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I would like to know how you would rigorously introduce the trigonometric funcions ($sin(x)$ and relatives) to first year calculus students. Suppose they have a reasonable definition of $\mathbb{R}$ (as Cauchy closure of the rationals, or as Dedekind cuts, or whatever), but otherwise require as few concepts as possible.

Some approaches I can think of are:

  1. The "geometric way": $\sin(x)$ is the ordinate, on the usual unit-radius "trigonometric circle" in the $xy$-plane, of the end point of a circle arc of length equal to $x$.
  2. The "power series way": define $\sin(x)$ as $\Sigma_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$.
  3. The "complex exponential way": let $\exp:\mathbb{C}\rightarrow\mathbb{C}^{*}$ be the unique homomorphism of groups (blah blah), and define $\sin(x)$, $x$ real, to be the imaginary part of $\exp(ix)$.
  4. The "differential equation way": $\sin(x)$ is the unique function $u(x)$ of class $\mathcal{C}^{\infty}$ such that $u''+u=0$, $u(0)=0$ and $u'(0)=1$.

Unfortunately, it seems to me that each of the above approaches has some drawbacks (developing the elementary properties of trigonometric funcions from some of these definitions may be not so straightforward), and need some "non elementary" notions to be introduced, where by "non elementary" here I mean notions involving e.g. the concept of limit or of derivative. One would like the standard functions like $\sin(x)$, $\cos(x)$ and $\exp(x)$ to be already available to the students before introducing limits, derivatives or integrals, let alone power series or differential equations.

Edit (example): when I was a first year student, the reals had been introduced axiomatically (in disguise) as an [but it was implicitely assumed that it was unique] ordered field with the "sup" property; but this is irrelevant: a lot of undergrads see the definition of $\mathbb{R}$ via Dedekind cuts [which is the definition is usually given in second or 3rd year of high school]. Then $\sin(x)$ was introduced as in 1 (geometric way). Then limits, continuity etc. were introduced (so, it made sense to ask "find the limit of $\sin(x)$ as $x\rightarrow0$"). My point is that the "geometric" definition 1 is actually cheating, as it already requires limits and differentiation: what is the "arc lenght" of the circle otherwise?

Edit: btw, I don't have to teach calculus to anybody now, I just asked myself this question by reading other m.o. questions related to teaching.

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I'm having a hard time imagining a calculus class in which one introduces the Cauchy closure of the rationals, the function $\sin(x)$, and the concept of limit in that order. – Mark Meckes Sep 27 '10 at 16:47
I didn't mean to imply that such an approach was circular or inconsistent, but that I can't imagine why anyone would feel it made pedagogical sense. To discuss what order I would follow in part requires clarifying what kind of class we're talking about. In the US, "first year calculus" is usually a very different class from "undergraduate real analysis", and it sounds like you're talking about something more like the latter. In a real analysis class, I don't feel any need to introduce the functions you mention until there's plenty of general machinery to make any of your approaches precise. – Mark Meckes Sep 27 '10 at 17:03
In a freshman calculus class, on the other hand, I have no problem with relying on an informal geometric definition of $\sin(x)$. – Mark Meckes Sep 27 '10 at 17:04
I'd like to point out that high school teachers somehow manage to introduce ("define" is really not the right word since it carries too much baggage us mathematicians) the sine function on the domain from zero degrees to 90 degrees without worrying about all these things. U suggest that you should do the same in a freshman class. Of course, if a student complains to you about the lack of a proper definition, get that student into a proper real analysis course ASAP. – Deane Yang Sep 27 '10 at 17:48
In what country do high school students learn about Dedekind cuts? – Qiaochu Yuan Sep 27 '10 at 17:49

7 Answers 7

I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Edit: my favourite introduction to the exponential function goes as follows (after having defined the derivative): having differentiated polynomials, one gets the natural urge to differentiate something like $2^x$. If one goes through the definition of the derivative, one gets as an answer some limit (if it exists) times the function itself. Repeating this with $3^x$ gives the same result, but a different limit. A natural question then is: can we choose a base to make this limit 1, so that the derivative of the function gives you the function back? If you go through the algebra, you will arrive at the number $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$.

Edit 2: To address the recent edit of the question: I don't think that using the 1st method is cheating. You can introduce $sin(\alpha)$ as the ratio of the side opposite to the angle $\alpha$ to the hypotenuse in a right-angled triangle. They already know from high school that this is well defined and I don't think that that proof uses disguised limits either.

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Introducing the number e in relation to continuously compounded interest is one way which avoids differentiation or integration, and I think is the way most people see it the first time. – Steven Gubkin Sep 27 '10 at 17:07
@Steven: that approach still uses (implicitly or explicitly) the idea of a limit, so it's not clearly any better by the OP's criteria. – Mark Meckes Sep 27 '10 at 17:10
To introduce $\mathbb{R}$ there's no need of limits: at high school (second or third year) we were given the (perfectly rigorous) definition of $\mathbb{R}$ via "Dedekind cuts". – Qfwfq Sep 27 '10 at 17:16
You are right, I almost forgot about Dedekind cuts as a way of defining the reals. Personally, I don't like it, since it also seems to go out of the way to avoid doing the natural thing - limits of Cauchy sequences. In particular, it's not immediate from the Dedekind cut definition that reals have decimal expansions, at least much less clear than from the Cauchy sequence definition. But I suppose it's a matter of taste. I do believe that the order outlined in my answer should work, if you choose to proceed in this way. – Alex B. Sep 27 '10 at 17:22
I totally agree with Alex' answer. The goal of a Calculus course is to get the students in touch with some concepts in the fastest and most accurate way, to prepare the ground. It is not to build a formal system from scratch at the beginning of every fall. If the student has a clear understanding of what is going on, that some definitions need justification, and that circularity must be avoided, then I would be quite satisfied. – Piero D'Ancona Sep 28 '10 at 10:25

When I was first taught analysis, I do remember things being in a slightly strange order. What annoyed me most was how we were expected to do an exercise involving say, the sine function before we had rigorously defined it. I remember thinking that we were being very careful and slow about all our elementary definitions of reals and sequences and series and yet jumping ahead and being asked whether or not $\sin\tfrac{1}{n}$ converged, but I turned out OK in the end. The point obviously was that everybody knew what sine was and what it did already and it would be silly to try to hide that fact and thinking about $\sin\tfrac{1}{n}$ would test our knowledge in a perfectly reasonable way.

When things were arrived at rigorously, the layers were added on to to our definition. e.g. Once power series are done 'properly', you can then say "here's a definition of sine". Once you point out that the theory is the same for complex numbers you could say "here's another" (the im part or exp as you say). Then once you have differentiation you can write down the ODE and say here's another.

Looking back now though, I'd say don't be too hard on yourself as a teacher: Perhaps your students needn't have all the answers at every step of the way. It's OK if you "cheat" in a couple of places, surely? They might come away once or twice feeling like you haven't told them everything but you just can't hope to. Hopefully they invest enough effort along the way that at the end they have all the different facts/definitions to reconcile.

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I think ‘cheating’ a little like this can be fine, so long as you’re completely clear to the students about when you’re doing so. Without this, it can be (as Spencer says) both annoying and confusing for them. I had a lecturer in undergrad who took this sort of approach, very effectively, explaining it at the beginning of the course with something along the lines of: “Yes, we will sometimes be dishonest. But when we are, we will at least be honest about it!” I wish I could remember who it was! – Peter LeFanu Lumsdaine Sep 27 '10 at 18:18

I am fond of distinguishing between the "pre-rigorous", "rigorous", and "post-rigorous" phases of mathematical education, see

For the "pre-rigorous" stage (which, in the US, is basically everything up to undergraduate calculus), I don't see a pressing need for necessarily introducing and working with a concept (e.g. the sine function) before the rigorous foundations for that concept have been introduced; an informal appeal to Euclidean geometry should suffice at this stage.

Things do get more interesting at the "rigorous" stage (which, in the US, roughly starts at a good undergraduate real analysis class), when students already have plenty of pre-rigorous exposure to real numbers, limits, special functions, etc. but are now ready to revisit these concepts from a rigorous foundational point of view. In my own textbook at this level, I proceed by this route:

  • Define rational numbers
  • Define Cauchy sequences of rational numbers, and equivalence of Cauchy sequences
  • Define reals as the space of Cauchy sequences of rationals modulo equivalence
  • Define limits (and other basic operations) in the reals
  • Cover a lot of foundational material including: complex numbers, power series, differentiation, and the complex exponential
  • Eventually (Chapter 15!) define the trigonometric functions via the complex exponential. Then show the equivalence to other definitions.

But certainly one can proceed in a different order to the above.

At the post-rigorous level, one can view of course trig functions as special cases of much more general operations, such as the exponential operation on a Lie algebra...

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Lakatos, in his article "What does a mathematical proof prove?," among other places, argued for a similiar trichotomy---for him, it was "pre-formal proofs," "formal proofs," and "post-formal proofs." His development of the notion of informal mathematics (a single piece of which may admit many formalizations) may provide a framework for discussing questions like this. (How should one formalize the informal sine function for freshmen? And how should one judge various candidate formalizations?) – user2490 Sep 29 '10 at 3:12

A number of textbooks introduce the exponential function $e^x$ as the inverse of $\log x$, after defining $\log x$ as the integral of $1/x$. The properties of $\log x$ are found from the definition as an integral. It would be consistent with this approach to define $\arcsin x$ or $\arctan x$ via integrals and deduce their properties.

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This is the path my high school calculus class took (which was closer to an analysis class), and I think there is a certain beauty to it. Showing the exponential function is it's own derivative in using these definitions is especially pretty. – Steven Gubkin Sep 27 '10 at 21:40
Yes, and this way of doing things generalizes nicely to studying elliptic functions too. – J. M. Sep 28 '10 at 0:09
It's a very pretty way of introducing $e$, but it requires even more analytic machinery than to define it via differentiation, since you need some notion of integral. The OP would like to introduce these functions as early as possible, so this might not be the best way to go for him. – Alex B. Sep 28 '10 at 1:57

The geometric "definition" of the trigonometric functions (approach 1 above) rests not so much on the general definition of arc length but on the more restricted notion of circumference. In this context, even the notion of circumference is really only an encoding of the notion of revolution.

Since we have the real numbers, it is safe to assume that we also have set theory and so we have at least the cartesian coordinates of the euclidean plane - though not the polar coordinates. In some sense, the question could be posed as "How can you rigorously derive polar coordinates from cartesian coordinates without using the calculus?" In the following we also assume it is fair to assume euclidean plane geometry, including Pythagoras' theorem.

As a transformation of the plane, a full revolution can be trivially defined, but is rather uninteresting and for our purposes, not very useful. However, a half revolution

$\pi : {\mathbb R}^2 \rightarrow {\mathbb R}^2$

can be defined as the composition of a reflection in the $y$-axis with a reflection in the $x$-axis. Now integral fractions of a half revolution can be defined: let $q \ge 2, q \in \mathbb Z$ and define

$\frac {\pi}{q} : {\mathbb R}^2 \rightarrow {\mathbb R}^2$ via $(\frac {\pi}{q})^q = \pi$

i.e., as that transformation of the plane which when composed with itself $q$ times gives back the transformation $\pi$ itself. That such a transformation exists, we conveniently leave as an exercise for the students, as well as the subtlety that there are two such transformations corresponding to the clockwise versus counter-clockwise rotations.

Now let $p \ge 1, p \in \mathbb Z$, and define

$\frac {p}{q} \pi : {\mathbb R}^2 \rightarrow {\mathbb R}^2$ as $\frac {p}{q} \pi = (\frac {\pi}{q})^p$

which gives us any rational fraction of a full revolution, by choosing $p$ to be even.

Though it may seem like going from underpants to profit, considering real numbers as equivalence classes of cauchy sequences of rational numbers, it now seems possible to define real fractions of a revolution without recourse to any machinery that has not already been introduced to the student.

Consider the family of circles centred at the origin. Any ray emanating from the origin is the image of the positive $x$-axis under the transformation $(2a)\pi$ for some $a \in \mathbb R$. Such a ray will intersect each circle in a point $(x,y)$, giving rise to the family of similar right triangles defined by the points


Since the triangles are similar, $\frac {y} {x}, \frac {y} {x^2 + y^2}$ and $\frac {x} {x^2 +y^2}$ are all constant over the family of triangles that is parametrized by $a$, and we can define them as $tan(a), sin(a)$ and $cos(a)$ respectively.

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A similar approach can be used to define exponential functions (although not, as far as I can see, the natural base e). In both cases, there is some working showing (essentially) that the functions that we're defining are continuous. Personally, I prefer Dedekind cuts to Cauchy sequences, and that's fine, since the only continuity that we need here is order-theoretic (although the trig functions have some kinks), which we can deal with ad-hoc without any general notion of limit. – Toby Bartels Apr 4 '11 at 3:46

Let me point you to the article: Circular Reasoning by Fred Richman The College Mathematics Journal Vol. 24, No. 2 (Mar., 1993), pp. 160-162 Some main observations

1) Archimedes said that $\sin x < x < \tan x$ based on an axiom about relative lengths of convex curves ( If two plane curves C and D with the same endpoints are concave in the same direction, and C is included between D and the straight line joining the endpoints, then the length of C is less than the length D.)

2) One can define radian measure using area rather than arc length (this is the approach of Apostol)

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I was quite satisfied with the approach my instructors took with me.

In the first semester course in calculus, we covered some properties about conics, polar coordinates, and don`t go past differentiation, where $\sin(x)$ can be differentiated with the limit definition, independent of how you first define sine. Thus we kept our approach at the high school level, relating the sine and cosine functions to the unit circle.

In second semester calculus, one starts learning about power series, and it is now natural to present the sine and cosine functions in their power series representations, where with the addition of a little complex analysis (also introduced in second semester calculus) can be related to the power series for $e^x$ and indulge Euler's identity.

The differential equation version was never really emphasized, though certainly mentioned after you first find the derivatives of sine and cosine and (hopefully) notice a pattern right away. (I believe this was something along the lines of an assignment problem asking for the $n$-th derivative of $sin(x)$, where $n$ was quite large.)

I suppose this doesn't really answer the question though.. If I were teaching a calculus course to student who had already seen a construction of $\mathbb{R}$, I would go with the power series definitions. The unit circle should really be review from high school, Euler's identity should really focus more on the connection of the functions instead of being a first glance, and the differential equation, well, you can see this from the power series representation anyway!

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