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:
- 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 $x$.
- The "power series way": define $\sin(x)$ as $\Sigma_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$.
- 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 $\exp(ix)$.
- 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.
Best Answer
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:
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...