Geometry and Trigonometry

Abstract

Geometry is the oldest of the mathematical sciences. Its age-old theorems and the sharp logic of its proofs make you think of the words of Andrew Wiles, “Mathematics seems to have a permanence that nothing else has.”

This chapter is bound to take you away from the geometry of the ancients, with figures and pictorial intuition, and bring you to the science of numbers and equations that geometry has become today. In a dense exposition we have packed vectors and their applications, analytical geometry in the plane and in space, some applications of integral calculus to geometry, followed by a list of problems with Euclidean flavor but based on algebraic and combinatorial ideas. Special attention is given to conics and quadrics, for their study already contains the germs of differential and algebraic geometry.

Four subsections are devoted to geometry’s little sister, trigonometry. We insist on trigonometric identities, repeated in subsequent sections from different perspectives: Euler’s formula, trigonometric substitutions, and telescopic summation and multiplication.

Since geometry lies at the foundation of mathematics, its presence could already be felt in the sections on linear algebra and multivariable calculus. It will resurface again in the chapter on combinatorics.