# Why are hyperbolic functions given "short shrift" at "low" levels of math?

Starting with "precalculus," students learn trigonometric functions. After they've spent a semester or more learning how to differentiate and integrate these trigonometric functions in calculus, students are then introduced to "hyperbolic" functions, but mainly "in passing." (At least, that was my experience.)

While there is some mention of hyperbolic functions at somewhat higher levels, such as multivariate calculus and linear algebra, by far the greater emphasis is on trigonometric functions. Hyperbolic functions don't get (more or less) equal time until about complex variables.

Is there any pedagogical reason for this, or is it just the relative frequency of (low level) applications that produces this pattern?

• Hyperbolic functions rarely came up for me in elementary courses (elementary differential equations being the most notable exception), but I found them to be nice source for classroom examples, homework problems, and test problems. In precalculus I used $\sinh x$ and $\cosh x$ to examine the even and odd function properties, the behavior near $x=0,$ the behavior for large negative $x$ and for large positive $x,$ etc. Also, verifying various hyperbolic identities made for good exercises in algebraic work with exponents. Jul 14, 2014 at 19:41
• I found hyperbolic functions useful in similar ways in calculus 1, such as in verifying/discovering differentiation formulas. In calculus 2, during Taylor series stuff when $e^{i \theta} = \cos{\theta} + i\sin{\theta}$ shows up, I got lots of exercises for students by making use of the various formulas relating hyperbolic functions and trig functions when complex numbers are involved. There are also the various hyperbolic trig substitutions you can include. Incidentally, I always made it a point to define all needed hyperbolic functions on any tests where this kind of stuff appeared. Jul 14, 2014 at 19:47
• Going back to precalculus again, I used to show how to solve algebraically things like $\sinh x = 2$ (one solution), $\cosh x = 2$ (two solutions, with opposite signs, but you needed properties of logarithms and rationalizing numerators to verify this from the values themselves), and $\tanh x = 2.$ In all cases you let $u = e^x$ and get a quadratic equation for $u.$ The equation is easiest to solve in the case of $\tanh x$ (because no linear term involving $u$ shows up). The nature of the solutions (one, two, of opposite signs, etc.) can be deduced from earlier work with hand-drawn graphs. Jul 14, 2014 at 19:54
• Frequency-domain analysis and Fourier series. Trig functions are periodic and have nice orthogonality properties. Hyperbolic functions... ? Also simple harmonic motion. For elementary math, it's still more interesting to describe natural occurrences, even if we haven't yet developed the math to show why these are the solutions to second-order ordinary linear differential equations. Jul 14, 2014 at 23:27

There are a few reasons I can think of, not all of them strictly pedagogical.

• The trigonometric ratios (as opposed to functions) have a direct relationship to measurement of triangles, which are an everyday concept, and so the introduction of these functions via triangles is a natural thing to do during high school. Hyperbolic trig functions do not relate so directly to everyday geometry and so are not as easy to introduce so early.

• The fundamental definition of the trig functions involves the geometry of a circle, which is familiar to students, whereas the hyperbolic trig functions relate to the geometry of a hyperbola, which for many students is not familiar at all!

• The graphs of the trig functions have repeating shapes with nice maxima/minima that allow for intense investigation of shifts and dilations. The graphs of the hyperbolic trig functions have no major distinguishing features other than their even/odd-ness and one important point (minimum for cosh, inflexion point for sinh and tanh).

• The definition of the hyperbolic trig functions using the exponential function is not at all obviously related to the geometry of the hyperbola, and many teachers/lecturers do not even know the connection. I myself only learned this year where the "angle" is in the hyperbola geometry. I still don't know why hanging ropes make cosh curves.

• (Inverse) Hyperbolic trig functions seem to be mainly used to provide integrals of functions that inverse trig functions can also be used for, so I think many people choose to ignore them because they know it can be done with ordinary trig functions.

• A couple of the major connections between hyperbolic and ordinary trig functions involve concepts that the students are just not familiar with until they have done more maths -- Taylor series and complex numbers, for example.

So that's several reasons why they are not introduced earlier.

However, I do wish more time was spent on them at the point they are introduced, though, regardless of when that introduction happens. In my experience, students find them quite difficult to assimilate when they are introduced with a wave of the hand and then promptly forgotten about. A little discussion of the connections between them and the trig functions, and a proper investigation of their trig identities and how much easier they make some integrals would make them a bit easier to swallow!

PS: Here is a paper about the history of them: http://www.jstor.org/discover/10.2307/3219227

• Yes, hyperbolic functions start to make sense once you get to computing the trajectories of "hanging ropes" in the calculus of variations. But most math students don't get that far (I've barely scratched the surface myself). Jul 14, 2014 at 19:32
• This might sound a bit silly, but for the purposes of using hyperbolic functions as a source for examples for things like even/odd functions, solving equations quadratic in $e^{x},$ and the other things I mentioned in my comments under Tom Au's question, I found it was usually enough to begin by mentioning the hyperbolic trig. function keys on their calculators, and saying here's what they stand for and now let's see what we can discover about them using some of the algebraic and graphical techniques we have available. The fact that special calculator keys (continued) Jul 15, 2014 at 14:40
• (continuation) are designated for them pretty much "proves" their importance in mathematics, although you will definitely want to have 3 or 4 uses written down in your notes because there will always be someone who asks for specifics. Jul 15, 2014 at 14:42

I am of the personal opinion that until complex numbers and manipulation of infinite series is introduced that it is pointless to introduce hyperbolic trig functions as I had learned them in that order and fared just fine.

That being said the rest of the math worlds seems to disagree insisting on learning hyperbolic-trig indentities and integrals before showing most students what these things actually are.

The reason they aren't seen too much though and given the 'fair treatment' is because it's not pedagogically useful to focus too much on them though since what ends up happening is that students need to be taught a plethora of trigonometric-style tricks without much (if any) intuition or proof when they are first introduced.

By studying Trig first, followed by taylor series, and complex exponentials one much more naturally arrives at the theory of hyperbolic-trig and has an algebraic means of understanding exactly why they behave the way they do.

For people that insist on geometric intuition: the diagram below:

http://en.wikipedia.org/wiki/Hyperbolic_function#mediaviewer/File:Hyperbolic_functions-2.svg

With the use of some heavy algebra (and knowledge of tools such as arc-length formulas) can help them derive all the trigonometric-style tricks

I would think this is a useful method too to learn them but have never seen this done, EVER, in a class.

Personally, I think the best order of learning is: differential and integral calculus for simple algebraic functions followed by taylor expansions, then followed by an introduction to trigonometry whereas as soon as some of the basics have been learned, students are more than well equipped to derive taylor expansions for the trigonometric functions, determine their complex exponential formulas, etc, etc...

Lastly hyperbolic trig, if anything should be something students derive on their own attempting to solve their associated differential equations.

Lot's of great discussion in the other answers, and the comments by Dave Renfro were on point as usual. But, I'll add my two cents here:

I think the reason that hyperbolic functions are largely ignored is multifaceted and likely to worsen as more and more educators buy into tempting arguments to downplay topics like trigonometric substitution and partial fractions because those calculations are not interesting and can be easily done with technology. Also, most calculators don't have hyperbolic trig functions, so why would you use them as an example when using the calculator is baked into the teaching of many.

My own experience with calculus was that hyperbolic functions were not emphasized and perhaps not even mentioned. Now, probably if I went back and looked I could find hyperbolic sine and cosine somewhere in the textbooks I used. But, it didn't really get my notice until in the midst of my course on Special Relativity. The professor used cosh and sinh and tanh as follows: $$\tanh \phi = \beta \ \ \ \ \& \ \ \ \ \cosh \phi = \gamma \ \ \ \ \& \ \ \ \ \sinh \phi = \gamma \beta$$ where $$\beta = v/c$$ and $$c$$ is the speed of light whereas $$v$$ is the speed of the inertial frame. Here $$\gamma = \frac{1}{\sqrt{1-\beta^2}}$$ so the assignment of rapidity is reasonable as $$\cosh^2 \phi - \sinh^2 \phi = \gamma^2-\gamma^2\beta^2 = \gamma^2(1-\beta^2)=1.$$ The quantity $$\phi$$ is called the rapidity. As a student, the rapidity was fascinating, new, and followed the simple rule rapidities add. In contrast, velocities do not add in Special Relativity. Much to the horror of our professor, we explained didn't know what "cosh" and "sinh" were. That was my introduction. Later, as I taught Calculus and Differential Equations, and Complex Analysis I learned that not only were cosh and sinh essential topics, they are necessary to take a complete view of the exponential function.

Let me give an example from the problem of inverse Laplace transforms. $$\mathcal{L}^{-1} \left( \frac{s}{s^2+4s+13}\right) = \mathcal{L}^{-1} \left( \frac{s+2-2}{(s+2)^2+9}\right) = e^{-2t}\cos (3t) -\frac{2}{3}e^{-2t}\sin (3t)$$ Typically, when the denominator factors over $$\mathbb{R}$$ then partial fractions is used in the standard course. But, there is no need for a problem which parallels the one above. In fact, $$\mathcal{L}^{-1} \left( \frac{s}{s^2+4s-5}\right) = \mathcal{L}^{-1} \left( \frac{s+2-2}{(s+2)^2-9}\right) = e^{-2t}\cosh (3t) -\frac{2}{3}e^{-2t}\sinh (3t)$$ So, with both cos and cosh as well as sin and sinh, the technique of completing the square leads to a completely symmetric treatment for the non-repeated root case.

It turns out there are generalizations of both trigonometric and hyperbolic trigonometric functions which appear in connection with the exponential over an algebra which is characteristic of a given $$n$$-th order ODE. The component functions of this exponential form a fundamental solution set. Cosine and sine and cosh and sinh are just the lowest order examples of this pattern.

There is much that can be said about Calculus II as well, we see trigonometric substitution used as a method to eliminate radicals. Hyperbolic functions can be leveraged to make the same reduction. There are problems which are simple with the hyperbolic technique which are quite complicated in the trig. technique. Likewise, there are other problems which are easy in trig. substitution which are dastardly in the hyperbolic approach. Both methods should be used as the reinforce one another.

There is a whole theory of hyperbolic trigonometry which mirrors the commonly known theory of circular trigonometry. I think Saul Stahl's text A Gateway to Modern Geometry is a good place to look if you want to learn more. I suspect working through hyperbolic trigonometry would give a deeper understanding of the usual circle-based trig.

I have a background in chemistry, through Ph.D. Some dabbling in physics and engineering as well (classes and work). Not a complete jock, but have a sense of the general STEM needs. I almost never encountered hyperbolic trig functions, other than the treatment I saw in AP Calculus BC, c. 1982 (Thomas Finney). They probably showed up somewhere. But I remember them less than harder topics like Bessel functions or series solutions of ODEs or Laplace transforms.

Given the above, I think the limited treatment was fine. You could even make an argument to cut it more. Pedagogically, I felt that the placement in TF calculus was fine (after trig, after exponentials, before series, before ODEs). But if anything it was just a chance to do more manipulations. Not something that I thought would connect to the real world the way exponentials and trig do. I wondered maybe I'd see them more later in my STEM career. But...no.