Trigonometry

Abstract

The word trigonometry comes from the Greek for “triangle measurement.” More specifically, it means the study of relationships between the size of sides and the size of angles in triangles. Euclid says very little about this. He has theorems about equal angles and the sum of angles, and one angle being twice another or simply larger than another, but he never actually measures angles. He does not represent angles by numbers, nor does he represent them by lengths or areas. This suggests that angle measure may be a deep concept, perhaps beyond the scope of traditional geometry. The Greeks had some inkling of this when they tried unsuccessfully to construct the area bounded by the unit circle, the problem they called squaring the circle. In modern terms, squaring the circle amounts to constructing the number π, which is both the area of the unit circle and half its circumference. It is also the natural measure of the straight angle, formed by two right angles, so constructing π is in fact a fundamental question about the measurement of angles.