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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".

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    $\begingroup$ 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. $\endgroup$
    – Gerald Edgar
    Dec 11, 2018 at 13:33
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    $\begingroup$ 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. $\endgroup$
    – Andreas Blass
    Dec 11, 2018 at 15:31
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    $\begingroup$ 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. $\endgroup$
    – Rusty Core
    Dec 11, 2018 at 16:47
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    $\begingroup$ 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. $\endgroup$
    – Dave L Renfro
    Dec 11, 2018 at 19:45
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    $\begingroup$ 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. $\endgroup$
    – smci
    Dec 11, 2018 at 22:32

4 Answers 4

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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.

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    $\begingroup$ 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. $\endgroup$
    – Steven Gubkin
    Dec 12, 2018 at 12:47
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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...

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    $\begingroup$ @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. $\endgroup$
    – Steven Gubkin
    Dec 11, 2018 at 13:45
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    $\begingroup$ 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. $\endgroup$
    – Steven Gubkin
    Dec 11, 2018 at 13:53
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    $\begingroup$ 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. $\endgroup$
    – Steven Gubkin
    Dec 11, 2018 at 13:55
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    $\begingroup$ 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. $\endgroup$
    – Steven Gubkin
    Dec 11, 2018 at 13:56
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    $\begingroup$ In my heart, cosh and tanh are equally welcome. $\endgroup$
    – James S. Cook
    Dec 11, 2018 at 16:05
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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.
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    $\begingroup$ I'm surprised no one posted this obligatory link: theonion.com/… "Nation’s Math Teachers Introduce 27 New Trig Functions" $\endgroup$
    – robphy
    Dec 12, 2018 at 0:13
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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...

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    $\begingroup$ " Not even the co-functions"... How about cosine then? $\endgroup$
    – Zuriel
    Dec 11, 2018 at 20:44
  • $\begingroup$ 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. $\endgroup$
    – guest
    Dec 11, 2018 at 20:50
  • $\begingroup$ 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. $\endgroup$
    – alephzero
    Dec 12, 2018 at 10:43
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    $\begingroup$ 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 :-) $\endgroup$
    – Jyrki Lahtonen
    Dec 15, 2018 at 8:45
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    $\begingroup$ celestial navigation is recovering because of anti-gps warfare ... naval-academy-brings-back-celestial-navigation $\endgroup$
    – kjetil b halvorsen
    Dec 16, 2018 at 14:22

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