Trigonometry: A Complete Introduction

The Easy Way to Learn Trig

Hugh Neill

• Publisher: Teach Yourself
• Year: 2018
• ISBN:
  9781473678507
• Topic: Matemáticas
• Subtopic: Trigonometría

Trigonometry: A Complete Introduction is the most comprehensive yet easy-to-use introduction to Trigonometry. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge. The book covers all areas of trigonometry including the theory and equations of tangent, sine and cosine, using trigonometry in three dimensions and for angles of any magnitude, and applications of trigonometry including radians, ratio, compound angles and circles related to triangles. Everything you will need is here in this one book. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions.

1

The tangent

In this chapter you will learn:

• what a tangent is

• the meanings of ‘opposite’, ‘adjacent’ and ‘hypotenuse’ in right-angled triangles

• how to solve problems using tangents.

1.1 Introduction

The method used by Thales to find the height of the pyramid in ancient times is essentially the same as the method used today. It is therefore worth examining more closely.

You can assume that the sun’s rays are parallel because the sun is a long way from the earth. In Figure 1.1, it follows that the lines RC and PB, which represent the rays falling on the tops of the objects, are parallel.

Figure 1.1

Therefore, angle PBQ = angle ACB (they are corresponding angles). These angles each represent the altitude of the sun.

As angles PQB and ABC are right angles, triangles PQB and ABC are similar, so

The height PQ of the pyramid is independent of the length of the stick AB. If you change the length AB of the stick, the length of its shadow will be changed in proportion. You can therefore make the following important general deduction.

For the given angle ACB, the ratio

stays constant whatever the length of AB. You can calculate this ratio beforehand for any angle ACB. If you do this, you do not need to use the stick, because if you know the angle and the value of the ratio, and you have measured the length QB, you can calculate PQ.

Thus if the angle of elevation is 64° and the value of the ratio for this angle had been previously found to be 2.05, then you have

1.2 The idea of the tangent ratio

The idea of a constant ratio for every angle is the key to the development of trigonometry.

Let POQ (Figure 1.2) be any acute angle θ°. From points A, B, C on one arm, say OQ, draw perpendiculars AD, BE, CF to the other arm, OP. As these perpendiculars are parallel, the triangles AOD, BOE and COF are similar.

Figure 1.2

Spotlight

So if OE is double the length of OD then BE will be double the length of AD.

Now take any point Y, it ...

Table of contents

• Cover 
• Title
• Contents 
• Welcome to Trigonometry: A complete introduction
• Introduction
• 1 The tangent
• 2 Sine and cosine
• 3 In three dimensions
• 4 Angles of any magnitude
• 5 Solving simple equations
• 6 The sine and cosine formulae
• 7 Radians
• 8 Relations between the ratios
• 9 Ratios and compound angles
• 10 The forms a sin x and b cos x
• 11 The factor formulae
• 12 Circles related to triangles
• 13 General solution of equations
• Glossary
• Summary of trigonometric formulae
• Answers
• Copyright