Can someone please fully explain hyperbolic trigonometric functions to me?
First a little background. I first was exposed to trigonometric functions in a physics class in my senior year in hs. I was immediately fascinated by the subject. In fact, I even used it in a practical application. There was this tree in front of my neighbor’s house across the street from me. And I always wondered how tall it was. I used trig to find it out. (Well, it gave me a thrill at least–at least at the time [remember I was still relatively young then;)].)
Anyways, then came hyperbolic trig functions. We barely spent anytime on that subject at all, because it apparently wasn’t considered essential at the time. But we were told that unlike being based on the circle, it was based on the hyperbola. But I have a hard time even imagining something like that now. How can angle measurements be based on a hyperbola? I know, the angle does sweep around in the (unit) circle. I understand that fully. But hyperbola? What am I missing here?
Please make your explanation(s) as simple as possible. And thank you in advance:)
On the unit circle x[sup]2[/sup]+y[sup]2[/sup] = 1, any central angle can be interpreted as:
the amount of rotation needed to bring one ray (the initial side) in line with another ray (the terminal side). With a protractor we find this number in degrees, radians, or any other angle units our protractor is divided into.
the length of the arc cut off by the two sides of the central angle.
twice the area of the circular sector bounded by the unit circle and the two sides of the central angle.
As you noted, the first interpretation is hard to generalize to the unit hyperbola x[sup]2[/sup]-y[sup]2[/sup] = 1. The second two interpretations fare slightly better. It takes a bit of calculus to make the connection between arclength s and Cartesian coordinates (x,y) = (cosh s, sinh s), not to mention the relation to a bounded area.
I prefer to think of it in terms of the functions themselves, rather than the geometric representation. You know, of course, that the derivative of sine is cosine, and the derivative of cosine is negative sine, and that hence the fourth derivative of sine gets you right back to sine (and likewise for cosine). That’s kind of interesting. Well, what if you wanted a function (or set of functions) such that the second derivative gets you back to the original function? Turns out, that’s sinh and cosh.
They hyperbolic trig functions satisfy cosh(x) = cos ix and sinh(x) = -i sin ix, and thus satisfy analogous addition formulas, convenient differential equations, etc. to those of trig functions. (This also gives a convenient way to prove the addition formula, double-angle formula, etc.: they all derive from the fact that exp(x+y) = exp(x) exp(y), which is easy to prove by, e.g., the fact that a sufficiently nice linear differential equation with boundary conditions has a unique solution.) There’s no new “information” there, though; they’re just occasionally more convenient forms of sin and cos, same as with tan and similar functions.
You may be wondering why we have these functions at all, as in why we have a special name for them and they’re on calculators. After all, regular trig functions are useful for finding things like the height of that tree, so what are the hyperbolic functions good for?
I’m not a mathematician (I am an EE), so take this for what it’s worth. In mathematics, there will be some patterns of functions and numbers that crop up frequently, and those tend to get their own special names. The numbers pi and e are so named because they just come up so often when you’re doing math. The trig functions are the same - besides measuring triangles, they just fall out all over the place when you’re doing, for example, differential equations for systems like vibrating strings.
Hyperbolics are the same way, although to a lesser extent. I recall from college 30 years ago solving the equations for a vibrating membrane, and while trig functions naturally occur where you have a vibrating string, with a membrane (like a drum head) you get hyperbolic functions. However, that’s the only time I recall ever actually coming across them, and since I got out of college 30 years ago I have not seen them again.
ETA: I hope I’m remembering right about hyperbolics applying to membranes. On second thought, it could be Bessel functions. Or maybe I just came across Bessel functions as they apply to FM modulation. It’s been a while.
Yeah, I think it’s the Bessel functions you’re thinking about. The radial factor of the eigen-solutions to the wave equation for a circular membrane are Bessel functions, while the angular factors are sines and cosines.
The point is, there are a lot of fun functions that show up in various places. Cosh, for example, shows up in shape of a weighted chain.
Another EE checking in. As mentioned upthread, normal trig identities are built around the unit circle (x[sup]2[/sup]**+y[sup]2[/sup]=1) while hyperbolic identities are built around the hyperbola (x[sup]2[/sup]-**y[sup]2[/sup]=1).
They have applications in electromagnetic theory, transmission lines in particular. Not the transmission line as in hanging the wires that distribute power (but that too as that’s the catenary curve) but transmission line theory as in dealing with complex (not complex as in it’s complicated but complex as in real and imaginary numbers) impedances. I also understand that they also crop up in heat transfer, fluid dynamics, and relativity.
One area where their particularly useful is relativity.
In 3-D space, the line element of the metric in Cartesian coordinates is: ds²=dx²+dy²+dz² and rotations of the coordinates in, for example, the x-y plane are given by x’=xcosθ-ysinθ and y’=xsinθ+ycosθ where θ is the (trigonometric) angle of rotation.
In special relativity the line element of the metric in Minkowski cooridnates is: ds²=dx²+dy²+dz²-dt². A rotation of the coordinates in the x-y plane remains as above, but a rotation of the coordinates in the x-t plane is given by: t’=tcoshθ-xsinhθ and x’=xcoshθ+tsinhθ where θ is the rapidity or hyperbolic angle. It’s important because this kind of rotation is equivalent to changing an observer’s velocity.
Hyperbolic function have the same complex form as the ordinary trig functions, only with real instead of imaginary arguments, so they can show up when you’re dealing with the wave equation when imaginary parts might get into the sines and cosines.
In the Timoshenko Beam Equation, for instance, which looks like your standard wave equation, but has a fourth power derivative with respect to distance, rather thsan a second order one (although the time derivative is still second order), a complete solution has both regular and hyperbolic sines and cosines, and you can see that the forms the bending beams take on are combinations of not only sines and cosines, but hyperbolic sines and cosines as well, depending upon the boundary conditions.
Of course, there’s more going on than just that cosine and sine satisfy f’‘’’ = f. They more specifically satisfy f’’ = -f; indeed, you can define them by “f’’ = -f if and only if f(x) = f(0) * cos(x) + f’(0) * sin(x)”. Thus, trigonometric functions are used to analyze situations where acceleration is negatively proportional to their value.
As you note, in exactly an analogous way, we can define cosh and sinh by the rule “f’’ = f if and only if f(x) = f(0) * cosh(x) + f’(0) * sinh(x)”. Thus, hyperbolic trigonometric functions are used to analyze situations where acceleration is positively proportional to their value.
[We could also define some “linear trigonometric” functions, let us call them cosl and sinl, by the rule “f’’ = 0 if and only if f(x) = f(0) * cosl(x) + f’(0) * sinl(x)”. Of course, cosl is better known as the constantly one function and sinl is better known as the identity function]
In all of these cases, we’re talking about differential equations of the form f’’ = kf; any such second-order differential equation can always be just as well analyzed by reducing it to the first-order differential equation f’ = q * f, where q is some fresh constant introduced with the property that q^2 = k. So trigonometric functions can also be understood in terms of the complex numbers [where we have some new constant i such that i^2 = -1; in that context, we can define cos and sin as the even and odd components, respectively, of e^(ix)], and hyperbolic trigonometric functions can also be understood in terms of the split-complex numbers [where we have some new constant j (which isn’t 1 or -1) such that j^2 = +1; in that context, we can define cosh and sinh as the even and odd components, respectively, of e^(jx)], while “linear trigonometric” functions can be understood in terms of the dual complex numbers [where we have some new constant d (which isn’t 0) such that d^2 = 0; in that context, we can define cosl and sinl as the even and odd components, respectively, of e^(dx)].
More generally, any second-order homogenous differential equation can be reduced to the study of equations of this form via the quadratic formula; whether the relevant discriminant is negative, positive, or zero will determine whether the relevant trigonometric functions are the ordinary (circular) ones, hyperbolic ones, or “linear” ones.
The last connection, which is the one the OP was most keen on, is via consideration of geometry. If we define the length of a vector <x, y> to be x^2 - ky^2, then movement at unit “speed” (as measured via this notion of length) along a unit “circle” (as determined by this notion of length) will satisfy the differential equation f’’ = kf. Indeed, if we identify the vector <x, y> with the value x + q * y, where q^2 = k, then movement along our unit “circle” at unit “speed” will be given by the differential equation f’ = q * f.
So, movement along an ordinary/Euclidean/Pythagorean circle (given by the equation x^2 + y^2 = 1), with velocity lying on that same circle, satisfies f’’ = -1 * f and is modelled by ordinary trigonometry, movement along a hyperbola (given by the equation x^2 - y^2 = 1) satisfies f’’ = +1 * f,and is modelled by hyperbolic trigonometry, and movement along a line (given by the equation x^2 + 0 = 1) satisfies f’’ = 0 * f and is modelled by “linear trigonometry”.
Instead of thinking directly in terms of unusual norms, you can also think simply in terms of area. One can consider the area swept out between the origin and one’s position as one moves along any of these curves to be a notion of angle; actually, it will be convenient to consider twice this area to be the relevant notion of angle. Then movement at a unit angular speed (that is, movement sweeping out one unit of area per two units of time) along one of these curves (the previously described circle, hyperbola, or line) will have coordinates described by the relevant trigonometric functions (ordinary, hyperbolic, or “linear”).
But, in the end, Chronos’s perspective is the one which I find most often most relevant to the appearance of any of these functions in mathematics: the key thing is second-order homogenous differential equations, which all reduce via the quadratic formula to second-order differential equations of the form f’’ = k * f, where k can furthermore be normalized so that all that matters is its sign and it is taken to be either -1, +1, or 0; the basis of solutions to these equations are then given by, respectively, ordinary trigonometric functions, hyperbolic trigonometric functions, or the constantly 1 and identity functions. These are all often useful because second-order differential equations come up so often (because, well, why wouldn’t they? That’s one of the simpler degrees that could come up…).
(So, I’ve lost the edit window, but the point is: Trying to understand hyperbolic trigonometric functions by visualizing hyperbolas is not actually generally that useful, even though you can do it. Indeed, the only reason we find it useful to understand “ordinary” trigonometric functions by visualizing circles is because we’ve been evolutionary trained to have extraordinarily well-honed in-built ability to reason about the Euclidean norm |<x, y>| = x^2 + y^2 and its properties, as this part of our physical universe’s arbitrary laws is so often relevant in our daily lives. We have much, much lesser in-built intuition for the norm |<x, y>| = x^2 - y^2. It’s not unnatural, therefore, to try to understand this unusual norm via its translation into Euclidean terms (which is what it amounts to to visualize hyperbolas), but, at least for me, this translation can add more muddle than clarity. Better, then, to just understand hyperbolic trigonometry on abstract terms, instead of worrying about shoehorning such understanding into the hardware acceleration we have for analyzing Euclidean geometry.)
I’m sure the OP’s now run away, screaming in horror. He asks a simple question, and everyone piles into a mathematical pissing contest. Indistinguishable’s last post was very useful, but the rest of it… ugh.
Minor corrections and rewording, and a major omission.
First, the minorly corrected and reworded passage:
And now, the major omission: As I said above, “whenever we have a constant c such that c^2 = +1, we can define cosh and c * sinh as the even and odd components, respectively, of e^(cx)”. We can do this using the split complex numbers, but also, we can take c to just be 1 (or -1, but 1 is simpler). cosh and sinh are just the even and odd components of the natural exponential; that is, cosh(x) is (e^x + e^(-x))/2 and sinh(x) is (e^x - e^(-x))/2.
The reason these definitions are so useful are the ones noted above (they give a convenient basis for describing and analyzing functions which are proportional to their second derivative), but seeing them straightforwardly defined in terms of more familiar functions may be helpful, and I should have done that before.
Catenaries are cool. I remember in high school (nearly 40 years ago!) as the teacher showed how with a suspension bridge, the cable falls into a parabola, since the weight supported is proportional to distance along the X axis. In contrast, with a chain or rope, the weight supported is proportional to the distance along the chain or rope. He used this to show the usefulness of using ds rather than dx and dy, and I found it fascinating.
I learned a bit more about the magical catenary arch, seeing stuff by Gaudi in Barcelona. An interesting property of catenaries is that they have only compressile (for arches) or tensile (for chains) forces. So, using them for an arch or building frame means you can use the least amount of material – especially when that material is like cemented stone, which is very strong for compressile stress but not for transverse stresses.
Thanks! That succinctly explains its application in the thread about accelerating to light-speed travel!
