Fuck trigonometry
I meet a lot of people, at airports and music festivals. I’m a friendly, talkative kind of person (at least when I haven’t slept in a wet tent for two nights straight). When they ask me what I do for a living, I often say I’m a mathematician. Then they pretty much always tell me how much they hate math. When I ask why, they often suggest it was trigonometry that killed any interest they might have had for the subject.
I agree, trigonometry sucks. Fuck trigonometry. It’s a terribly unmotivated subject, and as a student you are expected to memorize double angle formulas with no proofs. It’s never clear why you’re learning it, except for the possibility of later memorizing the integrals and derivatives of said functions. It’s almost a case study in how to make someone feel like math is meant to be mysterious. A few comments before you decide I’m being unfair.
First, yes, trigonometric functions are needed in Fourier Analysis, which is hugely important nowadays for the sake of music files and information compression. But by the time you’re working with Fourier Analysis, you have more mathematical technology, and in particular you know the magic formula which makes all mysterious double and half angle formulas super easy, namely Euler’s Formula:
Understanding this formula, the Fourier Analyst is equipped to work with trig functions without all the mystery, and absolutely no memorization. In fact my mom did me the favor of explaining the above formula to me when I was in high school so I could avoid all the memorization. It helped, but even then I had memorized it, and it wasn’t until college that I understood it.
Next, this isn’t the first time I have pooped on trig. I wrote a post about statistics and algebra teaching in high schools a while ago, and in that post I suggested chucking trig from the curriculum. A few people stood up for trig in the comments. Here’s what they said.
- Someone mentioned it’s needed in “shop class.” But then they went on to explain the shop classes no longer exist.
- Someone mentioned that trig functions are great examples of periodic functions. While that’s true, we don’t need to go deeply into the subject – never mind double angle formulas – to explain that. Even just talking about an ant walking around the unit circle would be sufficient, especially if we asked for the x- and y- coordinates of the ant at a given time. Enough said. We could end the lesson with a historical remark along the lines of, by the way these functions have names, they’re called the sine and cosine functions, and you’ll learn more about them when you learn about the complex plane and Fourier Analysis.
- Someone mentioned that they use trig functions every day at work in the physical sciences. Again, I’m willing to bet they also know about the complex plane.
- Nobody mentioned that ship captains needed trig, but again we have GPS now.
When I mentioned my hatred of trigonometry to my husband, he countered with an argument that wasn’t mentioned so far. Namely, that we have really no reason to teach high school kids any given thing, so we just choose a bunch of things kind of at random. Moreover, he suggested, if we remove trig, then meeting people at an airport would just elicit some other reason for hating math. We’d be simply replacing trig with some other crappy topic choice.
I disagree. Although it’s not obvious that everyone needs higher level algebra in their daily lives (although they most definitely do need to solve systems of linear equations), it’s still more defensible to teach them to factor quadratic polynomials than it is to introduce arctan. And even though most people end up memorizing the quadratic formula, it is at least derivable using a simple completion of the square. In other words, it’s at least clear that there is an explanation, even if you don’t have it at your fingertips.
mathbabe 
…well, but tell us how you really feel!?
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Completely agree. High-school trig, at least the way I was taught, makes you think a proof is the sequential application of memorized rules.
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You’re spot on. In A level Maths (UK pre university level) there’s loads of the stuff, who cares about sec for example (well except for the chat up line Hey babe you’re 1 /cos c). Many students in UK could do A level maths and have studied almost no probability for example.
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But some students *want* to memorize formulas, at least if the alternative is having to think! Once you have memorized a few trigonometric identities, you can use them to prove other identities, and doing this provides the pleasure of solving a puzzle. Not everybody likes puzzles; but a purpose if not *the* purpose of school should be to give young people the opportunity to find out what they really do like.
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Fair enough if they have met all the “important” stuff but at this level, they often have not encountered key ideas.
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My favorite comment about trigonometry was from Nick Katz who, while explaining Katz-Sarnak over the course of a semester, said something along the lines, “Trigonometry is the just the representation theory of U(1). Doesn’t that make it easier?!” Decomposing tensor products of representations into characters is basically all of the trigonometric identities, so yeah, it sort of did.
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I agree that teaching trig without Euler’s formula is pointless, but instead of throwing out trig why not introduce the complex plane and Euler’s formula? And before you argue that heads will explode, I will counter that every high school student I have tutored (plenty of whom hated math and trig in particular) thought this this was pretty damn cool. Even if they do nothing other than memorize the formula as a short cut to save them from having to memorize a whole bunch of other identities, they will have learned something useful.
Having said that, I’ll concede that probability is far more important than trig, but this should be taught *much* earlier than trig — I would start as early as elementary school when they are learning ratios and fractions.
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Sine and cosine functions describe simple harmonic motion (https://en.wikipedia.org/wiki/Simple_harmonic_motion), which is everywhere: it describes (approximately) the motion of a pendulum or spring, the flow of alternating current (https://en.wikipedia.org/wiki/Alternating_current), the length of a day (http://www.allotment-garden.org/vegetable/garden-day-length.php) — essentially any situation where a system stays near (but not exactly at) an equilibrium position.
This is a very analogous phenomenon to that of exponential growth and decay. The same function appears again and again in a range of otherwise very different real world systems, even ones governed by (seemingly) quite different physical or other laws. And for sine and cosine there is also the link with geometry. Not boring at all!
I agree that spending a lot of time on trig formulas is pointless though. The main one to keep is sin^2 + cos^2 = 1, which is of course just the Pythagorean theorem, and also very pretty and easy to memorize. And the role of other trig functions should be limited, as they are just plain less important (although not completely useless! I especially like arctan, which is the integral of the Lorentz or Cauchy distribution https://en.wikipedia.org/wiki/Cauchy_distribution)
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You forgot surveying! Back in the day (early 70s) when hand-help calculators were just getting started and very expensive, I spent a summer on a surveying crew. We used trig all the time. My crew chief was very proud of his new HP45 ($500!), and used it instead of the large books of tables, but we still needed a fundamental understanding of trig to know the proper formulae to use.
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Yes, surveying, and more generally civil and (static) mechanical engineering need trig, and do not need Euler’s formula. Basically, if you build structural stuff, you need trig. Only electrical engineering really needs Euler’s formula, along with some parts of mechanical engineering like dynamics and thermo.
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You’ve got GPS and your navigating the high seas. Your GPS goes down for whatever reason, perhaps satellite problems. Now it’s like you’re in your grocery story and the cash registers malfunction, and the teenager behind the counter can’t add up the price of the milk and cereal that you want to buy.
I’ve also seen my wife’s Physics class notes. Plenty of trigonometry.
Frankly I love trigonometry and I love seeing how most problems can be solved using multiple mathematical methods. And I love e to the minus i pi.
Now have some breakfast or whatever before you get grouchy about some other aspect of math that you don’t like. 😉
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I thought this was going to be a belated Aunt Pythia post where I was going to learn some crazy new undulations.
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Come to think of it, I’m also disappointed it wasn’t that.
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Trig is great. Your complaint is only about how it is taught (memorization). Teach it with a focus on proofs (the way I was taught, and the way geometry is typically taught), and it’s a great place to develop and exercise proof-writing skills and let students experience the elegance and beauty of mathematics.
In fact, it’s a rare field where a student could legitimately come up with a theorem that he or she wasn’t shown first — there are just so many trigonometric identities that work, and many of them only take a couple of steps to prove.
E.g. once you cover definitions and the first identity, you can ask students to prove tan^2 x + 1 = sec^2 x. And then show both an algebraic proof and a geometric one. It’s completely accessible, and it’s fun, because things just fit, like a jigsaw puzzle. I think it’s much harder to find easy provable nuggets in, say, probability or statistic.
As for motivation, it is, of course, practical, and historically, trig came pretty early, as a way to measure things in geometry (e.g. Law of Cosines), and then largely motivated by astronomy and large distances on Earth. There are plenty of good problems to add an “applications” aspect to the motivation.
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I agree with this comment. Trig was part of our geometry curriculum and all formulas including the double angle formulas were taught as proofs from first principles. While general students will certainly get by with the “ant walking the unit circle” version, people in technical fields will use trig repeatedly throughout their careers.
IMO, trig should always be taught with a healthy dose of protractors and rulers on graph paper. Sine waves, for example, can be very easily introduced by drawing various angles on graph paper and simply measuring the vertical and horizontal components and computing the trig functions by simple division. It is critical that students have this kind of “hands on” experience to gain an intuitive feel for trig quantities when the heavy math comes in. It is the divorce of the math from the intuitive understanding that causes students so much grief.
Regarding the memorization of formulas without proof, I think there is exactly one formula that should be taught that way – the area of the circle. This formula is typically taught somewhere around 5th or 6th grade along with formulas for other basic geometric shapes. Unlike those other shapes, though, the formula for the area of a circle requires a deeper mathematical background with some familiarity with limits. Nevertheless, the formula is simple and the circle is certainly an important geometric shape, so I think this formula can be taken on faith until students have the mathematical maturity to understand its derivation.
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My son – one of them – once created a nifty animation that more or less proves the formula for the area of the circle from the definition of pi. He was in 5th grade when he made the animation, so I’m guessing he understood the statement and proof well before that.
I’ll try to get my hands on it, but it amounted to approximating the area of a circle by cutting it into many many (say N) very thin triangles of height r and of base b. The area of all the triangles is then N*b*r/2, but we know that N*b becomes very close to 2*pi if N is big enough.
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The method you mention is one of the standard derivations. The other involves mentally dividing the circle into a number of thin rings and then “opening them up” into thin rectangles and stacking them one atop the other to form a triangle. Both methods require at least an intuitive understanding of limits, so the derivation is usually deferred to an advanced algebra or pre-calculus high school course.
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Agreed that it involves limits, but clearly limits are intuitive! Especially compared to memorizing trig formulas!
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This proof? https://en.wikipedia.org/wiki/Area_of_a_disk#Rearrangement_proof
Agreed, it is very convincing. On the other hand, this is a rather convincing proof of \pi = 2:
(OK, the bad drawing helps it be convincing, but why should truth depend on a good drawing?) It *is* possible to explain to a child the distinction between approximating an area and approximating a length, and how the latter can be messed up by curves of unbounded variation… But I’m not sure if it conveys the “I can do that myself” feeling that trig or combinatorics can convey if taught well. If you want students to explore on their own, better take them to a place not riddled with pitfalls…
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Back when I was 10 or so, trig was one of the first things that got me into doing mathematics — I had accidentally gotten a 1970s German schoolbook (Lambacher-Schweizer, Ebene Trigonometrie, 12te Auflage 1979, 80pp — not to be confused with the Lambacher-Schweizers of these days, which are utter crap and differ from the old books in all but name) from a flea market, and it started making sense and making fun. My parents had been sprinkling me with maths for quite some years before, and it had helped me a lot (I guess I was some 3-4 years ahead of school in mathematics and was doing lower-level maths contests — DeMO, not BWM, in case you were surprised), but it hadn’t been enough to push me to explore mathematics on my own as opposed to just learning. The trigonometry book has done exactly that: I started inventing my own trig identities, obtaining results that were only hinted to in the book (I remember being really proud of proving that 1/a^2 + 1/b^2 = 1/h^2 for a right-angled triangle with legs a and b and height h) and creating my first proofs in imitation of the ones in the book.
Yes, you’re reading this right. The book had proofs. And exercises. And both the right-angled triangle and unit circle definitions of sin and cos, and sure as hell, periodicity was discussed. And an explanation of the geodetic intersection and resection techniques that make (or made) trig useful in mapmaking. And formulas that have been helping me in contests maths ever since because most other people have never heard of them, such as tan(A/2) = r/(s-a) = \sqrt{(s-b)(s-c)/(s(s-a))} in an arbitrary triangle (with s = (a+b+c)/2 and r = radius of inscribed circle). And have I already mentioned proofs? Three proofs for the cosine law, even: it looks like the authors didn’t think of proofs as a waste of space.
To what extent did it matter that this was a trig book, as opposed to an algebra book or an analytical geometry book? I don’t know. To some extent, yes, because trig is particularly suitable to give a student their first successful explorative experiences in mathematics, due to the amount of nontrivial but very accessible theorems, and the ease of finding new ones, along with the relatively manageable set of proof techniques that usually suffice to solve them. Elementary geometry can do the same, if properly taught, but it takes more preparation to be ready to find proofs independently (students nowadays don’t get this preparation anymore, since the inscribed angle theorem is gone); it was my second step from trigonometry. (And it is really boundless — I had to convince myself to bid farewell to elementary plane geometry when I was around grade 11 to get something else done!) Number theory lacks in approachability — you can follow your text and stand in awe of the beautiful results that took years to prove, but it won’t reward you for leaving the beaten path and exploring, unless you are a one-in-a-million genius. Calculus (whatever this means) is poor in explorable substance; at school level it is basically a mix of (fairly) straightforward and tedious computation and imprecise axiomatic pedantry (which is worse than no axiomatic pedantry at all). As to algebra, there is just not much to it in school, and whatever there is doesn’t lend itself to exploration (you’ve been told how to solve a quadratic equation; you probably don’t have a chance to solve the cubic one on your own; what are you going to do — solve a 2.5 degree equation?). Probability as it is taught in school is borderline impossible to understand, let alone make independent progress in; I still don’t understand even the discrete parts of it the way they are taught in school. (Why schools make their students learn set theory, only to later speak about “urns and balls”, eludes me.) Of course, the paradoxes inherent to an informal approach to probability (Monty Hall, two-child and the likes) are a subject that a certain kind of student takes delight in, but this is ultimately a different kind of delight than mathematics — in mathematics, we are supposed to resolve paradoxes, not to wrangle about them endlessly.
So, yes, trig has certain quantities that no other subject among those I’ve listed has, making it a particularly suitable subject for one’s first steps in mathematics. On the other hand, there are subjects I have *not* listed, and these might be even better suited. I believe that the more popular analogue of my “trig -> geometry -> mathematics” route is a “programming -> combinatorics -> mathematics” path (incidentally, recurrent sequences such as the Fibonacci numbers are more or less an algebraic version of trigonometric functions; the Binet formula is the analogue of sin(x) = (e^{ix} – e^{-ix})/(2i)). Back in my math-contest times (early 2000s in Germany), most successful participants without the luck of being born to PhD parents had either come from really good schools (with dedicated teachers — but this is only a necessary, not a sufficient condition; the whole hierarchy has to work in favor of the students), or had some sort of programming experience that was usually obtained independently (I think the share of German teachers that know how to code is still somewhere around 5%). I had my trig book that I wanted to add to; they had their NIBBLES.BAS that they wanted to mod. In hindsight, I find the latter to be the more accessible way; it is easier to download QBasic than a 1970s German trigonometry book, at any rate. If you have this sort of experience, of course you don’t need any trig by the time you’re doing maths.
And if you are learning from the last generation of textbooks, trig won’t do much for you anyway, because by now it seems to be reduced to a bunch of unproven formulas and calculator exercises. Of course, the same can be said of most other parts of the curriculum in Germany. From what I’ve heard of the US, you have more variance there, but I still would be surprised to see the Mollweide identities in a textbook (with a comment on their use in geodesics because they provide better error propagation than one of the standard formulas — sine law or cosine law, I don’t remember). Has trig been substituted by programming? That would be the only logical conclusion in a reasonable world, but I somehow don’t see much of it.
There is probably a special hell for people who write comments longer than the OP, but I’ve now written this all and I am not sure how much I can shorten it. I think that this kind of discussions are too much focussed on what to remove from an existing curriculum, and not enough on what a curriculum should contain. What is currently taught in schools (in Germany, at least) is already a result of generations of iterative reduction and dilution. Sure, it still takes the same 12-13 years (in Germany) as it used to, but much of that time is now wasted on fluff (exhibit 1: “growth models”, a pseudoscientific study of arithmetic, geometric and logistic functions/sequences before the notions of functions and sequences are introduced), useless skills (the use of “scientific” pocket calculators, which, if anything, teach you to think about programming in really old-fashioned ways), endless and brainless repetition (intersecting planes with lines, computing volumes, always with numbers, almost never with variables). In an actual class, teachers also have to spend an insane amount of time on “reminding” their students that a/b + c/d \neq (a+c)/(b+d) over and over, that a^(bc) is not a^b a^c, and many other things that have never been explained and so have to be committed to memory over and over. A lot of motivation has fallen prey to the shortening and dumbing-down: The sin (A + B) formula used to be applied in the proof of sin’ = cos, but now sin’ = cos is just a fact that you look up in a formulary (yes, that’s what the textbook says — you look it up in the formulary) and the geodesics is gone too, so a student indeed will wonder why the hell they need to know sin (A + B). Polynomial division and rationalizing the denominator are still around, but not enough algebra is being done to show the use of any of these. The whole situation reminds me of an ancient codebase which has been refactored, patched, “updated” too often and now is a mess that noone can make much sense of because the original authors are not around any longer. Maybe a rewrite from scratch would be good?
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sin and cos are the first functions one encounters which are not defined by an algorithm to compute them (well, sqrt is kind of like that too but less extreme, since there’s a plausible solve-for-the-next-digit algorithm). Instead one proves some theorems about similar triangles, which shows that sin and cos are well-defined. My teacher brought this up explicitly and it was a great moment. I think this was when I realized the difference between a function and an algorithm.
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I don’t understand what you mean.
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I think what argasrgarg is saying basically that students are unable to find (without a calculator) the value of sin x for almost any value of x (besides special cases like 30, 45, 60, 90 etc). For example, what is the value of sin 25? I have no idea (and of course we don’t define sinx via taylor series at this level, but actually, the taylor series is how sin x is rigorously defined, e.g. in a real or complex analysis course, and perhaps that is even how the calculator computes it – I still don’t know how the calculator spit backs the values so quickly!). And perhaps students feel that if they can’t even find the value of a function when they plug a simple number into it, then the function is something they don’t really understand. This reminds me of your post (which is my all time favorite blog post ever on any blog) on tensor products and how one doesn’t understand math, one just gets used to it. Maybe that’s part of what people are saying when they never understood trigonometry or that the sin, cos etc functions are mysterious.
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Ah! Thanks.
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I agree. Sin and Cos are defined with the unit circle. Its an absolutely a beautiful way to define a function.
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I agree that the time would be better spent learning probability (though of course there are limits to how deep you can get into it without calculus.) I’m a lot more doubtful about statistics, once you get anywhere being basic confidence intervals: You think people complain about learning trigonometric formulas, you just wait until you force them to learn Student’s t-test and chi-squared.
I would definitely ditch the quadratic formula before trigonometry. It would make almost as much sense to teach the formulas for solving the cubic and the quartic.
On the other side:
1. I think there’s a lot to be said for your husband’s argument. You gotta teach something specific, and whatever specific thing you teach is likely to be uninteresting and useless to a lot of them.
2. I’m not convinced how much adding Euler’s formula to the mix helps things. For one thing, though it gives you the addition formulas sin(a+b) and cos(a+b) and De Moivre’s theorem so on, it doesn’t give you the law of sines or law of cosines. For another thing,
the trig functions are a lot simpler in many ways than e^{it}. E.g. it’s easy to visualize the graph of cos(t), whereas the graph of exp(it) is a two-dimensional manifold in a four-dimensional space — not so easy to visualize. Third, you need calculus to prove it; and if you don[t prove it or justify it, you’re in the position of saying, “Here’s this cool formula which doesn’t make any sense, but allows you to calculate stuff.” That’s not the view of math we want to push.
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Trugonometry was by far my most favorite mathematics subject in high school. The math subject that I disliked were discrete probability. This was pretty common among friends of mine who ended up in physics and engineering. The friends who liked discrete probability more went into computer science. It seems to me that way too many people decide they hate mathematics because they hate one topic while those of us who go on in technical fields know better than to develop a bias against the entire field of mathematics just because we dislike one of the topics.
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There is almost no memorizing in trig since you can derive everything from basic principles. Secant is not just 1/cos its the length of a secant segment on the unit circle and tan is the length of a tan segment and cosec and cotan are also lengths on unit circles. Then Pythagorus gives all three trig identities. And Pythagorus should of course be taught earlier any 11 year old can learn it and see the geometric proof. So then the students have a nice aha moment. Working out the angle sum formulas is particularly beautiful if you use vectors and 2×2 matrices but can be proven without that. I see no reason to introduce the Euler formula but see no reason not to.
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Trig is essential for the trades, especially carpentry and woodworking related ones. Electrical and plumbing trades also use trig and machinists absolutely require it. In the USA high school shop classes may not be as visible as they used to be, but they still exist and are essential. Shop classes are no longer required for every high school student due to the focus on taking on crippling levels of student loan debt but there’s a strong argument to be made that shop classes should be required again. It’s just as essential to learn about hands on skills taught in shop class as it is to prepare for the standardized exam flavor of the day.
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Agreed. Shop classes will make a comeback in the not too distant future once we have to actually go back to work again and make useful products. I.e., once our current Potemkin economy has been terminally hollowed out and we’ve finished burning our way through the oil needed to ship stuff over from China.. Until then, understanding trig — or anything else that’s actually real and requires study — is probably optional.
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Also why is it unmotivated? Its instantly clear that trig is needed for carpentry and construction. Any textbook can easily include such topics. I do think shop should be brought back to school. I did all my carpentry at home and some kids miss out on that. We made dovetail drawers. I missed out on learning basic car maintenance and repair since we didn’t have a car.
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I thought that the “solving of triangles” was a nice application of trigonometry.
There were also a bunch of “rectangle triangle solving” which in retrospect is just similarity of triangles, but if presented as trigonometry applications, by lying a little, were also cool (stuff like measurement of height of pyramids by ancient greeks, and measurements of the earth, etc.) and could be presented together with the basic definitions as explicit motivation.
It is true that that won’t be used in the real world by most students (neither will almost anything in high school), but it didn’t strike me as unmotivated.
Also the Euler identity was introduced in my high school together with complex numbers (nothing beyond statement and formal manipulations of course), is it different in the USA?
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Euler’s formula is a thing of pure beauty.
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I didn’t have much trigonometry in High School (in Mexico); in fact, I never “solved triangles” until after I had finished the PhD and was teaching pre-Calculus at the University of Montana! All the trig I needed I learned JITT (“Just In Time Teaching”) along the way, as part of our Calculus sequence (which, granted, was 4 courses long, each course meeting eight hours a week). In that sequence we were taught several different ways to define the basic trigonometric functions (part of the unit circle, correspondence via triangles, etc) and their basic identities as we needed them to get included in the learning of Calculus. I don’t think it hurt me any to not know it.
So I agree wholeheartedly with you: we should take trigonometry out of High School and replace it with some basic statistics and understanding of probabilities (I’ve long argued there should be a course along the lines of “How to Lie With Statistics” for basic numeracy). I think trig is included because it *used to be* part of numeracy that could be generally useful (for people who got as far as High School, at any rate, which was not everyone). These days, it is *not* numeracy anymore: it is mathematics and needed for more advanced courses. So wait until you are ready for those advanced courses, teach it JITT, and replace it in High School with a course that is *true* numeracy.
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“How to lie with Statistics” is not really a replacement for trigonometry. These are just too different things. “How to lie…” (and, generally, any discussion of fallacious statistical reasoning) teaches things that students will likely need in real life, but it lacks positive results (aka “theorems”), algorithms and opportunities for problem solving. I don’t think hearing cautionary tales — however exciting and well-told — about mathematics misused would have ever gotten the point across to me that I can be a mathematician.
“How to obtain truth with statistics”, now that would be a good book. Any volunteers? 😉
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Also, sorry, but statistics is not representative (oops) for mathematics. In mathematics, things like the validity of models (almost) never matter; proofs and disproofs are (generally) final and cannot be debated away; questions (usually) have answers; you spend most of your time fighting lack of knowledge rather than misinterpretations or bias. These things all make mathematics what it is. If you replace too much of it by “numeracy”, students might not make the connection between the word “mathematics” and the kind of things mathematicians do, and might not find out that they have a talent for them.
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The point is that Trig was originally in the curriculum not as a representative of mathematics, but as a representative of *numeracy*. It no longer plays a role as numeracy (though in large part it is still taught as such). So it should be replaced with something that is useful *numeracy*, and understanding statistics (okay, it doesn’t have to be “How to Lie…” as a textbook) is a much more important numeracy skill these days. There are other places where you can do mathematics and teach it as mathematics.
One of the big problems with instruction in Middle and High School is the conflation of math and numeracy; each has value, but they way you teach them is different. Just like there is a distinction between teaching literature and teaching literacy (spelling, grammar, reading comprehension, etc). You don’t need to sacrifice one for the other: you need to identify which one you are doing so you can do it effectively. Teaching trig in High School used to be numeracy; it has remained in the curriculum mostly out of inertia rather than need; and while some programs have recognized that it no longer makes sense to teach it as numeracy and so have attempted to turn it into math, more often than not it is not taught like math (and certainly not effectively taught as math).
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Ah, I got your idea now, Arturo. But I’m not sure if I agree on the premise: I don’t think the book that I’ve mentioned in my really long post (and there were several other books similar to it in the 60s and 70s) really reduced trig to numeracy. It gave three proofs of the cosine law. The applications were probably of no direct use to 95% of the readers, who would become neither mapmakers nor navigators nor architects; but they provided intuition and context. For me, giving several different proofs of a theorem is the most obvious sign of regarding maths as an art. With modern textbooks, of course, I’d be happy if they included one…
What is currently taught in schools calls itself mathematics (out of habit), attempts to be numeracy, and succeeds at neither.
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That suggests to me that your book was one of the few that treated trig as math. Like I said at the outset, I didn’t have *any* trig until college (and I did have analytic geometry and some basics of calculus without transcendental functions in High School). That would have been one of the two standard curricula in Mexico at the time (80s; the two were one by the Ministry of Education, and one by the National University; I was under the latter), and so engineer, architects, etc. would also not have gotten trig until college. You can teach it effectively as math JIT, and you can teach it effectively as numeracy for those that need it as such JIT. Like you say: what is currently taught in schools under the single umbrella of “math” usually succeeds at neither. I think you could do a lot of nice math without involving trig (discrete math, *with proofs*, of course, for instance).
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Wow! who knew so many people were this emotional about trigonometry! 😉
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If all you learn is (real) exp, log and algebraic functions you will come out with the impression that there are no non-constant periodic functions and that, indeed, all functions are eventually monotonic (there is even a theorem to that effect). Business calculus has long been taught that way and it partly explains why most people seem so surprised with boom-and-bust cycles and think the economy should eventually reach some steady state of continued prosperity.
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Favorite calculus problem: integrate cos(x)sec(x)dx !
When my daughter asked about all those trig functions and identities we had a good time with the excosecant and the hacovercosine (https://en.wikipedia.org/wiki/Versine).
In the end I don’t think it matters which topic is taught, as long as you teach how to reason, or, how things “fit together,” and Euler’s formula is great in this regard.
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I got into trig when I was in high school. Mostly because I had a graphing calculator, and getting it to make pretty pictures was fun. I had an old book of programs in BASIC, and one of them was for making Lissajous figures. I figured out how to adapt it for my calculator’s programming language. It was a really simple program that just incremented a couple of variables by the parameters you gave it, and them plotted a line from the last (sin x, cos y) to the next. So with suitably small equal increments, you get a unit circle, with larger ones you get polygons and stars, and when x and y are being incremented by different numbers, you get Lissajous figures. And staring at all of these pictures and thinking about why they came from the parameters I gave the program helped me get what was going on in trig at a deeper level. Another time I wrote a program to simulate realistic worm trails on my calculator, which required trig to keep track of the worm’s bearing, and used a bunch of other interesting math as well.
More recently, I was designing a puzzle where pairs of pieces can meet at weird angles, and I had to figure out the proper width for the slots in the pieces so that they would meet at the correct angles. “Oh, hey, trig,” I thought, and even though I didn’t really remember how to do it, I knew which toolset I should be using, and I was able to figure it out.
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What killed math for me (mid-70s) was memorization, but rather than trig it was Calculus 2. At a state university all us engineering prospects were subjected to two years of weed-out courses “graded on a curve” and this was one of the last. It required memorization of a hundred or so methods of derivation if I remember right. It’s been a while.
I retook it & failed again, just plain outright HATED being asked to memorize stuff at that point in my life. Eventually got a BA by combining the other weed-out STEM courses I’d sailed through with anthropology, history, politics, literature.
Trig I guess I was taught from proofs. I I liked the puzzle-solving aspect. I very much wish I’d been taught music theory alongside geometry & trig. Cheers Mathbabe for provoking such an interesting thread of comments, also I enjoy hearing from you on the podcast (but you guys made such a mess of pretending to have anything worthwhile to say about water in California I haven’t tuned in since).
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I would like to point out that Fourier Analysis has been central to Electrical Engineering for several decades now (nearly a century), far before “music files and information compression”. Linear Time-Invariant systems are ubiquitous in signal processing and control, and sinusoids are eigenfunctions of these LTI systems. Most engineering undergrads initially struggle “signals and systems” concepts like impulse response and convolutions, even with plenty of motivating examples. Without high-school trigonometry, it would be a complete disaster.
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I agree with vr and would like to point out too how important high-school trigonometry is to Electrical Engineering.
Concerning it’s learning, I think that with appropriate time (3+3 months) of study everything becomes simple. The trigonometric identities of the same angle are all easily derived from the Pythagorean identity. From the addition formulas for sin and cos all addition and subtraction identities for the other direct trigonometric functions, the double- and half-angle are easily derived. The sum/difference to product formulas are not difficult to derive. The solution(s) to the linear trigonometric equation in cos and sin is/are easy to find. More difficult is the solution of triangles, but with practice, it becomes easy too.
In my answer to “Is there a more efficient method of trig mastery than rote memorization?”(http://math.stackexchange.com/q/65548/752), I’ve answered affirmatively and given some examples.
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Well, trigonometry (or its equivalent) did allow Eratosthenes to calculate the circumference of the Earth to within 2% by reading a papyrus, stepping out of his office and making one measurement. All without asking Siri. If that’s not cool, I don’t know what is.
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Quite the hornet’s nest you kicked!
I’ll skip previous citations from the trades except to point out the handy homeowner can’t build a fence or hang a cabinet without.
I appreciated the GPS comments! Fragile is an overstatement, as useful as the system can be.
Someone mentioned ship’s captains, far heavier users are pilots. Essential trig is taught, then reduced to shortcuts for almost every task. Calculating (in your head) the descent rate needed to maintain a slope that puts you at ground on speed at a fixed point in the distance. They learn to do it as private pilot’s, and even with the most modern flight decks you’ll frequently hear them mumbling the cross-check on the Approach.
I have actually help design the navigation systems they are cross-checking, and most the equations were passed down from ship sextant work as air navigation is three dimensional too. (I’m stifling myself here because if I get started on inertial navigation I do go on, but just to brag a bit, we have to account for the galaxy’s motion through space).
But as mentioned by many, like algebra, the coursework was taught rote to your high school teacher who largely teaches it by rote to the student.
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Omg please write a guest post explaining that stuff. At the end of it trig will be my favorite subject. Seriously.
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I’m one of those math teachers with “I love the Unit Circle” bumper stickers. All the rules are in there. I shared the kids’ frustration with “proving identities”, so we used to make them a writing assignment. Students had to describe their thinking with “action verbs” (mostly violent ones). They then associated different rules with the verbs they used. I had plenty of laughs reading those assignments.
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I wish there were more teachers like you!
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I don’t think it’s critical for kids to learn trig, but I think it can be fun. I told my students that trig is a branch of math that explores connections between circles, triangles and waves. They seemed to find that motivating.
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Seems like the bit of trig that is most widely used (used my the most people) – the bit that lets you use angles to find lengths you can’t measure directly can be taught very naturally with no or very little memorisation, assuming you remember Pythagoras from two weeks earlier in the curriculum. When I did it in grade eight, we spent half of that week’s math class outside calculating the height of school buildings and the tallest tree on the school grounds, so it was actually about the most fun bit of high school math, and I think most of the other kids in the class found it fairly impressive also.
OTOH. double and half angle formulas – yeah, they kinda suck,
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So I agree that memorizing a bunch of formulas is galling and I seriously can’t comprehend why any instructor would have students do that. When I teach college trigonometry I do give out a formula card to use all the time in class & on tests. (Just like I do in statistics.)
I would argue that the real payoff is being able to model harmonic motion at the end of the course; springs, rotations, tides, sunlight, etc. I would also emphasize that getting a new family of functions (sin, cos, tan) deepens the concept that any function must have an inverse with which to solve equations (beyond just add, mul, exp). In that regard if I had my druthers I’d pick that basic idea over factoring, to be honest. And the identity-proofs are not a bad way to dip ones’ toes into the requirement for writing your own proofs (before someone dives headfirst into a deep advanced college math program).
Coincidentally, I just gave a talk a few weeks ago on using math in computer games — my very first example being using the arctan() function in a game I wrote at age 14 to direct the energy-blast of a villain at the player in a 2D top-down superhero game. Which both motivated me to learn how trig functions were of use, and kicked off the career that I had in gaming (before switching to college lecturer).
I haven’t taught trigonometry in a few years, but now I’d be hard pressed to teach it the next time without using “tau” in place of “pi” throughout the course. I’ll point out that “tau day” is next Sunday, which in the house we celebrate with a dinner of tacos, tortilla, and tarts (http://tauday.com/).
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The more comments I get on this the more I’m convinced that some people teach trig in a truly compelling manner, and you’re one of them!
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+1 for formula cards. There is no good reason to force students to memorize formulas.
I can’t say I am particularly optimistic on \tau, though. I have probably seen all systems around — multiples of \pi, multiples of \tau, multiples of D (= \pi/4, in old Russian texts, maybe from “direct”?), degrees, gradians. I can’t find a compelling reason why one is better than the other, except that gradians are bad because of the difficulty of expressing useful angles such as 30° and 60° in them. In geometry, IMHO it’s D = 90° = \pi/4 that has the best claims to be the “basic quantity”, but writing everything as its multiples (the D system) causes the equilateral triangle to have angles 2/3 D, which is somewhat hard to memorize. For cyclotomic computations, \tau makes the most sense. Degrees are comfortable to some extent as one rarely ever has to use fractions of them. I don’t see a good way to avoid teaching at least two of the systems.
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I loved trig. As a biologist, my classes used trig to get through some chemistry and most physics (I took MCAT physics, not real physics).
Also, the whole idea of “I know this formula, let me use it to derive all the related ones” is really valuable, though I’m sure you could teach it with other things.
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We can make kids hate any subject/topic with poor teaching. Memorization without understanding is the first prescription for that. I wonder if the problem is with the subject/topic or with how it is taught.
I don’t use much Math in my work, but I teach my kids and try to motivate other kids who would listen. I have used trigonometry most frequently in two contexts: projection along another direction (multiply by cosine of the angle between the directions – run into this everywhere in physics) and the duality of time and frequency domains. At this point, I mostly remember the fundamentals in each topic and look up or derive everything else. I tell my kids the same thing – understand the concepts well and make connections with the other concepts they know. We derive the formulas at home that the teachers at my daughter’s school hand out at school for memorization and number plugging. As she hates memorizing stuff without understanding, we usually work on the topics before they come up at school.
Sine, cosine, etc. are just names for ratios in right triangles. Presumably, we could do the same math with similar triangles all the way, but it would be pretty hard to work with anything without names/labels.
Does it make any more sense to memorize Euler’s formula without knowing where it came from just because it is easier to memorize? I absolutely love it and know how to use it. However, it took me years of digging around to understand how to arrive at it ab initio. The search provided as much pleasure to me as learning and teaching math does. Now, I love math history along with math.
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Like trig or not, here is a typical problem from a so-called A+ mathematics nation:
https://docs.google.com/document/d/1uu3_MAPrA484ugVYxlkQyEzAEJ_sMYrf8sp2GjvKTlc/edit
It’s fairly difficult (and long), the only point here being that upper secondary students study much, much harder than those in, say, the US.
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Trig wasn’t bad…though I had a phenomenal teacher and took it over the summer (4 hours a day, 5 days a week for a month). Geometry SUCKED. I hated hated hated writing out proofs. I think a lot of it really just had to do with teachers though…
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I actually liked trig and still use it (for example watching the rate of change of the length of the day decelerate toward 0, reverse course at the solstice, then pick up speed in the opposite direction).
But I can’t say that means that everyone else should be forced to endure it.
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I am in such strong agreement on this! I use calculus, algebra and probability methods that I first learnt in high school every day, but haven’t had a single use for trig since I left high school. On the other hand, I was never taught advanced linear algebra in high school and had to teach myself it later, so there’s lots of other things that could usefully replace it.
To be fair I mostly work as a data scientist, so perhaps that skews the math that comes in useful.
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I think that your last statement is the most pertinent. Imagine if algebra had been removed from the curriculum during your student days because someone else hadn’t used it after high school. A lack of trig education would cut most people out of the opportunity of an engineering career.
In any case, the bottom line for all subjects is that they need to be taught well. Kids develop different interests, but they don’t need to go through any subject hating it because of poor teaching.
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“Someone mentioned that they use trig functions every day at work in the physical sciences. Again, I’m willing to bet they also know about the complex plane.”
I left the comment about using trigonometry in the physical sciences on a daily basis. I’ve known about the complex plane far before I learned the Euler formula (and its Taylor series derivation) in high school.
I still stand by this comment. I use trigonometry for a few things:
1. Relating things (forces, angles, etc.) in three dimensional space. Complex exponentials are pretty useless for this.
2. Various rigid body transformations. While quaternion and axis-angle are often easier, sometimes you have to use Euler angles and directed cosines.
3. Projecting various functions onto the Fourier basis. I do this exclusively with complex exponentials because it’s far easier.
In my line of work, I do *far* more of the first and second than the third, and I reiterate: complex exponentials are useless for the first. Trigonometry provides quantitative relationships between things that have angles. If I want to know what the normal restoring force for an 20-degree incline plane, I’m going to break out sine and cosine because they give me a direct answer.
I can’t tell if you’re mostly against how trigonometry is taught, or that the subject is taught at all. I think physics and engineering provide ample motivation. The double-angle formula can be taught from a geometric derivation, although many high school students don’t seem to have the patience for that, either. I think your husband has deep insight when he suggests that students would find some other math subject to hate on.
And, seriously, all you really need to memorize are the two general angle addition formulas and cos^2 x + sin^2 x = 1, and pretty much everything else drops out. The Euler formula would be a great thing to teach, as it summarizes most of this, but focusing solely on the complex plane neglects the numerous ways that trigonometry connects geometry with the quantitative physical world.
Last, and definitely least, given how weirded out everyone is when they first encounter Bessel functions, I’m quite glad that we’re repeatedly exposed to some class of special functions, and the trig functions make the most sense: they’re closely related to geometry, they pop up quite frequently, and for those of us who actually use them, all that rote work on identities in high school has made them second nature.
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“1. Relating things (forces, angles, etc.) in three dimensional space. Complex exponentials are pretty useless for this.”
Huh. Quaternions rock.
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I’m finishing a teaching certificate for high school math over the next months and found this to be very informative. Teaching maths, there has got to be a better way. Math illiteracy afflicts 5 out of every 3 people and that my friends is no laughing matter!
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Except for the MAA exams in high school (the 50’s) I had no idea that Mathematics was anything other than 30 minutes of homework each night doing exercises (There may have been a glimmer of the pleasure in geometry that came when I had an aha! moment, but most of the time was spent memorizing the axioms, definitions, theorems, and proofs). The one exception to this that I recall quite vividly was when my trigonometry teacher came in one day and said, “Here is an interesting trigonometry problem I just came across on a recent Russian exam.” Each night, for the next two weeks I would lie on the floor in my room after I finished my homework and think about that problem, getting up from time to time to do some calculations. In the end, I discovered the “golden triangle” which led to a solution via the pythagorean theorem. I wrote out a complete solution which took up a whole sheet of paper. The joy I felt at this point was as exhilarating as sex. I now had a much better of what mathematics was. When I showed the paper to my teacher, she took no more than 10 seconds to glance at it, saying “Yes,” and gave it back to me. We never said another word about it and it was never discussed in class. I still have that paper today and believe that solving that trigonometry problem was a major reason that I decided to major in mathematics after I got out of the Navy and went to college. I have spent the past 45 years trying to share the joy that comes from solving problems with my students. I think geometry and trigonometry were the most accessible areas for my students to see the big picture.
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