19

It seems that the following functions are not only excluded from a course in trigonometry, they are almost never taught in any course:

  1. Chord
  2. Versine
  3. Coversine
  4. Haversine
  5. Exsecant
  6. Excosecant

I could have asked the same question with the title "why have these functions lost their popularity" at math.stackexchange, but I fear that they will consider this question as "opinion-based".

Zuriel
  • 4,275
  • 20
  • 48
  • 11
    These things were useful when you had to look everything up in tables. But nowadays, with hand-held scientific calculators, we do not need them. – Gerald Edgar Dec 11 '18 at 13:33
  • @GeraldEdgar Even with the calculators, I bet almost all students don't even know what a versine is when they see it. – Zuriel Dec 11 '18 at 13:35
  • 5
    I don't teach these functions because I don't know what they are. I've heard of only half of them, but even for that half I'd need to look up their definitions. Students can look them up as well as I can. – Andreas Blass Dec 11 '18 at 15:31
  • 3
    On the one hand, you could get the answer from Wikipedia articles that you linked. On another hand, I learned something new, so thanks. Not sure why you included chord, which is not a trigonometric function, and is taught at school. As for "popular" functions, sine is useful when studying waves, cosine is useful to find a projection of a force, and tangent is the slope of tangent line, that is, a derivative. I guess finding a trajectory of a ballistic missile is more important nowadays than finding position of a brigantine. – Rusty Core Dec 11 '18 at 16:47
  • 1
    @RustyCore Chord is being used as a trig function here: namely the length of the chord on the unit circle which connects two points separated by an arc of $\theta$. It is $2\sin(\frac{\theta}{2})$ apparently. – Steven Gubkin Dec 11 '18 at 17:35
  • 1
    @PeterTaylor Thanks, fixed it – Chris Cunningham Dec 11 '18 at 17:53
  • @StevenGubkin Indeed, thanks. But frankly, if I were to find the length of the chord, I would not use the formula shown in Wikipedia https://en.wikipedia.org/wiki/Chord_(geometry)#crd Instead, I would drop perpendicular from O to the chord and calculate its length as twice the length of the opposite leg of the two right triangles, this seems much simpler to me. – Rusty Core Dec 11 '18 at 18:46
  • 6
    Even with the calculators, I bet almost all students don't even know what a versine is when they see it. --- I don't understand the reasoning behind of this statement. For example, even with calculators, I bet almost all students don't know what a vulgar fraction is when they see it. In fact, even with calculators, I bet almost all students don't know what a syntopogenic space is when they see it. My point is that I don't understand why one would expect students who have a calculator to be more likely to know what a versine is. – Dave L Renfro Dec 11 '18 at 19:45
  • I thought the discussion was about its definition and its use, not obtaining decimal approximations for its values, which is not something that needs a calculator key for or additional tables for (before calculators), since it's one minus the cosine. Regarding tables, it's the logarithms of the trig values you use when doing extensive calculations and the logarithm of $1-\cos\theta$ isn't immediately obtainable from the other trig functions or their logarithms. However, that's when the identity $1-\cos\theta=2{\sin}^2 \frac{\theta}{2}$ is useful. (Comment I'm replying to has been deleted.) – Dave L Renfro Dec 11 '18 at 20:33
  • 6
    Just because they were listed in lookup tables for Egyptian builders/astronomers several millennia ago, doesn't mean they're especially useful. They can be easily synthesized from a minimal set of more basic functions (or vice versa). They don't have massively interesting properties. – smci Dec 11 '18 at 22:32
  • 3
    For example, suppose you are doing $5$-figure calculations (something that was feasible with tables). Now $1-\cos\theta$, when $\theta$ is near $0$, obtained by subtraction, may not be very accurate; but $\log \text{versin};\theta$ could still be in your table, and very useful for $5$ figures. I have such things in a CRC book of tables I got in around 1965. – Gerald Edgar Dec 11 '18 at 22:46
  • 1
    @GeraldEdgar If the reason they aren't taught any more is that they were only useful when using table lookups instead of calculators, that still leaves the question, "Why do we not teach versine and haversine anymore but do still teach secant and cotangent"? – Ray Dec 12 '18 at 01:48
  • As 3Blue1Brown says, these functions are just " mathematical tattoos" – ruferd Dec 12 '18 at 18:33
  • @Ray: Regarding secant and cotangent, I think their teaching has to do with "completing the symmetry" for the trig. functions, in the sense that there are three analogous Pythagorean trig. identities, each trig. function has a co-function, each trig. function has a multiplicative inverse trig. function, and similar things. In my opinion, what should be eliminated is dealing with all 6 trig. function inverses and dealing with all 6 trig. function graphs. Also, secant and cotangent can be useful in basic closed form integration and students will see them sometimes mentioned in other places. – Dave L Renfro Dec 12 '18 at 18:47
  • @Gerald Edgar: I've got the 1972 20th edition of CRC (obtained in 1973) and the only place I can find versine is on p. 225, along with exsecant, coversine, and haversine. But the logarithms of versine do not seem to be given in any of the tables. Indeed, as far as I can tell, none of the other-than-the-usual-six trig. functions are mentioned anywhere else except for 4 identities involving sine and haversine for a triangle at the bottom of p. 240. It appears these other trig. functions were even less significant back then than I would have otherwise guessed. – Dave L Renfro Dec 12 '18 at 19:02
  • Incidentally, for those interested, here's how calculations were done using logarithms in my high school's 4th year math class in 1973-74. I took the class several years before anyone else in my graduating class, so by the time my peers were taking the class in 1976-77, calculators were on the scene and these logarithm calculations were being phased out, although still taught at that later time. I suspect they were not taught in this class after around 1978 or 1979. – Dave L Renfro Dec 12 '18 at 19:13
  • @DaveLRenfro Apparently our mileages vary. I have never understood the appeal of having separate names for $1/\sin x$ and $1/\cos x$. We don't have separate notation for $1/x$ either. Why would trig be any different? For me there are only three trig functions. I would prefer Mathematica to also use only three. – Jyrki Lahtonen Dec 15 '18 at 08:53
  • 1
    @Jyrki Lahtonen (and others): Of possible interest is Jenna Van Sickle's 2011 Ph.D. dissertation A History of Trigonometry Education in the United States: 1776-1900 (freely available, at least where I'm at). – Dave L Renfro Dec 18 '18 at 18:13
  • Thanks @DaveLRenfro – Jyrki Lahtonen Dec 18 '18 at 19:05

4 Answers4

22

More of a comment than an answer but: They are all composites of more basic functions. In fact, all of the trig functions could can expressed in terms of sine, linear changes in coordinates, and rational functions. For instance:

$$ \tan(\theta) = \frac{\sin(\theta)}{\sin( \frac{\pi}{2}-\theta)} $$

We certainly don't want children to have to memorize 20 different trig functions. That seems a bit silly. Do we even really need to teach all six ``standard'' trig function? Personally I tend to avoid $\csc$ and $\cot$ in my work...

Steven Gubkin
  • 25,127
  • 4
  • 61
  • 110
  • Then by the same reasoning, why do we teach $\tan$ and $\cos$ at all? – Zuriel Dec 11 '18 at 13:33
  • 8
    @Zuriel Indeed, that is the point. There is no mathematical need to teach them. The reason to teach them is because other people know them, and you don't want your students to be confused when they go to their physics classroom and their physics prof uses $\cos$. However, the physics prof is probably not going to use the chord function so we can safely ignore it. Perhaps in a hundred years we will have abandoned $\csc$. Good riddance I say. $\cos$ and $\tan$ have a special place in my heart however. – Steven Gubkin Dec 11 '18 at 13:45
  • Thanks! I understand that there is a need of $\cos$ in physics and other fields. But even for students majoring in pure mathematics, we teach them $\cos$ but perhaps never $\versine$. – Zuriel Dec 11 '18 at 13:48
  • 8
    There isn't a need for cosine in these other fields. I am just saying that the reason we use sine, cosine and tangent, but do not use versine, is the same reason that we use use glasses and don't use monocles: simply that monocles have gone out of fashion, and glasses have not. It is entirely for social reasons that we use some functions and not others. Your education must prepare you to have easy conversation with other people, so we must perpetuate some arbitrary choices to provide ease of communication. This is only one such choice. – Steven Gubkin Dec 11 '18 at 13:53
  • 1
    Other choices include the order of function composition, why we do not have a particular name for the antiderivative of sin(1+x^2), the order of operations, that $\sqrt(x)$ denotes the positive number whose square is $x$, etc. – Steven Gubkin Dec 11 '18 at 13:55
  • Thank you very much for your answer and comments! It is much clearer to me now. – Zuriel Dec 11 '18 at 13:55
  • 7
    Note that teaching students haversine, versine, etc will actually make it HARDER for them to communicate with others, because most other people have not learned these things. So they would have to find their common ground, and then work out the translations. – Steven Gubkin Dec 11 '18 at 13:56
  • 5
    In my heart, cosh and tanh are equally welcome. – James S. Cook Dec 11 '18 at 16:05
  • @JamesS.Cook Oh definitely. I do complex analysis, so the hyperbolic trig functions are my friends. – Steven Gubkin Dec 11 '18 at 17:32
  • In special relativity, where the "angle" (Minkowski arc-length along a hyperbola) is called the rapidity, with (v/c)=tanh(theta), then cosh(theta)=1/sqrt(1-(v/c)^2) is the time-dilation ("gamma") factor. Calculations in relativity are so much more intuitive and efficient with hyperbolic trig functions. – robphy Dec 11 '18 at 21:24
13

As several respondents have indicated, one could choose many different trigonometric functions to serve as the basic elements in terms of which other trigonometric functions are expressed and in (ancient) history other choices were made than those standard now. What we know now that was not known to the ancients that leads us to use cosine and sine is the central place in mathematics of linear ordinary differential equations and exponential functions.

The functions $\cos{t}$ and $\sin{t}$ are a basis for the space of solutions of $\ddot{x} + x = 0$, which is probably the most important differential equation in mathematics, because it is probably the most fundamental differential equation in physics. An alternative basis is $e^{i t}$ and $e^{-i t}$, so, said another way, the choice of $\cos$ and $\sin$ is made because these are the real and imaginary parts of the complex exponential $e^{i t} = \cos{t} + i \sin {t}$. The exponential is clearly fundamental, and it is convenient to express as much as one can in terms of it.

The alternative trigonometric functions do not have such properties and this means that to make extensive use of them it turns out to be more convenient to express them in terms of cosine and sine, or, equivalently, exponentials.

Also, this point of view makes apparent the parallelism with real exponentials, or rather the hyperbolic cosine and sine, corresponding to the differential equation $\ddot{x} - x = 0$.

Moreover $\cos$ and $\sin$ admit simply geometric interpretations and sets of functions such as $\{\cos{nt}, \sin{nt}: n \in \mathbb{Z}, n > 0\}$ exhibit useful orthogonality properties (with respect to integration) that make them well adapted for use in contexts like Fourier series. There are of course many systems of orthogonal functions. What the most useful ones have in common is an origin as the solution of a homogeneous second order linear ODEs with some parameter (in this case $n$) in its coefficients.

What one teaches to children is in part dictated by what is needed later by those who will study more, and in part dictated by what has proved most tractable in diverse contexts.

Dan Fox
  • 5,839
  • 18
  • 32
  • 2
    This is a very insightful answer. "Chord" would satisfy similar differential equations, but you would have unsightly powers of 2 everywhere in your Fourier expansions if you used it. – Steven Gubkin Dec 12 '18 at 12:47
6

Ask yourself if you would miss anything useful if you didn't know the functions you mentioned. I doubt it! I'm teaching high school math happily without having heard of them up to this point.

To the contrary, there are several downsides to teaching them:

  1. Quoting Steven Gubkin's comment: "Note that teaching students haversine, versine, etc will actually make it harder for them to communicate with others, because most other people have not learned these things."
  2. Cognitive overload. Obligatory xkcd.
  3. Personal opinion: Math should be more about (creative) problem solving than about applying memorized formulas. By not teaching above functions, students can solve more problems by creative usage of $\sin$, $\cos$, $\tan$ and Pythagorean Theorem.
  4. Calculators. If I express a solution using one of the fancy functions, I'm stuck with a calculator and can't come up with an approximate solution.
Jasper
  • 2,699
  • 16
  • 28
  • 3
    I'm surprised no one posted this obligatory link: https://www.theonion.com/nation-s-math-teachers-introduce-27-new-trig-functions-1819575558 "Nation’s Math Teachers Introduce 27 New Trig Functions" – robphy Dec 12 '18 at 00:13
4

They are almost never needed for applications problems in physics or engineering. Really sine, cosine and tangent are mostly what you need. Not even the reciprocal functions.

I'm talking with respect to how formulas and problems are normally written in those courses. Ask yourself, did you encounter those functions in any of your science course and see a need for math to cover them as a service?

I think the versine and such are useful in navigation. But even for celestial nav as of post ww2, it was mostly done with tables and worksheets that don't require you to use these functions (or really know any of the math in what you do). I believe there was a time when there was more need for them before celestial nav became so work sheet oriented. And now celestial nav itself is a dying art because GPS is so common. Ask the average QM to use a sextant and see how he does...

guest
  • 234
  • 1
  • 3
  • 1
    " Not even the co-functions"... How about cosine then? – Zuriel Dec 11 '18 at 20:44
  • keep cosine, but ditch secant. I meant the reciprocal functions not needed much. sec, csc, and cot. Sure make the monsters learn it in trig class. Good for them to learn t transform things. But outside of the big 3, they really won't see much in rest of their courses. – guest Dec 11 '18 at 20:50
  • It would be a shame to ditch secant. $1 + \tan^2 x = \sec^2 x$ and $d/dx(\tan x) = \sec^2 x$ are just as useful as $\sin^2 x + \cos^2 x = 1$, unless you plan to ditch tangent as well. But I agree that cot and csc are endangered species in the wild, though they thrive in the artificial environment of trig and calculus problem sets. – alephzero Dec 12 '18 at 10:43
  • I'm not asking to ditch them in trig class. Make the kids do all 6. That's good enough. I'm just being realistic that I don't see much of secant in engineering or physics. Whereas sin and cos are incredible common. (E.g. Fourier series). Tan less so, but still common. So...sure make em do all 6 in trig class. and a bit of all 6 in calc class. But...well...really sin/cos are the monsters. – guest Dec 12 '18 at 18:23
  • Did you mean "They are rarely needed"? – Jasper Dec 12 '18 at 20:58
  • Yeah, my bad. Will fix. – guest Dec 13 '18 at 00:30
  • 1
    I agree 100%. I never teach secant or cosecant either. Cotangent is so-so. Likely this is affected by the fact that back in the day my calculator only had keys for sine, cosine and tangent :-) – Jyrki Lahtonen Dec 15 '18 at 08:45
  • 1
    celestial navigation is recovering because of anti-gps warfare ... naval-academy-brings-back-celestial-navigation – kjetil b halvorsen Dec 16 '18 at 14:22