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March 18, 2016 10:26 AM

10 Secret Trig Functions Your Math Teachers Never Taught You
posted by the man of twists and turns (47 comments total)

How many trig functions are there?
Mnemonics in trigonometry
Need to brush up?
posted by the man of twists and turns at 10:28 AM on March 18, 2016

The case against algebra II
Fuck trigonometry
posted by the man of twists and turns at 10:39 AM on March 18, 2016

In some ways, it feels more like symbolic substitution than anything else (ie trig functions): that 1-cos(theta) is also called "versin(theta)". This lets you do some cleanup and what not in proofs, where with a few handwaves you've converted something ugly into something concise and pretty. (eg see sums that reduce to sin or cos with some simple arithmetic ).
posted by k5.user at 10:41 AM on March 18, 2016

aka "Ten MORE Trig Functions I Failed To Ever Learn".
posted by Greg_Ace at 10:41 AM on March 18, 2016 [4 favorites]

This is fantastic and why hadn't I heard of the author, Evelyn Lamb, before.

(And I know that my HP-41 had an e^x - 1 function for the same accuracy near zero problem the article describes for the trig functions. Oh, look, Wikipedia knows all about it.)
posted by benito.strauss at 10:42 AM on March 18, 2016 [2 favorites]

10 Secret C Trig Functions That Take Floats Instead of Doubles Your Teacher Never Taught You
posted by RobotVoodooPower at 10:43 AM on March 18, 2016 [11 favorites]

Clickbait for math geeks is a thing now, I guess.

I clicked.
posted by Abehammerb Lincoln at 10:46 AM on March 18, 2016 [7 favorites]

In college, we had a class called Numerical Methods that explained how to do computer calculations without losing precision or letting errors build up too much. It was eye-opening just how easily that following an ordinary algebraic equation could end up steering you into the path of serious computational errors quite quickly.
posted by CheeseDigestsAll at 10:48 AM on March 18, 2016 [9 favorites]

k5.user: it's simple substitution that lets you use lookup tables in pre-calculator mathematics.

haversines and secants, particularly, are important for navigation with charts and compasses when you don't want to do all those messy sqrt() and 1/x operations.
posted by Xyanthilous P. Harrierstick at 10:49 AM on March 18, 2016 [1 favorite]

Haversine, will travel.
posted by GuyZero at 10:50 AM on March 18, 2016 [1 favorite]

It's missing the chord function.
posted by leahwrenn at 10:52 AM on March 18, 2016 [4 favorites]

GOOGLE PROSTHAPHAERESIS
posted by wittgenstein at 10:53 AM on March 18, 2016 [3 favorites]

SOH-CAH-TOA 4 lyf
posted by Freelance Demiurge at 10:54 AM on March 18, 2016 [10 favorites]

By the way, there are the Hyperbolic trig functions,

sinh, cosh, tanh, sech, csch, and coth.

where all the identities look very similar, give or take a few minus signs.


If that's not weird enough for you, you can go to the land of the Elliptical trig functions,

sn, cn, dn, ns, nc, nd, sc, sd, dc, ds, cs, and cd

Things get hairy there.


If regular trig is the world where up is up, hyperbolic trig is a world where up is down, and in elliptical trig world up is either backwards, forwards, left, or right, depending on the elliptic modulus.
posted by benito.strauss at 10:57 AM on March 18, 2016 [7 favorites]

It's missing the chord function.

I heard there was a secret chord.
posted by jeather at 10:57 AM on March 18, 2016 [13 favorites]

Um... oh shit. These were absolutely taught to me (oh shit) 25 years ago when I took Geometry and Trigonometry. Have they been removed from the curricula? Is this what happened over the past two and a half decades? I mean, yeah... I was one of those guys from 'The Case Against Algebra II' article that sailed through those honors classes in math... but still. Everyone who wanted to go to college was required to learn Versine and Exsecant. I mean, the first thing we did was immediately transform them into their respective sine/cosine functions and eliminate them from the equation, but we sure as shit learned them enough to do so. I mean, if you handed me a 'reduce this trigonometric statement' problem, I'd reduce it to a sine / cosine statement, and I'd fail to recognize that I could convert part of it to some of the other properties. As stated, that's because of calculators and because we aren't table driven these days - and like Roman Numerals they're sort of antiquated, but hands down when you start getting back into number theory and dimensional analysis (which no one in their right mind likely takes as an unnecessary elective) there are some practical applications in using these again... but... ...well, if you just majored in finance or international business or hair design, or even a fair amount of engineering - its pretty unlikely that you are operating in the land of theoretical number spaces and are actively trying to find the fastest way between two points in a non standard number system, nor do you need to describe their periodicity... I mean... well... I get it. You don't need to know it... you don't want to know it... why should you know it... ... Liberal Arts.

And that's not to dis liberal arts - quite on the contrary. I think that the effort required to wrap your mind around dead trigonometric concepts is as relevant to liberal arts majors who want to some day be able to understand and speak to science nerds just as making sure that no one gets through the education system without having a fundamental appreciation for anything from Faulkner is probably a good idea. Faulkner never helped anyone build a bridge, but I implore you - make sure your tech guys are socialized and well read enough to function in society, they have a definitely lower likelihood of developing a robot army that goes stabby stabby to all of us that way.
posted by Nanukthedog at 11:02 AM on March 18, 2016 [2 favorites]

It's missing the chord function.

Wow, that is a much simpler formula than I was expecting. Cool.
posted by paper chromatographologist at 11:11 AM on March 18, 2016

These were absolutely taught to me (oh shit) 25 years ago when I took Geometry and Trigonometry. Have they been removed from the curricula?

Apparently, so. I took high-school geometry (at a prestigious prep school, mind) circa 1998 or so, and I don't know what the frig a versine is. Exsecants may have come up in one of my engineering courses, maybe, but not in high school.
posted by tobascodagama at 11:15 AM on March 18, 2016

w.r.t. "The case against Algebra II"; I'm absolutely on board with the idea of a re-vamp of high school mathematics.

The way it's taught at the moment appears to be leftovers of "Agrarian Mathematics" (how many bolts of cloth can i get for these bushels of wheat?, how much land do I actually own?, maybe some land and sea navigation) that got "fixed" in the 1950's and 1960's [1] in support of the Cold War idea that Everyone Must Be An Engineer! To Calculus With Thee!

The typical progression is still Algebra, Geometry, Algebra II, Trig/Pre-Calc. They justify keeping a horribly mutilated Geometry course as an excuse for "teaching logic and proof".

A better tactic would probably be Algebra, Algebra II, Statistics, and finish with some kind of simplified Discrete Math (for your logic and proof needs as well as 'Computer Science Math'). Then, we'd at least have another couple of generations of kids that understand the math needs of modern life.

(Probably a derail, but there's only so many trig jokes a thread can support.)
posted by Xyanthilous P. Harrierstick at 11:17 AM on March 18, 2016 [7 favorites]

Um... oh shit. These were absolutely taught to me (oh shit) 25 years ago when I took Geometry and Trigonometry. Have they been removed from the curricula? Is this what happened over the past two and a half decades?

No; they were mostly removed from the curricula roughly 40-50 years ago (yes, even in honors courses; that was my crowd as well (would have been taking those classes 30+ years ago)) In many years of competitive math, I never met anyone who had covered haversine, etc. in a class--we all just knew vaguely about them from other things (for example, if I remember correctly there are navigation scenes in Heinlein's 'The Number of the Beast' that mention some of these functions).

I posit your experience is *very* unusual.
posted by BlueDuke at 11:19 AM on March 18, 2016 [9 favorites]

These were absolutely taught to me (oh shit) 25 years ago when I took Geometry and Trigonometry. Have they been removed from the curricula?

You and I must be the same age, I took Trig in 1991 in High School, and they weren't in the curriculum or on any tests. These functions definitely weren't taught to us, but there was a very brief mention of them in the book, if I recall correctly, as useful in certain narrow circumstances which the article describes. Never used them, though, and only knew of them as curiosities a bit like snap, crackle, and pop.
posted by chimaera at 11:20 AM on March 18, 2016 [1 favorite]

I took Geometry (which included trigonometry in one giant mash-up) in 1993, and did not learn these functions - though I did learn about the hyperbolic trig functions.

Xyanthilous P. Harrierstick, I am a big math nerd and *loved* calculus, but I would love to see the typical math progression in high school incorporate statistics instead of jumping to calculus. (Even better, I would love to see some cross-over between science classes and statistics, since in my experience the best way to understand continuous statistics is to analyze your own lab data)
posted by antimony at 11:28 AM on March 18, 2016 [3 favorites]

Nanuk, where did you go to school? I took math a few years before you did, would track down this kind of stuff on my own for fun (lots of Martin Gardner), and I never heard of these functions until reading math histories many years after high school. I was going to guess that you might have grown up in Europe, possibly in one of the less accessible Alpine valleys, but your user page doesn't suggest that. Like BlueDuke says, very unusual.
posted by benito.strauss at 11:29 AM on March 18, 2016 [1 favorite]

Hey, this is neat! I love that these functions are useful if you are calculating things by hand, with tables. Like CheseDigestsAll, I also took a numerical methods class. And the things real computers doing floating point do to math are just horrifying. Floating point numbers aren't a field, so absolutely everything you know about real number algebra does not apply to floating point. There's a lot of subtle work required to do numeric calculations correctly. These extra trig functions are an easily accessible version of the same phenomenon.

I've actually implemented the haversine formula myself in Javascript, just cargo-culting in what I saw in a web page. I had no idea why it was the right thing and now I do. Yay!
posted by Nelson at 11:31 AM on March 18, 2016 [1 favorite]

Computers can't even represent all the integers.
posted by thelonius at 11:38 AM on March 18, 2016

antimony: I'm a huge math nerd, and I *hated* Calculus -- Calculus as taught in the high schools and colleges of the United States -- namely "Calculus and Analytic Geometry". It's that latter part that is unnecessary for most people. The core idea of Calculus could be a half-semester elective course with no loss.

I just wish we had a population that was minimally versatile in basic statistics, so that they can understand articles in the news that mention things like polls, pharma effectiveness, the spread of diseases, climate change, etc.

I'm not asking that the general population build rockets to mars or cure cancer; just that they can follow along and nod appropriately when someone else does.
posted by Xyanthilous P. Harrierstick at 11:39 AM on March 18, 2016 [2 favorites]

A better tactic would probably be Algebra, Algebra II, Statistics, and finish with some kind of simplified Discrete Math (for your logic and proof needs as well as 'Computer Science Math'). Then, we'd at least have another couple of generations of kids that understand the math needs of modern life.

That's "replacing everyone should be an engineer!" with "everyone should be a programmer!" though. I'm not sure that's a whole lot better. Trig is still useful for a lot of people as long as the building trades continue to exist. Houses still need to be built somehow, and lights installed.

OTOH, more stats are more good, and I fully agree that at least simple forms should be taught as part of the basic high-school package. Statistics illiteracy is significantly damaging to real people's lives. Anti-vaxxers are at least in part, stats-impaired, for example.
posted by bonehead at 11:42 AM on March 18, 2016 [4 favorites]

That's "replacing everyone should be an engineer!" with "everyone should be a programmer!" though.

Sadly, yes, to some degree. I mostly throw it in there for the logic and proof components of math pedagogy, and less about the combinatorics and set theory bits; and I'm doing so because Discrete Math has not (yet?) been ruined by Education.

Have you seen what passes for "Proofs" in a high school level Geometry course? It's "I apply rule 1, rule 8, rule 4b." In fact, you're not even allowed to write your proofs in long form english; it's literally a table of applied "proof functions" in the proper order.
See here for example.

It's really not the Geometry vs. Discrete Math part that makes me Sad/Angry; it's the destruction of the long form argument of proof. That takes thought and creativity, and I really don't care what subject is used underneath for examples to work with. (This will never fly, naturally, because you can't apply standardized grading rubric to long form proofs.)

That I also happen to consider computer sciences more valuable today than agrarian geometry (and to a lesser degree, engineering) is definitely a bias and conceit I'll admit to.
posted by Xyanthilous P. Harrierstick at 11:57 AM on March 18, 2016 [2 favorites]

Hmm. When I was in school, what I noticed was that Algebra II was almost exactly the same as Algebra I, so I stopped paying attention in class or ever doing any homework (because math is far too boring to do it twice). When I aced all the tests, they decided I was some kind of math genius, and moved me ahead a year, so I missed trig/precalculus. It put me a distinct disadvantage in later on, because people just expect you to know stuff like that. I won't say that's why I didn't end up doing math (I made it through several years of college calculus, linear algebra, and logic) but it would have been useful to know.

Now I'm looking at the diagram in the article and thinking, "Huh. That's what those mysterious functions mean." Delightful post!
posted by surlyben at 12:18 PM on March 18, 2016

I mostly throw it in there for the logic and proof components of math pedagogy, and less about the combinatorics and set theory bits; and I'm doing so because Discrete Math has not (yet?) been ruined by Education.

Yeah, I agree with this. If the main/only reason we're teaching Geometry is so kids can learn how to construct proofs, then Discrete Math is obviously way better at that. Geometry for its own sake is probably still useful in some regards, though. As prep for Physics I, if nothing else?
posted by tobascodagama at 12:36 PM on March 18, 2016

It might help avoid threads like these. Physical problems like that can be made obvious PDQ with enough experience with word problems and free-body analysis, but those are both skills that take a lot of time to develop.

Translating real-world issues into math frameworks that allow for solution, is a really challenging skill, which takes many years practice and lots of challenges to learn. It's a tremendously useful skill in any numerical field, which yes, is science and engineering, but probably more importantly applicable to most people in terms of business and financial decisions. Things like: "Is it better to increase the mortgage payment or put money away for retirement?"
posted by bonehead at 12:45 PM on March 18, 2016 [3 favorites]

Damn. That unit circle diagram is illuminating. I've never seen a diagram representing all the functions like that. It was always in terms of a triangle for sine, cosine, and tangent, and then purely relational expressions for the rest of them. Makes so much sense.

This coming from someone who did well in math and physics at university.
posted by scelerat at 12:55 PM on March 18, 2016 [1 favorite]

That opening diagram of the circle was in the booklet I got with my first slide rule, when I was 12 or so. (Minus the 'secret' ones)
Didn't make any sense to me until I took trig 4 years later, but then I thought it was very cool.
posted by MtDewd at 1:00 PM on March 18, 2016 [2 favorites]

I was fine with trig as a teenager - it has more obvious and straightforward applications than a lot of math (plus some less obvious one later on that have turned out to be more relevant to my life). Some of the stuff you have to memorize is pointless these days though.
posted by atoxyl at 1:20 PM on March 18, 2016 [1 favorite]

That's "replacing everyone should be an engineer!" with "everyone should be a programmer!" though.

An argument can be made that 30 years from now, basic programming and computer knowledge is going to be equally as essential to running a business, any business, as finance or statistics are now.
posted by The Pluto Gangsta at 1:40 PM on March 18, 2016

I thought (hoped?) this was going to be a clickhole link.
posted by jjwiseman at 1:49 PM on March 18, 2016

Havercosine? But I barely know 'er!
posted by pwnguin at 2:05 PM on March 18, 2016

One of these weird old trig functions was used for building railroads. Can't remember which one. I believe it was used to compute the radius of a turn by running a straight rope between two points and measuring the distance from the turn apex to the rope. Someone did a project on these functions in one of my college math classes, was really interesting. Not the only example of a math tool that was really useful back when you had to look everything up in tables.

> I am a big math nerd and *loved* calculus, but I would love to see the typical math progression in high school incorporate statistics instead of jumping to calculus.

I agree. Trig is incredibly useful; I use it constantly. But the current math curriculum is left over from the days when preparing for mechanical/electrical/civil engineering was the main motivation. Nowadays I think it is a lot more important to teach high schoolers how to critically read charts/statistics in the media and decide if they are bullshit.
posted by scose at 2:07 PM on March 18, 2016 [4 favorites]

An argument can also be made that in 30 years, the closest any business people are going to come to computer programming is Microsoft Excel; same as today.

Using a spreadsheet; particularly one as powerful and complicated as Excel counts as computer programming in my book. I took an Optics class once where we did all our ray tracing work with spreadsheets. And then there's this guy that implemented AES in Excel.
posted by Xyanthilous P. Harrierstick at 2:13 PM on March 18, 2016 [2 favorites]

"Everyone should be a programmer" is, for better or worse, the world we've landed in. Curricula must change from generation to generation; right now a logic-heavy curriculum is appropriate. Something else will be appropriate later — likely something involving how to survive on a scorching hot half-drowned planet with no remaining unburnt petrocarbons.

also this half-drowned planet may or may not have a certain amount of radioactive fallout covering it, depending on whether or not the United States government falls into the hands of genuine psychopaths anytime soon.

okay, I'm biased here: if the "everyone should be a programmer" curriculum were in place when I was a kid, I'd probably think of myself as someone who's good at math rather than borderline terrible. This is because I tend to be great at math that doesn't involve numbers, and so the computery stuff comes to me naturally. but keeping track of all the damn numbers in physics-related math breaks my brain altogether.
posted by You Can't Tip a Buick at 2:45 PM on March 18, 2016 [2 favorites]

There are no trig functions. There are just different combinations of exp(iθ) and its evil twin exp(-iθ).
posted by doop at 3:08 PM on March 18, 2016 [4 favorites]

An argument can also be made that in 30 years, the closest any business people are going to come to computer programming is Microsoft Excel; same as today.

If you want people to be good at Excel, then incorporate basic linear algebra into the curriculum - not because of the matrix math, but to introduce kids to the idea of systems of equations.
posted by antimony at 3:13 PM on March 18, 2016 [3 favorites]

If you find you really hate transcendent numbers this might help: Rational Geometry
posted by wobh at 4:51 PM on March 18, 2016 [1 favorite]

OK, seriously. I use this stuff every day. I just spent half a day trying to understand that rational trigonometry link and I can't decide whether to be mad at somebody or to be concerned that maybe he's suffering from some kind of mental break. Because I don't care where here's a professor, that dude is the crankiest crank to ever crank a crank.

If you want to spare yourself the descent into madness, he basically rules out any calculations that might lead to an irrational number. It's unclear if he's aware that he has basically redefined distance^2 and sin^2 as his new base units of length and angle, and it's equally unclear if he realizes that his problem with irrational numbers doesn't actually go away just because he chooses to move the moment of reckoning to the very end of the calculation. I gave up when he asserted that, if you can't calculate an answer to something yourself, with pencil and no other assistance, to arbitrary precision, then you do not actually understand the underlying mathematical concept.
posted by range at 7:51 AM on March 19, 2016 [2 favorites]

One of my favorite minor almost-trig functions is the integral of tanh, log cosh. Away from the origin it quickly approaches the two straight lines |x| – log 2, but of course it has a convenient derivative everywhere.
posted by mubba at 9:56 AM on March 19, 2016 [1 favorite]

I hated trigonometry and linear algebra in high school. Twenty years on they've become treasured tools.
posted by humanfont at 6:44 PM on March 19, 2016

I'm sorry you didn't like that one, range. I've got others.
posted by wobh at 6:59 PM on April 2, 2016