Freshman's definition of sin(x)?

Question

I would like to know how you would rigorously introduce the trigonometric functions ($\sin(x)$ 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 length" 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.

Answer 1

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

http://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

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...

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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

${(0,0),(x,0),(x,y)}$

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.

Answer 6

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 http://www.jstor.org/stable/2686787 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)

Answer 7

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!