Axioms of Trigonometry

Question

On Wikipedia it gives a picture of all trigonometric functions of an angle laid atop the unit circle, 1. Obviously there are other trigonometric identities, but what I'm wondering is, does Trigonometry have a list of axioms, or is it just a special case of analytic geometry? And if so, how does it fit into the rest of mathematics, because I seem to see it everywhere.

Answer 1

I'd say the modern viewpoint is that the trigonometric functions are best viewed through the lens of complex analysis. From this vantage point, there's no real "axioms of trigonometry." In particular, we define:

$$\cos(z) = \frac{1}{2}\left(e^{iz}+e^{-iz} \right) \qquad \sin(z) = \frac{1}{2i}\left(e^{iz}-e^{-iz} \right)$$

Note that all of the transcendental functions appearing above can be defined as solutions to certain initial value problems:

• The exponential function is the unique smooth function $f:\mathbb{C} \rightarrow \mathbb{C}$ satisfying: $$f'(z) = f(z), \qquad f(0) = 1$$

• The cosine function is the unique smooth function $f:\mathbb{C} \rightarrow \mathbb{C}$ satisfying:

$$f''(z) = -f(z), \qquad f(0) = 1, \,f'(0) = 0$$

• The sine function is the unique smooth function $f:\mathbb{C} \rightarrow \mathbb{C}$ satisfying:

$$f''(z) = -f(z), \qquad f(0) = 0, \,f'(0) = 1$$

This means that the exponential function is an eigenvector of complex differentiation (with eigenvalue $1$), and that sine and cosine are eigenvectors of twice-iterated complex differentiation (with eigenvalue $-1$).

That notwithstanding, Euclidean geometry that can certainly be given quite an abstract treatment; see my question here, and in particular, be sure to check out Audin's Geometry. This book easily deserves a 5-star rating. But I'd say the trigonometric functions "come first" so-to speak, and they exist independent of geometry.

Answer 2

As trigonometry is normally conceived and taught at the school level, there's no need for axioms:

The basic concepts of trigonometry are points, rays, triangles (and ratios of their side lengths), and circles, whose axioms come from Euclidean (or spherical, or hyperbolic) plane geometry.

Alternatively, as you note, trigonometry can be expressed in Cartesian (i.e., coordinate) geometry, whose properties rest on axioms for the real numbers.

That is, trigonometry is really a collection of results about ratios of sides of right triangles, locations of points on the unit circle, certain "analytic" functions, and the web of interrelationships between these viewpoints.

That said, there are multiple approaches to trigonometry (Euclidean-geometric, coordinate-geometric, analytic...), each with their own definitions and theorems.

It's difficult to enumerate all the ways trigonometry fits into mathematics, because there are so many places where circles appear openly, or lurk beneath the surface. (Fourier analysis comes immediately to mind.)