Chapter 8 · Introduction

Chapter 8 — Introduction to Trigonometry

A ladder against a wall, a kite on its string, a tower seen from far away: each hides a right-angled triangle. Trigonometry is the study of the ratios of its sides — ratios that depend only on the angle, never on the size.

A $5$ m ladder leans against a wall with its foot $3$ m from the wall. It reaches $4$ m up the wall, so the height reached is $\tfrac45=0.8$ of the ladder's length. Now take a $10$ m ladder and lean it at exactly the same steepness. It reaches $8$ m up the wall — again $0.8$ of its length. A $2.5$ m ladder at that steepness reaches $2$ m, and $0.8$ again. The number $0.8$ does not belong to any one ladder. It belongs to the steepness, that is, to the angle the ladder makes with the ground.

That is the whole idea of trigonometry (from the Greek words for "triangle" and "measure"). In a right-angled triangle, the ratio of any two sides is fixed once the acute angle is fixed. Give these ratios names — sine, cosine, tangent and their reciprocals — and you can find the height of a tower, the width of a river or the length of a kite string from one angle and one length. In this chapter you define the six ratios, work out their exact values for $30^\circ$, $45^\circ$ and $60^\circ$ from two triangles you already own (the set squares in your geometry box), and prove three identities that connect the ratios. Chapter 9 then puts them to work on heights and distances.

Right triangle: opposite, adjacent, hypotenuse→Ratios $\sin A,\ \cos A,\ \tan A$ and their reciprocals→Exact values at $0^\circ,30^\circ,45^\circ,60^\circ,90^\circ$→Identities such as $\sin^2A+\cos^2A=1$
🎯 By the end of this chapter you can…
  • name the hypotenuse and the sides opposite and adjacent to an acute angle, and define the six trigonometric ratios from them;
  • explain why the ratios depend only on the angle, using similar triangles, and find all six ratios when just one is given;
  • write down, and use, the exact values of the ratios for $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$, and see where they come from;
  • prove the identities $\sin^2A+\cos^2A=1$, $1+\tan^2A=\sec^2A$ and $1+\cot^2A=\operatorname{cosec}^2A$;
  • use the identities to simplify expressions and to prove other trigonometric identities step by step.
Teaser — Two Triangles, One Angle

A small and a large right-angled triangle share the angle $A$ at the left-hand corner. Change $A$ with the slider (or press Play). Both triangles tilt and their sides change, yet the ratio opposite ÷ hypotenuse is the same in the small and the large triangle, and so is opposite ÷ adjacent. The ratio is a property of the angle alone.

35°
small: opposite ÷ hypotenuse—
large: opposite ÷ hypotenuse—
small: opposite ÷ adjacent—
large: opposite ÷ adjacent—

Sections in this chapter

8.1

Trigonometric Ratios

Name the sides of a right triangle, define the six ratios, and learn to find all of them from just one.

$\sin A=\dfrac{\text{opposite}}{\text{hypotenuse}}$ · $\tan A=\dfrac{\text{opposite}}{\text{adjacent}}$
Open section →8.2

Trigonometric Ratios of Some Specific Angles

Build the table of exact values for $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$ from two familiar triangles, and use it.

$\sin30^\circ=\tfrac12$ · $\tan45^\circ=1$ · $\cos30^\circ=\tfrac{\sqrt3}{2}$
Open section →8.3

Trigonometric Identities

Prove the three identities that link the ratios and use them to simplify expressions and prove other identities.

$\sin^2A+\cos^2A=1$
Open section →
✅ Before you start
You need, from Class 9 and earlier: the Pythagoras theorem for a right-angled triangle ($\text{hypotenuse}^2=\text{base}^2+\text{height}^2$), the angle sum of a triangle ($180^\circ$), square roots such as $\sqrt{169}=13$ and $\sqrt3\approx1.732$, and the idea of a ratio. From this course you need similar triangles — in particular the AA criterion of Section 6.3 (see Section 6.1 for the meaning of corresponding sides) — and skill with equations and with fractions in algebra (Section 3.2). Warm up: a right-angled triangle has legs $5$ cm and $12$ cm. Find its hypotenuse (answer: $13$ cm).
← 7.2 The Section Formula8.1 Trigonometric Ratios →