Introduction

The Meaning and Origin of Trigonometry

The word trigonometry comes from two Greek words: trigon (meaning a triangle) and metron (meaning to measure). So, at its root, trigonometry is the study of measuring triangles. The subject was first developed to solve geometric problems involving triangles — for example, finding the height of a mountain without climbing it, or the distance to a ship at sea.

Over centuries, its use expanded far beyond pure geometry. Sea captains used it for navigation, surveyors used it to map new lands, and engineers used it in construction. Today, trigonometry is essential in fields as diverse as seismology (studying earthquakes), designing electric circuits, describing the state of an atom in quantum mechanics, predicting ocean tides, analysing musical tones, and much more.

What You Already Know

In earlier classes, you studied trigonometric ratios for acute angles — angles less than $90^\circ$. You defined these ratios using the sides of a right-angled triangle:

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
• $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{1}{\tan \theta}$

You also learned trigonometric identities (like $\sin^2 \theta + \cos^2 \theta = 1$) and applied these ratios to solve problems involving heights and distances.

What This Chapter Does

In this chapter, we take a big step forward. Instead of restricting trigonometric ratios to acute angles in a right triangle, we generalise them to trigonometric functions that can handle any angle — positive, negative, or even angles larger than $360^\circ$. This shift from ratios to functions opens up a much wider world of applications. We will study the properties of these functions, their graphs, and the relationships between them.