Introduction

Introduction

We are already familiar with algebraic equations such as $x^2 - 5x + 6 = 0$, which are satisfied by only finitely many values of $x$. In this chapter we turn instead to trigonometric equations -- equations involving one or more trigonometric functions of an unknown angle. Because every trigonometric function repeats itself after one full period, a trigonometric equation behaves very differently from an algebraic one: once a single solution is found, infinitely many more (obtained by adding multiples of the period) satisfy it too.

This chapter first pins down the solutions lying inside a single period -- the principal solutions -- and then builds a single formula, the general solution, that captures every solution at once. It then turns to a second classical application of trigonometry: solving a triangle, using polar co-ordinates, the Sine Rule, the Cosine Rule, and the Projection Rule. Finally, the chapter introduces the inverse trigonometric functions, along with their properties and principal values. Trigonometric functions -- and the equations built from them -- go on to play an important role in integral calculus, studied later.