Note: Those of you who would rather read this in French may
transfer to <http://www.sauv.net/raimi.php>, where a couple of French math
warriors have translated it very nicely.
WHY LEARN TRIGONOMETRY?
The following question came into a public email list called
"math-teach", mainly read and written by school teachers of math and
professors of mathematics or of mathematics teaching, some time in 1996:
> You are teaching a group of skeptical high school students trigonometry and
> they need to know "Why do we learn Trigonometry?"
> Unacceptable answers:
> 1. It's the next unit in the book.
> 2. The curriculum committee says you have to.
> 3. It's on the SAT.
> 4. Mathematicians find it "elegant."
> 5. In case you ever need to know the height of a flag pole.
When this question first came up I regarded it as a joke, the humor
being contained in the list of four popular and one fatuous answers to
the question. The true answer to the question, "Why trig?" seemed obvious
to me, and I assumed it was obvious to the person who posed the question.
I answered light-heartedly, making reference to another fatuous question
of about 1942, when I was a physics student:
That question, in imitation of the sort of thing that too often
appears on exams intended to sound practical and true-to-life, was, "How
would you use a barometer to measure the height of a building?" Talk
about open-ended questions! The expected answer, as any experienced
high-school test-taker knew, was that you measured atmospheric pressure
with the barometer at the bottom and the top of the building and applied a
scale (found in a book somewhere) that tells you how pressure falls off
with altitude. Airplane altimeters used this principle, which was thought
to be of utmost practicality in the days before radar and World War II.
But "How would we use the barometer to measure the height of a
building" gave rise to a lot of merriment among us teachers of math or
physics (as I later joined those ranks), and some of the following
proposals remain in my memory:
1. Measure the length of the barometer and apply it repeatedly and
vertically up the side of the building. Multiply its length by the number
of times it is needed to reach the top.
2. Carry the barometer to the roof and drop it to the ground. The time
it takes to crash is related to the height by the well- known Galileo
3. Sell the barometer and hire a lawyer with the proceeds. Send the
lawyer to the County Real Estate Records Office for the desired
I must have seen a dozen equally valid answers, "practical" as
anyone committed to current educational theory might ask. It is not hard
to make fun of current educational fashions, in every era. And of idiotic
examination questions, in particular, there is no end.
So I mentioned this to the math-teach people, though lightly,
thinking my point had been made: Ask a silly question, you get a silly
answer. But more recent communications to "math-teach" on the thread of
"Why trig?" have persuaded me that the question was serious, and intended
to elicit real explanations such as might appeal to high school students
condemned to study that apparently obsolete subject, and to their parents.
Despite all the efforts of the scientific professions in the years since
C.P. Snow first deplored the Two Cultures phenomenon as he saw it in 1960,
much of the "educated" world still regards science and mathematics as
trades like lace-making or piloting an airplane, activities of value for
which we pay when we have need, but certainly not something the rest of us
should learn anything about in detail. So I have a few serious replies of
First, I agree that it is a pity such questions come up at all.
Those (math teachers) who liken the question to "Why read Shakespeare?"
are quite right. Trigonometry is part of the equipment of an educated person,
as much as history, literature, biology and the rest. Students usually don't ask
what good Shakespeare is, or the story of the American Civil War, or
Hitler. References to such things pervade everything we read, or see in
the movies, or hear our relatives talking about, from early childhood; and
so we -- as children -- simply understand that knowing about them is like
knowing how to speak and read and write at all. Similarly, children
tend to know the necessity of counting change when making a purchase with
a five dollar bill, and counting in general, since they see it around them
every day, just as they see people speaking English.
When I say "it is a pity" that the trig question comes up at all,
what I really mean is that children usually get the mistaken idea,
illustrated by this question, that there are two different justifications
for learning things, by which the things themselves are classified:
The first class consists of things that are obviously to be
learned as part of general culture they see about them as soon as they
learn to read. They need to know what is a big bad wolf, a princess, and
change from a five-dollar bill. These may or may not be practical things
(they will never be frightened by a wolf, or see a princess), but they are
so ordinary they bring up no questions of the form, "Why must we learn
Then there are things that are not seen every day. Trigonometry
and Shakespeare seem to be examples of the second sort. These are just as
much part of our culture (I will say more about this below) as the big bad
wolf and the five dollar bill, but since they are not often seen in early
years they tend (in the schools) to be regarded either as impositions on
childhood and its ineffable spirituality, or as needing a different sort
of justification from what is so obviously relevant in earlier childhood
A second kind of justification therefore gets invented, the
so-called "practicality" of mathematics in particular, as if mathematics
were different from other forms of literature in furnishing the alert mind
with ways to understand and maybe control the universe. It is a curious
and unfortunate feature of our present civilization that nobody thinks it
necessary to question or defend the value of the corresponding knowledge
of literature or music on similar grounds.
If we defended the reading of Shakespeare on the grounds that
novelists and poets need it to help them write their own books, we would
be as bad off as we now are in defending mathematics. Why study
Shakespeare when you just know you're not going to be a professional
We don't defend Shakespeare on such crass grounds of practicality,
because the question is not even posed. If it were, if someone really
were to ask, "Why study literature? What do I need it for?", we would say
in all seriousness, "There is simply no telling when or whether you will
need it. You need it every day. It is part of your intellectual
There might be those who are still not satisfied. Certainly the
analogous answer in the case of trigonometry is not widely believed. We
might go on (in the case of Shakespeare, or other literature, say) with
something like this:
"Literature is like some obscure muscle in your hand: if it is
weak, your hand won't work as well, but it would be very hard to specify
just which task would be rendered just how much more difficult by its
weakness. You want your hand to be strong in all its parts, and to be
able to call on reserves of strength whenever you need them, even while
not at the time recognizing which reserves they are.
"Thus an athlete exercises his whole body, and not just those
muscles he fancies are the ones he will be using in next week's game.
With proper exercise his whole body is more capable; that is enough
explanation, and it works. Why isn't the same argument also given for
the mental exercise of charting the geometric relationships by which we
all view the world?"
This argument is not, by the way, the "transfer of training" thesis
popular a hundred years ago, saying that strengthening the mind with
mathematics also strengthens its other abilities ipso facto. The
analogous physical argument would say that exercising the fingers somehow
carries over into stronger abdominal muscles. Mental exercise was once
thought to carry over in this sense, but later psychologists made
experiments to show this was not so, and that discipline in algebra did
not, for example, strengthen the power to memorize phone numbers.
Actually, that question is still not entirely settled, but it is not
needed, either. We don't have to justify teaching trigonometry on the
grounds that it helps you argue cases in court if you become a lawyer.
(Actually, I myself believe it does. I have a daughter who is a lawyer,
and who studied calculus in college, and she is not sorry she did.)
So "transfer of training" is questionable, and measuring flagpoles
is laughable. And to say that engineers and scientists need trigonometry
is not enough either, since those with no intention of becoming scientists
and engineers remain unanswered. If there is an answer, it cannot be
The true answer is that trigonometry is part of our culture, and
should be as visible in daily life as anything in Dickens or
Shakespeare. Why can we not make it so? Why is our daily vocabulary so
poor that while we can speak of Communism, or the national debt, no
newspaper ever prints an algebraic formula and no columnist has ever
been known to mention the secant of the angle of inclination? These
things are within our power, and would increase our ability to describe
the world and make reference to its properties quite as much as does our
ability to quote Mr. Bumble or Hamlet.
We would even see the world a little differently. One who has
learned trigonometry walks a little taller as a result. Like any other
part of mathematics, or literature, or history, trigonometry furnishes the
mind with frameworks that render the experienced universe more
understandable, every day.
Part of it is practical, to be sure, but it is a disastrous mistake
to present this part as if the only, or most important value. How can we
present it otherwise, then? Merely telling the kids they will walk a
little taller won't do the job. (It might even be called HEIGHTISM.)
The subject itself has somehow to be presented in such a way as to
convey the lesson of its own interest. Just as an athlete told to
exercise only needs to know his body feels better and stronger as a
result, without necessarily testing the results of each day's exercise
by counting his score that day on the golf course, the intellectual
exerciser should get his exercise in a way that makes him feel
mentally stronger and happier. If the exercise is given without context
it will usually fail. In the past, with most people, it has failed.
It sometimes happens that age and experience alone will provide
the context, and that a person who has been out in the world a while will
see the beauty, and indeed utility, of something like trigonometry simply
because he has more to hang it from, as it were. After World War II, for
example, the returning veterans came to college three years older, on
average, than the usual college students, and their work showed it. There
was never a generation of students like that one. Alas, to delay education
until it is that late entails a great loss of time, and we must find
I would like to illustrate with a story about my father, who came here
from Poland as a young man, never having had any mathematical or
scientific education of any kind. He liked the idea of my being a
mathematician, but did not understand what that was, except dimly.
Something like what Einstein did, maybe, but he knew nothing about
Einstein's work either.
My father was a rough sort of amateur carpenter; as a small
businessman he often built things for his store: shelving, counters,
window displays. One time, when he was already sixty years old, he came
to me with the following observation:
In building a vertical frame for a set of shelves he would brace the
vertical plank with a diagonal piece of wood, one end nailed to the floor
a foot away, say, and the other to the vertical object. The brace was not
necessarily at a 45 degree angle, but depended on certain limitations of
the rest of the structure.
When he attached it at a point 8 inches up from the floor, he
explained to me, the diagonal piece would have to be longer than the 12
inch baseline, of course (he measured it as being about 2 and a half
inches longer than one foot). But if he wanted to attach it twice as far
from the floor, i.e. 16 inches up, the diagonal piece had to be more
than twice as much longer. He measured it as about 20 inches from the
floor, that is, 8 inches longer than the 12 inch base line, and not 5
inches longer, which he would have expected from having attached it twice
as high as the other one. Why? His language was not technical, of
course, but what he was saying was that the increments of diagonal length
were not proportional to the increments of height.
Now this was not really a practical problem, even though it began
with carpentry. He had had no difficulty all his life making scale
drawings and cutting pieces of wood accordingly. He had never heard of
trigonometry. But here he was, at age 60, suddenly seized with curiosity
about a phenomenon he had not previously thought about analytically. Why
should it be that in making a brace to attach 16 inches from the floor you
increase the 12 inch baseline more than twice as much as you have to
increase it when attaching one at 8 inches?
Of course what he had discovered was (if one wished to put it so)
the behavior of the cosecant of the angle his brace made with the floor.
More simply, he was observing the Pythagorean theorem in action, where
even without invoking angles one can chart the disproportions as height
I say "of course" here to emphasize that what is being described
is no more difficult than the speech Polonius gives Laertes, and should
be as much part of our vocabulary, if we are to make easy reference to the
world as we know it, even as we daily see it with our own eyes. To the
question "why" the cosecant function behaves that way there was really no
answer, except that Euclidean space is that way, the Pythagorean
relationship being in effect the definition of the Euclidean space (also a
philosophical observation of cultural interest!). But I was able to show
him some other examples of the relationship that he had observed, with a
few diagrams to illustrate the ratios as the angle approaches zero and as
the angle approaches 90 degrees.
Well, he found that very interesting. He had thought this was
the sort of thing mathematicians might know about, but hadn't been sure.
What really astonished him was my telling him that not only had these
facts been known for several thousand years, but that the results were
tabulated in as much detail as the tables of interest payments on
mortgages, and had been used by astronomers since the time of Ptolemy, two
thousand years ago, where they were essential in the prediction of the
conjunctions of planets and the like. And that furthermore, without
drawing any pictures or scale diagrams I could, by means of tabulated
information, predict the length of his brace for any desired height of
attachment, whether he used a baseline of 12 inches or any other!
He had no need to see the tables of sines and cosines -- for his own
rough carpentry such information was of no value -- but I showed him a
book of tables anyhow. (Today, with calculators available, I would have
had to use something else to prove to him that his observations were both
important and old.)
Think what he might have learned, and how much more he would have
appreciated about the history of mankind, if he had had a few weeks in his
youth to calculate a few triangles, with a teacher and textbook presenting
the problem in the way he had come by himself, in sixty years, to
appreciate it. He had read, in his romantic youth, of the ancient Code of
Hammurabi; why could he not also have learned about Pythagorean triples
and their consequent trigonometric tables as found on the famous Tablet
#322, also from Old Babylonia, now residing in the Plimpton collection at
Columbia University? We no longer live by the laws of Hammurabi, but the
Plimpton tables are as valid as ever.
If I had told him about flagpoles and shelf bracing for their
sakes only, he would not have been interested; he has known as much as he
needed about shelves and flagpoles since childhood. He understood quite
well, as our discussion went on, that his shelves were merely the language
in which the relationships that so fascinated him were made vivid.
Even without the actual numbers, the behavior of the diagonal in the
vicinity of zero and ninety degrees was more interesting to him than all
the rest put together. This was perhaps the only time in his life that he
had an insight into what mathematics is.
What good did it do him? I wish he were alive today, to answer that
question for you himself.