The best way to introduce trigonometric functions in a rigorous analysis course

Question

This is something I have always had issues with. Generally, three approaches are used:

1. The geometric path: this follows the standard way how you would introduce these functions in school. The problem is that it is almost impossible to do this in a rigorous way and you have to introduce difficult concepts (like arc-length) what you would normally do later on.
2. The power-series road: the advantage is that this is quick, elegant, and you can prove all the properties in a quick and not too painful way. The problem is that for many students this is ad-hoc, the definition is unmotivated, and there is no connection with the geometric meaning.
3. The functional equation road: to define them through functional identities and continuity, and then prove properties. This is halfway motivated because at least addition formulae can be assumed from school, but, as I had it this way as I was a student, I have some reservations because I remember that this approach left some odd feeling.

If your students had a calculus class, then clearly the second approach can be used and this is what I do with those students who have this class after a calculus class.

But generally, in my university, it is common that first year students start their studies with a rigorous analysis course. Which approach do you consider as the best in this case, where I cannot assume much familiarity with calculus knowledge and also do not want to wait till power series appear?

Answer 1

I suppose my preference would be

    4. The differential equations road.

That is, define $x(t) = \cos t$ and $y(t) = \sin t$ to be the solutions to the following initial value problem: $$ \frac{dx}{dt} = -y,\qquad \frac{dy}{dt} = x,\qquad x(0) = 1,\qquad y(0)=0. $$ This assumes that you've proven the existence and uniqueness theorem for solutions to ODE's, which I would certainly want to cover in any two-semester analysis course.

This treatment has the following advantages:

1. It is 100% rigorous.

2. It is related to geometry. The differential equations essentially say that $(x,y) = (\cos t,\sin t)$ is a unit-speed parametrization of the unit circle. Indeed, the idea of a unit-speed parametrization of the unit circle is essentially a rigorous definition of "angle", so this is really very close to the geometric path.

3. It leads directly to the power series for $\cos t$ and $\sin t$. Specifically, it's very easy to show that the power series for sine and cosine are solutions to the given initial-value problem, so they're equal to $\cos t$ and $\sin t$ by the uniqueness theorem for ODE's. Actually, this is the most coherent explanation I know for why the sine and cosine are analytic functions.

4. A very similar treatment is available for $e^x$, so you can use the same approach for all of your transcendental functions.

5. It provides a nice application of the existence and uniqueness theorem, which otherwise seems strangely disconnected from the rest of analysis.

Answer 2

This aspect is discussed in great detail in §7.2 of A Course in Calculus and Mathematical Analysis by Ghorpade and Limaye. The discussion must be accessible to students undergoing their first rigorous course in analysis.

The discussion is well-motivated: the authors point out in §7.1 that one can integrate all functions of the form $x^r$ for all rational $r \neq 1$ in terms of functions of the same kind. So to integrate $x \mapsto 1/x$, they introduce the logarithm function and study its properties. Now, of course, one has to verify that this is a genuinely new function: they show that $x \mapsto \ln(x), x > 0$ is not a rational function (cf. Lemma 7.26, loc. cit.): in fact, it is not even an algebraic function (cf. Proposition 7.27, loc. cit.).

Now, the next natural question is integrating inverses of quadratic polynomials. If the polynomial factors into (necessarily) linear factors, one might use partial fractions. However, if the polynomial does not factor over $\mathbf{R}$, then, we are stuck again! One may show using the method of partial fraction that it suffices to learn to integrate the function $x \mapsto 1/(1+x^2)$. They then introduce the arctangent function as the integral of $1/(1+x^2)$ and go on to prove its familiar properties. The other trigonometric and inverse trigonometric functions are defined from just this function (by algebraic formulas or by using the monotonicity to establish the existence of an inverse, as applicable). Once again, the authors prove that the trigonometric functions are transcendental (cf. Proposition 7.29, loc. cit.).

Finally, let me reproduce the following quote from the linked book:

The approach outlined above gives not only a precise definition of the logarithmic, exponential and the trigonometric functions, but also a genuine motivation for introducing the same. In most texts on calculus, the trigonometric functions are ‘defined’ by drawing triangles and mentioning that angles are ‘measured’ in radians. The main problem with this approach is succinctly described by Hardy [...], who writes: “The whole difficulty lies in the question, what is the x which occurs in cos x and sin x.” Hardy also describes different methods to develop an analytic theory (cf. [...]) and the approach we have chosen is one of them.

Finally, I would like to point out that this method is very similar to how we introduce the number systems to students (by showing them equations "in those systems" you cannot solve by staying there!).

The authors also point out:

Failure to be able to integrate a function has often led to interesting developments in mathematics. For example, a rich and fascinating theory of the so-called elliptic integrals and elliptic functions arises in this way.

I hope this helps.

Answer 3

My answer goes a bit sideways: I would say that it is not necessary to rigorously define the trigonometric functions in a rigorous analysis course that uses them. And in fact, this applies much more broadly (e.g. one can do a course on integration without using any particular integration theory).

In any case, we never, ever define everything we use in courses. There always are black boxes, as we do not go back to founding all mathematics from axioms before many years of study (if ever). And this ok: of course we learn how to add and multiply numbers before we rigorously construct them. We should do rigorous proofs before having founded everything from axioms.

An important point is to make as clear as possible what you do not prove, and what you ask your student to accept (until they know more and can clear it up). It is in principle possible to use any approach to trigonometric function:

Consider what you need from them and, more importantly, what kind of proofs you feel are most useful to show in your course, and choose the definition for trigonometric function accordingly.

If the chosen definition feels unnatural to you, or unrelated to previous definitions student have seen, then call it a "starting property" instead and ask your student to accept that sin, cos, etc. has this property and that it gives a characterization of the given function (of course, try to give them a feeling why this property is not unexpected given their current knowledge).

But it would be a loss to get constrained into proofs or considerations that are not central to the course by a definition.

Answer 4

Isn't the "functional equation road" the same thing as the "differential equations road"?

You can take an axiomatic approach to defining sine and cosine. See Apostol's Calculus book, page 95. Many other authors have done a similar thing and the background isn't too steep. You simply declare that there exist two functions satisfying a few properties (which can be geometrically motivated). Then later in your course you prove the existence portion in a rigorous fashion. I don't see anything wrong with this.

Answer 5

I would use polar coordinates. "Everyone" is familiar with working in Cartesian coordinates. Not everyone knows how to translate these into polar coordinates. But the fact of the matter is that polar coordinates make (easily) accessible a number of applications and "spaces" that would be difficult to manage using Cartesian coordinates.