The Bend in the River

Trigonometry

I feel the need to expound some kind of philosophy about trigonometry.

Trigonometry is, somewhat unfairly, viewed as a collection of formulae that one is subjected to and that have no inherent interest on their own. While this is true of many modern treatments of the subject, including most (if not all) of those aimed at the secondary school level, I do think that trigonometry as a theory has many nice properties that make it an attractive introduction to mathematical thought for people with only rudimentary knowledge of the subject.

1. Trigonometry as a subject is easy to motivate: in surveying, in computer graphics, and basically any other time we want to approximate a well-behaved surface in a way that is both easy to compute and easy to improve, the use of triangles is ubiquitous.
2. Trigonometry has very few formal prerequisites. There is no need to deal with the real numbers (as in calculus), or with the arithmetic of polynomials (as in classical algebra).
3. Conversely, the few prerequisites that trigonometry does have are interesting in and of themselves, and a proper treatment of trigonometry will present a new framework on which to hang some of the nice elementary theorems of plane geometry.
4. The foundations of trigonometry are easy to hand-wave away without too much guilt: the continuity of the trigonometric functions, for example, is non-trivial to prove with bare hands — but if we define the trigonometric functions by circle-segmenting, then the continuity follows intuitively from the continuity of the circle (which we can take to be obvious).
5. There are no difficult theorems. By this, I mean two things: (a) all the theorems are quite `small’ and very easy to state (usually a simple equality, and easy to draw pictures of); (b) all the proofs are elementary enough that the motivated secondary school student is able to generate them with only a little effort.
6. On the other hand, most theorems can be proved in a variety of different ways, which allows us to give aesthetically `nicer’ proofs than the elementary angle-and-length-pushing arguments with relatively little additional effort.
7. The theory that is built up is self-contained, elegant, and can be summarised nicely in a short collection of classification theorems. (It can even be summarised on the back of an envelope.)
8. Finally, the theory has some very beautiful applications within mathematics (for example, in complex analysis), and can be re-built from a variety of foundations in the future (for example, with the use of power series), each of which exhibits the theory in a slightly different way.

I think it is obvious, but nonetheless I must point out that the majority of the arguments above hold iff the theory is presented in a way that has two features:

• The student must do a non-negligible amount of work, both stating and proving theorems. (Luckily in this subject it is even easy for students to come up with their own theorems to prove!)
• The subject must be presented with the correct balance of computational and theoretical examples, and to avoid dryness this balance will be significantly in favour of theory-building to avoid dryness. (Computations are not to be avoided, they just must be non-trivial.)

The Victorians did a lot of trigonometry, most of it with a very Victorian flavour (series of trigonometric functions, identities with integrals, and so forth), and a lot of it is accessible without too much effort: the only sacrifice needed is to spend less time on the silly, unmotivated, and formulaic problems that seem to stand in for doing proper mathematics; and above all, to spend precisely as much time that is necessary, and no more.