^2)dy] - 1/2_cothè_sinh^2(è) Now, substitute y = sinhö \<==\> dy = coshö\*dö. y = 0 corresponds to ö = 0 and y = sinhè corresponds to ö = è. Now the area is: A = Integral[(coshö_coshö_dö), {0, è}] - 1/2_cothè_sinh^2(è) = INT[cosh^2ö\*dö] - … but cosh2ö = 2cosh^2ö - 1 (another interesting property of the hyperbolic trigs – sharing similar identities, although not for all identities, like when you have an implied or explicit product of two sines, which is linked to cosx - isin(x) because (isin(x))^2 = -sin^2(x). anyways… A = INT[1/2\*(1 + cosh2ö)] - … = 1/2[ö + 1/2sinh2ö] on [0,è] - 1/2coshè/sinhè\*sinh^2(è) = è/2 + 1/4sinh2è - 1/2coshèsinhè = è/2 + 1/4(2coshèsinhè) - 1/2coshèsinhè = è/2 And yes, I know I didn’t type all that out too well (especially on writing out my intervals, and skipping it altogether several times), but it’s still understandable and the result is interesting. A summary of the hyperbolic trigs: - The area between the graph and a line from the origin are related to that of a circle. - The hyperbolic trigs have identities very similar to those of circular trigs (you can find one from the other using Osborne’s Rule). - cosh(x) is the even part of e^x; sinh(x) is the odd part of e^x. - You can express the inverse hyperbolics in terms of logs using a bit of algebra. - The antiderivatives of several common functions involve hyperbolic trigs (look up an antiderivative table). … and there are more I’m sure. Also, the curve y = cosh(x) is called a caternary, and is the shape of a hanging heavy wire – it minimizes the potential energy of the wire. At least, that’s what our prof said. There are also some interesting relationships if you take the solid of revolution about the x-axis of cosh(x). ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [November 29, 2002, 5:49am UTC](https://boards.straightdope.com/t/hyperbolic-functions-defined/139144/8 "2002-11-29T05:49:38Z")
My copy of _Standard Mathematical Tables_ (Chemical Rubber Company) has a geometric definition of hyperbolic functions on p 431. It is too complex to describe in words and I can’t put a picture here. However, the Chemical Rubber volume is well known in the technical field and you should be able to find one, if you don’t already have it.