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# 352

Bath Time! - December 9, 2010
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Wow. That is so awful, I'm voting it 5 :)

I agree - but I figured sometimes 'awful' turns into 'awfully good'.

I think you mean Awefull. I love this comic.

can someone once and for all explain to me what the heck are those hyperbolic thingas needed for and what they represent?

Hey JK,
I hope you already know that exp(x) may be represented as an infinite sum, the Taylor- or McLaurin series. Doing the same for sinus and cosinus gives you that:

exp(ix) = cos(x) + i·sin(x)

Now that is, although useful, not exactly what we want. Something to split up the exponentional function in the even part (x^0, x^2, x^4, x^6…) and the uneven part (x, x^3, x^5, x^7…) would also be nice. And that’s exactly what sinh and cosh do:

exp(x) = sinh(x) + cosh(x).

d/dx sinh(x) = cosh(x)
d/dx cosh(x) = sinh(x)

There is also something like sin^2(x) + cos^2(x) = 1:

cosh^2(x) - sinh^2(x) = 1

There’s also:

sinh(ln(Φ)) = 0.5 with Φ = (1+sqrt(5))/2.

Oh, what you need it for…for example a chain, fixed to two points and pulled downwards by gravitational force roughly follows something with cosh(x), but I don’t know the exact formula anymore.

Have fun :)

Here are some formulae for it: http://en.wikipedia.org/wiki/Catenary

But maths is much more fun if you don't try to find applications for it. :)

It isn't really math if you include an application.

...unless it's a grant application.

But anyway, it's still maths, just not very pure.

well what about the branch of applied math? ODE's, non-linear ODEs, PDEs and other dynamical systems?

thanks for the explanation. now i want comics about sinuses and cosinuses... :D

A while back, a comment pointed out that the bird on the right of the banner for Dark Side of the Horse, named Sine, is flying in a sine wave...

Does that count?

(I did also reply that if there was a second bird, it could be a CoSine...)
http://www.gocomics.com/darksideofthehorse

A geometric explanation:

Sine and cosine are the "circular functions." They relate an angle, or equivalently, an arc-length on the unit circle, to the coordinates on the circle at that point. Hopefully you've seen a picture like there is here: <http://en.wikipedia.org/wiki/Unit_circle>.

Well, the unit circle is x^2 + y^2 = 1. The "unit hyperbola" is x^2 - y^2 = 1, and if you go an arc-length s away from the x-axis, you'll reach the point (cosh s, sinh s). (Of course, you can add +/- signs to each of those coordinates.)

Hence the term "hyperbolic sine," etc.

Why they're useful, at least in these days of physics and calculus, is better explained by the above reference to decomposing the exponential function.

Ah Hyper :)

Did anyone notice:
bath(a,b) = int_a^b ((cosh(x)-(-cosh(x))dx) = 2*(sinh(b)-sinh(a)
?

Nope, you forgot the bat’s head!

anyone notice that bath is not even a function?

I noticed.

Is this why whenever I use a PDF based on the absolute value function in [-1,1], bound from below by a caternary, my labmate (let's call him Bruce W.) "suddenly remembers" he has to run home to feed his cats?

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