### Introduction

Our discussion so far has been limited to right-angled triangles where, apart from the right-angle itself, all angles are necessarily less than $9{0}^{\∘}$ . We now extend the definitions of the trigonometric functions to any size of angle, which greatly broadens the range of applications of trigonometry.

#### Prerequisites

- have a basic knowledge of the geometry of triangles

#### Learning Outcomes

- express angles in radians
- define trigonometric functions generally
- sketch the graphs of the three main trigonometric functions: $sin,\phantom{\rule{1em}{0ex}}cos,\phantom{\rule{1em}{0ex}}tan$

#### Contents

1 Trigonometric functions for any size angle1.1 The radian

2 General definitions of trigonometric functions

2.1 The sine and cosine of an angle

2.2 Sine and cosine in the four quadrants

3 Graphs of trigonometric functions

3.1 Graphs of sin $\theta $ and cos $\theta $

3.2 The graph of $tan\phantom{\rule{0.3em}{0ex}}\theta $

4 Engineering Example 4

4.1 Optical interference fringes due to a glass plate