8 - Trigonometry

Summary

The main concerns of this chapter are to define and then to investigate the notions of sine and cosine in Minkowski space. These functions are connected to the two concepts of perpendicularity that we have called normality (Definition 3.2.2) and transversality (Definitions 4.6.2 and 5.5.2). We then go on to show that these trigonometric functions retain some features of the Euclidean case. In particular, there is a sine formula for triangles and a variety of trigonometric identities. Both functions are defined in a natural way by evaluating a linear functional at a vector. The formula for cosine is the more self-evident; because it is desirable for the sine function to be related to area and volume by the usual formulas for the volume of a parallelotope, its definition depends on the choice of area function. As far as possible we shall leave this undetermined, and speak of the function σ, the isoperimetrix I, its polar I° and the normalization Ĩ of I, introduced in Definition 5.3.6, that has the property that dμ(Ĩ) = μ(∂Ĩ). In some cases we shall need to be specific about the choice of area function. The last two sections deal largely with two-dimensional subspaces of a Minkowski space X and so form a sequel to Chapter 4.