Introduction

Introduction

Trigonometry studies the relationship between the sides and angles of triangles. The name comes from the Greek words trigonon (triangle) and metron (measure) — literally, 'triangle measurement'. Ancient civilisations needed exactly this: the Egyptians used an early form of trigonometry to build the pyramids, Greek astronomers used trigonometric ratios to work out distances they could not measure directly, and Hipparchus (190–120 BC) is credited as the founder of the subject for formulating its general principles.

What you already know. For an acute angle θ inside a right triangle, with the sides labelled relative to θ:

and the three reciprocal functions

Where this chapter goes. The previous chapter introduced the idea of a directed angle — an angle that can have any measure at all, not just between 0° and 90°, and that can be positive or negative depending on the direction of rotation. This chapter extends the definitions above so that sinθ, cosθ, tanθ (and their reciprocals) make sense for any real angle θ, by describing them in terms of the coordinates of a point on a circle rather than the sides of a right triangle. That single change of viewpoint is what lets us later talk about the sign of sin300°, the exact value of cos(−60°), or the domain and range of tanθ as a function on all of $\mathbb{R}$.