Introduction

Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. The name itself comes from two Greek words — trigonon (triangle) and metron (to measure) — so trigonometry literally means "measuring triangles."

The subject has a long history. Greek mathematicians first used trigonometric ratios to find unknown distances, while the Egyptians had already used a primitive form of it to build the pyramids around the second millennium BCE. Aristarchus used trigonometric reasoning to estimate the distances to the Moon and the Sun, and Eratosthenes was the first to calculate the Earth's circumference using a unit called the stadion. Hipparchus is generally credited as the founder of trigonometry as a systematic subject, and his ideas fed into Ptolemy's influential theory of astronomy — the same Ptolemy whose reflection on tracing "the windings to and fro of the heavenly bodies" opens this chapter.

India's role was pivotal. The astronomer-mathematician Aryabhata defined sine, cosine, versine ($1-\cos$) and inverse sine, and packed his results into just 108 verses — including a formula for the area of a triangle. Brahmagupta, Bhaskara I and Bhaskara II carried this work further.

Trigonometry only became a subject in its own right — separate from pure geometry — through the work of Johann Bernoulli and Leonhard Euler; Euler in particular connected trigonometric functions to complex exponentials, one of the deepest links in all of mathematics. Later, Joseph Fourier's work on trigonometric series (what we now call Fourier series) made it possible to describe almost any periodic phenomenon as a sum of sines and cosines, with applications from vibration analysis and electrical engineering to acoustics, optics, signal/image processing and quantum mechanics.

Trigonometry today. Trigonometric functions are now understood as ordinary mathematical functions of an angle (or, as this chapter develops, of any real number), used across nearly every branch of mathematics and its applications — including two things you rely on every day: the GPS in a car or phone fixes your position using trigonometric calculations, and medical scans such as CT and MRI, used in detecting tumours, are built on sine and cosine functions.