Why cosines?

Why Do We Learn About Cosine Functions? asks Forrest Hinton on The Quick and the Ed. I’ve waited more than 40 years for the answer — which he has not got.

Hinton flashes back to his Algebra III classroom in a Raleigh suburb. A girl asks why she has to learn about cosines. What’s the point?

He asks her what she wants to be when she grow up. She answers: a beautician.

I went on to describe how the business cycle oscillated between recession and expansion, much like a cosine function, and how she would have to follow this trend as a small business owner. The young woman was unimpressed.

He continues with “a vague, cop-out” answer:

On the whole, mathematics is a useful subject because it teaches you how to think logically, problem solve, and justify the things you do. It also stretches your mind as you deal with abstract complexities and forces you to be concerned with small, but important, details. Now, let’s take a look at example three…

There’s a lot that high school juniors might learn, Hinton writes. Teachers should understand why they’re teaching certain pieces of knowledge and skills, so they can motivate students to learn.

Are they learning it merely because it’s tradition? Will 10% of the Algebra III class eventually use this in their professional lives? Are students, in reality, learning to generally solve problems and confront complexities?

I took Algebra II/Trig because I needed it to apply to selective colleges. (In those days, only math-science types took AP Calculus.) Algebra was fine, but I hated trig. I was running out of enthusiasm for jumping through hoops and trig seemed more like a strategy for torturing 16-year-olds than an essential subject. I repeatedly asked my teacher why those of us with no aspirations for math, science or engineering majors had to learn sines, cosines, logarithms and so on. What was the point? He couldn’t answer the question. At the end of the year, he gave me a higher grade than I deserved. I was so surprised that I asked him if he’d made a mistake. He said he’d factored in my classroom participation. I guess he liked being asked, even though he had no answer suitable for a future English major.

About Joanne


  1. I’m working on specifications for a system which handles maps, and I’ve been using trig identities for the first time sin more than 30 years.  Surprisingly, they came right back (Wikipedia is a huge help).

    I can’t say what cosines, tangents and logarithms will do for an English major.  What I can say is that what you don’t know about these things when you leave school, you will almost certainly never suddenly see applied in front of you in nature, architecture, finance or anything else.

  2. Charles R. Williams says:

    A case can be made for well-educated non-science students understanding the sine function and logarithms because they are a natural tool for modeling certain data. We don’t need cosine to solve right triangles or to model data but it does give a sense of completeness to name every ratio of the sides of a right triangle from the perspective of a given vertex.

    A beautician doesn’t need any of this. True mastery of middle school math would be more useful. We need to understand that a student who tailors his education to low-skilled career aspirations will probably never have a serious shot at a technical career. For the most part we are unwilling – on paper at least – to allow for this. So future beauticians are taught cosines and we cook up phony real world applications to try to motivate them.

  3. I hated trig as well and only took it because it was a pre-requisite to calculus (which I needed because I was pre-med at the time). I’ve never used it and only very rarely used calculus in my career (and that was by choice as there was a work-around that did not require calculus).

    I’ve never understood the reasoning behind why trig is a pre-req for calculus as I never used anything I learned in trig in my calculus classes. My DH, who studied engineering, used it in his physics coursework but I had no intention of majoring in engineering or physics.

  4. Tom Linehan says:

    A couple of times I have had math professors in college level classes who, while teaching a topic, would comment in passing that they did not know any practical applications. I would be sitting there with my mind spinning thinking of all the practical applications. I approached both to explain the potential uses. One was very interested. We had an extensive discussion. The other was not.

    For those who do not need to know much math in their occupations and in everyday life, there are larger issues in which an understanding of math is necessary to learn beyond the most elementary level.

    In Education a good knowledge of statistics helps understand what the research means. For instance, when is a correlation high enough to possibly be more than a coincidence? What does Statistical Significance mean? What does the Effect Size tell us?

    In economics, why is the margin (slope) important? Why may a change in the margin be more important? Yet the answer to these types of questions makes figures such as a change in the national debt more comprehensible.

  5. Cranberry says:

    It might be that the daughter of a beautician would change her mind about becoming a beautician. She might decide to become a lab technician, for example. It is much more cost-effective for her to learn Algebra III at public expense, in high school, than at her own expense, after she develops an allergy to the hair dyes.

    In a larger sense, it would be very unfair to deny public school students basic math preparation for professional careers just because they can’t conceive of life beyond high school graduation.

  6. My answer: well, you never know. Few mature people I know ended up on the career path they expected to. I was supposed to be an aerospace engineer, but I spent more years writing software than engineering. These days, by the time they’re 50 a lot of folks have had two or three careers already.

    I took a job as a video game developer and had to spend a week refreshing by knowledge of linear algebra and trig because 3-D graphics are all matrix multiplication and trig. A couple weeks ago, I had to write code to calculate the variance of a set of data points. While trying to teach myself Latin (an ongoing project that isn’t going particularly well), I found that I had to go back and completely relearn high school grammar.

    You just never know. The only knowledge you’ll be certain never to apply in your life are the things you never learned.

    By the way: if you go past basic calculus, trig starts to pop up more an more as necessary to solving certain equations. It makes more sense to teach it before calculus than to try and add it in to a later calculus course.

  7. The question is somewhat like asking a coach ‘why do calisthenics?”

    Too often, education is only seen for first level effect: being able to directly apply learning to the vocational enterprise of the student. I think this is misguided.

    Build a fit body with calisthenics, build a fit mind with math (and literature and …). The student will need both to do whatever he wants to do with his life. How we, as teachers and coaches, aid a student in building a fit body and a fit mind may not be optimum and may not be as well suited to an individual’s need as we might wish but we should keep in mind that it is a method and process that has been learned through very much experience.

  8. The question of what math courses should be taught is a good one, one that few commenters are touching on.

    I am a big beleiver in teaching math. Clearly algebra is important. But I am not so sure about trig or even calculus — at least not how we teach it. Statistics & Probability really should be taught much more, and perhaps much earlier. And then we get into modelling, like researchers need to do.

    Logorithms and trig used to be very important and basic tools for measuring and figuring, but with modern coputers — including the calculators in our smartphones — they are FAR less important for most people. The fact that they remain important and useful in some fields is not sufficient to preserve the old place, if you ask me, as part of the core sequence. Serious work on statistics and data analysis would be much more useful, both in the practical sense of preparing students for potential future careers and for the preparation of citizens and future voters.

  9. Miller Smith says:

    The trig functions are just like any other funcition-they give you methods of determining an unknown that cannot be easily directly measured.

    Now there may be very few people who care, but that is not an excuse for the math teacher to not have a clue as to why functions exist. Sad…very sad.

  10. Cardinal Fang says:

    You use the cosine to compute the third side of a triangle if you have the other two sides and the angle between them. People do end up doing this in real life; wanting to know the distance between two places is not exactly unusual.

  11. Cardinal Fang says:

    And I’ll add: the number of students who will need trigonometry in later life is enormously greater than the number of students who will need to do literary analysis. If we’re throwing out high school subjects on the basis of usefulness, literary analysis should be the first to go.

  12. Why would you purposefully NOT want to know something? It’s not like you’ll run out of brain.

  13. Why would you purposefully NOT want to know something? It’s not like you’ll run out of brain.

    But time spent in formal education is limited. As a would-be physician, it would’ve been a much better use of my time to take something like statistics in high school than trigonometry. I didn’t wind up going to medical school as it turns out but I didn’t going into engineering either. I found the statistics class I took in college to be very helpful, but trig was a complete waste of a year.

  14. By the way: if you go past basic calculus, trig starts to pop up more an more as necessary to solving certain equations. It makes more sense to teach it before calculus than to try and add it in to a later calculus course.

    What percent of students who take trig ever go beyond basic calculus? Why not teach it as part of an advanced calculus or physics course to only those who actually need to learn it?

  15. I run into something vaguely similar with the pre-meds; they don’t want to learn anything about botany. “I’m going to be a doctor,” they say, “Why should I have to know about PLANTS.”

    I admit I’m occasionally tempted to say something like, “But when society collapses, you will be grateful to know what a willow tree looks like, so you can brew aspirin from its bark.” But I’m afraid that would come back to bite me on teaching evaluations.

  16. Cardinal Fang says:

    Crimson Wife: “I’ve never understood the reasoning behind why trig is a pre-req for calculus as I never used anything I learned in trig in my calculus classes.”

    I’m speechless.

    You must have had a very, very strange calculus class, then. And when I say very strange, I mean a calculus class that left out the calculus part of calculus. In about week two of a calculus class, the professor will start talking about the derivative being the slope of the line tangent to the graph. He’s going to start talking about triangles, and sines, cosine and tangents, and the student who doesn’t know trigonometry will be totally lost, if that student wasn’t already totally lost.

  17. Cranberry says:

    Why not teach it as part of an advanced calculus or physics course to only those who actually need to learn it?

    How do you identify those who actually need to learn it? I’m trying very hard not to be snarky here. I feel that the heterogeneous classes which range from special needs to gifted make it difficult to teach anything, but the critics of tracking do have a point. In the past, students tracked into lower-level classes were (are?) minorities, women, and the poor. There’s no way around that. How do you justify limiting educational opportunity? Is it preferable to schedule the daughter of a working class family into a study hall rather than Algebra III, because, well, you can’t see her becoming an engineer?

  18. Chartermom says:

    What a perfectly appropriate posting for me. My 16 year old just finished pre-calc (which included cosines) and it kicked his rear end. It particularly bothered him that they were taught things in the class for which there were not any word problems because there weren’t any practical applications (or so my son complained). That is the material he struggled with the most.

    Next year he’s taking AP stats and I’ve assured him that he will find much more use for those. And then he will likely take Calc his senior year. (As to why Stats before Calc — the current recommendation being passed down from college students is take Calc your senior year as you will need to retake it in college anyway and better to have it fresh in your mind.)

  19. It particularly bothered him that they were taught things in the class for which there were not any word problems because there weren’t any practical applications

    I’m speechless.

  20. Cardinal Fang- my university had 2 calculus tracks. One was for STEM majors, and the other was for everyone else. I took the latter and don’t remember anything trig-related being discussed. I pulled the course catalog off my DH’s bookshelf (why he’s kept it there 12 years after he graduated is a good question but it’s coming in handy now). I see the track I took described as “more conceptual than traditional courses”. So there you have it. But it satisfied the med school admissions requirement, which is what I cared about at the time.

  21. Cardinal Fang says:

    Here’s a simple word problem involving cosine: The post office is at the intersection of Main Street and Elm Street, which meet at a 30 degree angle. The library is also on Elm Street, 1200 feet west northwest of the post office. MacGuffy’s Sweets is 2000 feet west of the library on Main Street. University Avenue goes straight by the library and MacGuffy’s, connecting the two directly.

    Jimmy and his mother are at the post office dropping off a package. His mother parked the car there, and she and Jimmy are going to walk over to the library for Story Hour, then to MacGuffy’s for ice cream, then back to the car. How far will they walk?

  22. But, few math teachers have all that much understanding of science – so, know few practical applications of their trade.
    (I’m not knocking math teachers – there’s a special place in heaven for those who teach it).
    I’m an example of someone who saw no need for the math – I was going to run a business. Later, in college, I was a History major – again, little need for math.
    However, as a hedge against unemployment, I also got a General Science certification. Naturally, I quickly found work as a science teacher. Over time, I added on coursework, and am finally a Physics/Chemistry teacher.
    BOY, do I wish I had paid more attention in math class!

  23. Cardinal Fang says:

    Crimson Wife– you were cheated. Your class might have been called calculus, but you didn’t learn even the rudiments of calculus. I’m surprised a med school that required calculus would accept a pretend course.

  24. I checked the current pre-med requirements and it says that “Over 50 schools require one or two semesters of college math. 16 medical schools require at least one semester of calculus.” So it looks like students don’t even need to take any calculus for most med schools. I think it would be difficult to get a decent grade in college-level physics without having taken calculus, though.

  25. georgelarson says:

    I am in favor of teaching trigonometry, but almost all two dimensional trig problems can be solved quickly geometrically. All you need is paper, ruler and protractor. It would be best to teach both.

    You don’t really understand Calculus till you pass Real Analysis which usually has Calculus as a prerequisite.

  26. I’m always pretty saddened to see arguments for ignorance.

    I’ve learned a LOT of stuff over the years for which I’ve probably had little use. But you never know. I just like learning. The “when am I ever going to use this” whine is tiresome and I would think below adults who profess to tell the rest of us something about education. It’s sort of like the “this is stupid” complaint I get from the teenagers. Usually, it means nothing more than, “I feel stupid because I don’t get this.”

  27. Roger Sweeny says:

    Cardinal Fang,

    You don’t need cosines for your problem, only the Pythagorean Theorem.

    A little trig is useful for many people, but ceolaf (8:56 am) is exactly right about how much less useful trig and logs are nowadays and how much more important understanding statistics is.

    If you’re talking about courses to REQUIRE future citizens to take, trig has much less to offer than practical statistics.

  28. I didn’t say that trig is “stupid”, I just said that it’s only really needed by a small percent of those forced to study it. I learned it well enough to get a “B” on my final and have never used it since. And I don’t think I’m alone in saying that.

  29. Wow, I can’t believe the answer about learning trig as a way to teach logical thinking and problem solving was called a “a vague, cop-out”.

    Why do we still learn Latin?

    How about asking why I, a math/science person should ever study literature because understanding emotion and the human condition play no (direct obvious) part in my job.

    Learning logical thinking and problem solving can be one thing taught by studying difficult mathematical subjects, it is every bit as important as studying humanities.

  30. Ben- most kids don’t study Latin. And studying Latin is not made a pre-requisite to studying a different Romance language the way trig is made a pre-req to studying calculus.

  31. Ex-PhysicsTeacher says:

    I’ve generally been happy to learn whatever I’m reasonably able, but I, too, draw the line at literary analysis/criticism, whatever In my senior year of HS we were required to do a research paper on some work. I chose “Kubla Khan” because I really liked Rush and they had a song inspired by the poem. I went to the university’s main library and found loads of “analysis” of the poem. These authors were ascribing all sorts of conscious meanings and motives to a poem supposedly inspired by a drug-induced nap. I saw them essentially lying to fill up the pages of a book, and whatever respect I had for these folks, which was little to begin with, completely evaporated.

  32. Roger S., Pythagoras is no help in that problem; at a minimum, you need the Law of Cosines.

  33. Ben, and these were the same folks saying hip-hip-hooray to classical education a few days ago. Kinda funny, right?

  34. George Larson says:

    The reason Trig is a prerequisite for Calculus is you cannot do trigonometic substitution or transformation from rectangular to polar coordinates without Trig. You need Trig to understand the dot product of two vectors and the cross product of two vectors.

    You can learn enought trigonometry to do this in an hour.

  35. I love math. I am indifferent to the cosine function, although earlier today I posted at my site a solution to an old Japanese problem that required cos and sin.

    I am irked by the question and by most of the answers. You can’t get a right answer if you ask a wrong question. We may question whether we (average people) ever need any subject, except some language of communication and basic arithmetic. Why so special emphasis on cosine? Who needs chemisty or history?

    On the other hand, statistically, any attempts to justify the study of mathematics on pragmatic grounds are doomed, because very few of adults use any kind of mathematics and every one knows that.

    The suggestion that math somehow helps brain development is a complete nonsense. First of all, there are better ways to master logic and reasoning. Search the web for the name of Harold Fawcett or Louis P. Benezet. Both ran experiments on a large scale whose results have been and are being overlooked.

    By far a better question would be what is the purpose of education, how should the system of education be structured to satisfy its purpose or whether the current system makes any contribution towards that goal.

    That’s a tough question, indeed.

  36. How we, as teachers and coaches, aid a student in building a fit body and a fit mind may not be optimum and may not be as well suited to an individual’s need as we might wish but we should keep in mind that it is a method and process that has been learned through very much experience.

    This is so exactly right that I’m stunned anyone would think otherwise. What the heck is wrong with people?

    However, I certainly agree that we should only teach trig to those with the cognitive ability to understand it, and that’s a whole lot fewer people than currently take it.

  37. Four years ago I wrote a post called When Are We Ever Gonna Have To Use This? The closing paragraph is as true today, and as germane to this discussion, as it was the day I wrote it.

    To understand that post, you must know a little bit about history, too 🙂

  38. You need Trig to understand the dot product of two vectors and the cross product of two vectors.

    That’s not Calculus, that’s Linear Algebra. Most folks who are forced to study trig are never going to take the latter course.

  39. Diana Senechal says:

    I agree with Alexander Bogomolny. If we start judging subjects by their utility, most of them will have to go.

    This does not mean that we should teach subjects arbitrarily or “just because” they have been taught through the decades and centuries. The big question is: why do we teach mathematics beyond the basic skills that people need for everyday life? Why, indeed?

    Perhaps for the beauty and challenge of it. Perhaps because it allows us understandings we would not otherwise have. But as Alexander says, it is a tough question. What do we teach these subjects for in the first place? There seem to be several ways of thinking about them: (a) teach them for their inherent value, with or without utility; (b) teach them because they have been taught before and are required; and (c) teach them because they are useful.

    Of course it is hard to separate inherent value from utility altogether; what is valuable to us is generally useful in some way, though not necessarily in an immediate or practical sense. Nor is it right to ignore the practical. We all need things that we can do in the world.

    But as Rob pointed out, we never know what might be useful in the future. Or rather, sometimes we do know. We know that certain topics are important for advanced studies, for understanding of certain ideas. Students often live in the short term. It is the responsibility of teachers and schools to help them see beyond that. Perhaps the beauty of a subject and its long-term value are not too far flung.

  40. Cardinal Fang says:

    Crimson Wife, trig is a prerequisite for calculus because most students studying calculus don’t study the watered-down Calculus for Poets you took, but real calculus, which most certainly needs trig. The AP Calculus AB curriculum, a standard first year college calculus curriculum, uses trig.

    Roger S, I don’t understand how the Pythagorean Theorem can help us solve a triangle if none of the three angles are right angles, which they aren’t in the problem with Jimmy and his mother. How do you solve this problem? Like Engineer-Poet, I use the Law of Cosines.

    Anytime you have circles, cycles or triangles, you’ll need trig. Here’s another trivial problem that requires trig (you could use a cosine but I’d use a sine for this one): Emily and her boyfriend Jason get on a Ferris wheel with a 100 foot diameter. Emily is flirting with Jason, but when the wheel has gone around a third of the way it stops to let on another group. Emily looks down, sees how high she is above the ground, and screams. How high up is she when she screams?

  41. Allison says:

    Crimson Wife,

    Can you tell me the definition of a derivative, without using cosine?

    I cannot. I cannot imagine what course you took that claimed to be calculus.

    Here are some other simple word problems using sine and cosine:
    A cellular tower needs to be built. Given a broadcast wattage of W, a signal can reach a phone at distance r away from the tower. How tall should the tower be to reach my home, 100 yds away from the base of the tower (assume flat earth in local region)?
    Given a tower of 100 ft, what’s the farthest the signal can reach (assume flat earth).

    Now, include the curvature of the earth. solve again.

  42. Okay, so I dug out my DH’s old calculus textbook (mine has been long since sold). He took the STEM track course. Looking through his book, it does include trigonometric functions and vectors. The stuff that I remember doing mostly were straight forward differentiation & integration problems.

  43. Cardinal Fang says:

    Allison, to define a derivative without cosine I’d just use the standard limit definition: the derivative of f(x) at a is the limit of (f(a+h)-f(a))/h as h goes to 0, if that limit exists.

  44. Allison says:

    But the point is that the limit is the slope of the tangent line.

    Really, how can a calc class ignore the slope of the tangent line?

    How is that possible?

  45. I’m sure there are lots of people with degrees beyond bachelor’s who don’t need cosines.  MD’s in clinical practice sound like folks who’d never think of trig.  But if they got into the research or epidemiological end, calculus concepts like the logistic curve might become far more important.

    The problem is, if you don’t have that background when you get to that point, you’re suddenly out of your depth.

  46. “That’s not Calculus, that’s Linear Algebra.”

    Ever hear of Vector Calculus?

    BTW, all the trig functions have their own derivatives and integrals too… Which surely you saw in your Calculus class. You know,

    if f(x) = sin (u), then f'(x) = (du/dx) cos (u)

    that sort of thing…

  47. For some of us, there is a difference between analytic geometry and trigonometry. Slope of the tangent is the former, while cosines are the latter.

    Trig is not at all necessary to understand calculus, but it provides useful problems to solve. Calculus is nothing more than the mathematical despcription of the basic laws of physics–that’s why Newton invented it.

    I’m more intrigued by Ben’s idea that understanding emotion and the human condition have anything to do with literature. Now THAT’s a connection I don’t see. What I learned about emotion and the human condition came entirely from experiences, not reading. And I read a lot.

  48. Cardinal Fang says:

    But trig will come up in calculus. Even if the problem doesn’t initially have any trig functions, those pesky trig functions creep in. The rational function 1/(1 + x^2) looks simple enough, but its integral is arctan.

    Moreover, a calculus student should be able to recognize and set up problems involving trig functions, because they come up all the time in simple calculus situations. If the student is going to actually use calculus, she’ll need trig.

    And if she isn’t going to use calculus, rather than taking an imitation calculus class she should study probability and statistics.

  49. My son was asking this same question this year while taking trig.

    I probably use trig and geometry a few times each year, mostly for surveying kinds of questions. Trig: How big a tarp do I need for the tree house if I rig it diagonally, want a two foot overhang, and want at least a 30º slope so it doesn’t fill up with leaves? Geometry: How much cement do I need to buy for a 12″ diameter foundation, 3′ deep, with a 4×6 post in the center?

    I don’t think I’ve ever solved a quadratic equation for a real-world problem.

    I agree that probability and statistics would be far more useful than trig for understanding daily life.

  50. Roger Sweeny says:

    Cardinal Fang,

    You are right. I was wrong. The Pythagorean Theorem will not solve the problem you presented.


  1. […] This post was mentioned on Twitter by kriley19, Joanne Jacobs. Joanne Jacobs said: New blog post: Why cosines? http://bit.ly/a9PGWO […]

  2. […] Why cosines? "Joanne Jacobs I took Algebra II / Trig, because I needed it to apply to selective colleges. (In those days, the mathematical science types just took AP Calculus.) Algebra was fine, but I hated trig. I was running out of enthusiasm to jump through hoops and trig seemed ….. He took the trunk tracks progress. Looking through his book, he includes trigonometric functions and vectors. The things I remember were doing mostly front differentiation and integration issues. Cardinal Fang … […]