**NEW EDITION**with a new chapter (No. 11) on 'Easy Trig Solutions in Degrees'. See 'Preface' and 'Contents' tabs below.

Price $45 (including postage)

## Description

This book shows an original and highly effective way of unifying many branches of mathematics using Pythagorean triples. A simple, elegant method for combining these triples gives unexpected and powerful general methods for solving a wide range of mathematical problems. There are applications in trigonometry (you do not need any of those complicated formulae), coordinate geometry (2 and 3 dimensions), transformations (2 and 3 dimensions), simple harmonic motion, astronomy etc., etc.

## Reviews

*I feel strongly that the methods you have pioneered, such as "Triples", etc; will catch on all around the world....*

Gary Adamson, U.S. mathematician.*I bought your books on Triples and Astronomical Applications. I am at the moment pursuing the book on triples. I like it very much. In fact I noticed the mathematics of addition , subtraction of angles using triples. It makes the laborious proofs and steps needed for the derivations using coordinate geometry (as is usually done nowadays) look superfluous. It is so simple and easy to understand. It is simply enjoyable. Thanks for all the ideas you have put forth because it has opened up vast vistas for imagination.* Raji Sharma, Vedic Maths teacher, India.*An eye-opener!**If you are attracted to the beauty and simplicity of Mathematics, and if you are drawn to the elegant patterns within Pythagorean Triples then this book will take you on an amazing voyage of discovery. Following the introduction of a few simple arithmetic methods for working with triples, there follows a host of traditional mathematical topics worked using the new 'triple' method. The efficiency and elegance of the mathematics is often breathtaking. Solutions which traditionally may take many steps of complex algebra just fall out in a few straightforward applications of the triple method.**At the core of the book is Mathematics from the Indian Vedic culture (c2000 BC to c600 AD). This is a new territory for me (and I suspect for most) but is a departure that I found refreshing and stimulating. I thoroughly recommend this text for anyone with a knowledge of Mathematics at approximately A Level standard and a love for the beauty of the subject.*

Des Prez (Ireland), 14 May 2009

## Preface

The unchanging laws of number have always been a source of delight and inspiration. Particularly attractive are the Pythagorean triples which have so many elegant and interesting properties. But these triples are also of great practical use: through the theorem of Pythagoras the triples link the three main branches of mathematics: number, algebra and geometry.

This appears to be the first time that these triples have been developed into a useful structure, having applications in trigonometry, transformations in 2 and 3 dimensions, coordinate geometry in 2 and 3 dimensions, solution of triangles and equations, complex numbers, hyperbolic functions, simple harmonic motion, astronomy etc. Many more applications are also likely to appear. This book shows how the triples (and their 3-dimensional equivalent, quadruples) can be developed and applied and how they form a unifying thread linking many areas of mathematics.

In this latest edition of the book (2017) the chapter on Hyperbolic Functions has been replaced by a chapter on Easy Trig Solutions in Degrees. This shows how we can get a good first estimate of trig functions and their inverses (to at least 2 significant figures) and is therefore of practical use. The entire book has been reformatted and many minor modifications made.

This book also serves as an illustration of Vedic Mathematics: a mathematical system which has been rediscovered by Sri Bharati Krsna Tirthaji (1884-1960) from ancient Vedic texts and is expounded in his book (see Reference 1). This system is based on sixteen formulae which are said to give one line answers to all mathematical problems. Being based on fundamental principles these Vedic formulae are therefore conspicuous in any structure that is developed in a simple and natural way. As the triples idea is introduced and extended in this book the operation of these formulae is evident. The formulae are expressed in word form (for example, By One More than the One Before) and as they arise in the text they are indicated by italic type. An index of these formulae will be found at the end of the book.

The diagram below gives a guide as to how the chapters in this book depend upon each other, so that Chapter 8 for example, can be understood by first reading only Chapters 1, 2 and 6.

## Contents

** PREFACE vii****1 TRIPLES 1**

The Triple Theorem 2

Some Historical Background 3

Notation for Triples 4

Equal, Prime and Complementary Triples 5

Some Perfect Triples 6**2 TRIPLE ARITHMETIC 7**

Addition of Triples 7

Double Angle 9

Triple Subtraction 10

Quadrant Angles 13

Triple Geometry 15

Angles of 30°, 60°, 45° etc. 16

Half Angle 17

Simplifying Calculations 20

Summary 21**3 TRIPLE TRIGONOMETRY 22**

Introduction 22

Inverse Functions 25

The General Triple 26

Solution of Trigonometrical Equations 30

Further Trigonometrical Equations: A 32

Further Trigonometrical Equations: B 34

Further Trigonometrical Equations: C 35**4 TRANSFORMATIONS IN A PLANE 39**

Transposition of the Origin 40

Rotations 40

Spirals 44

Integration of cosx etc. 45

Rotation of Lines and Curves 46

Reflections 48**5 COORDINATE GEOMETRY 53**

Length of Perpendicular 53

Foot of Perpendicular 55

Angle between Two Lines 56

Equation of a Line 58

Further Examples 59**6 CODE NUMBERS 61**

Grouping Triples 61

Code Numbers 63

Geometrical Significance of the Code Numbers 64

Code Number Pairs 65

Code Numbers as Triples 66

Algebraic Formulation 67

Converting Code Numbers to Triples 68

Converting Triples to Code Numbers 69

Addition and Subtraction of Code Numbers 69

Code Numbers of Code Numbers 70

Code Numbers of Complementary Triples (CT) 70

Code Numbers of Supplementary Triples (ST) 72

Code Numbers for 0°, 90°, 180°, 270° 72

Relation between Code Numbers and Angles 73

Further Examples 74

Summary 76**7 SOLUTION OF TRIANGLES 77**

The Angle-Deficiency Formula 77

The Sine Formula 81

The Code-Number Formula 84**8 FURTHER APPLICATIONS OF TRIPLES 87**

Solution of Equations 87

Complex Numbers 89

Conics 91

Difference and Sum of Two Squares 96

Incircles and Circumcircles 98

The Golden Triple 100**9 ANGLES IN PERFECT TRIPLES 101**

Revision 101

Triples and Their Angles 103

Finding the Angle in a Given Triple 106

Further Applications of Code Numbers 111

Finding a Triple with a Given Angle 112

A Refinement 114**10 SINE, COSINE, TANGENT AND INVERSES 118**

Near Triple, Small Triple 118

Sine, Cosine and Tangent 119

Inverse Cosine and Inverse Sine 121

Inverse Tangent 125**11 EASY TRIG SOLUTIONS IN DEGREES 129**

Proportion 130

Finding a Side 131

Small Angles 134

Finding an Angle 135**12 APPLIED MATHEMATICS APPLICATIONS 141**

Simple Harmonic Motion 141

Projectiles 147

Forces in Equilibrium 151

Work Done by a Force and Moment of a Force 154**13 THE TRIPLE METHOD 155**

Range of Application 156

Deriving the Conventional Formulae 157

Two Comparisons of the Conventional and Triple Methods 159**14 QUADRUPLES 161**

Introduction 161

Quadruple Generators 163

Obtaining the Code Numbers of a Perfect Quadruple 163

The Coordinate Axes 164

Quadruple Subtraction 164

Comparative Densities of Perfect Triples and Perfect Quadruples 165**15 APPLICATIONS OF QUADRUPLES 166**

Coordinate Geometry 166

Work and Moment 170

Rotation about Coordinate Axis 171

3-Dimensional Rotation of Curves 172

Rotation 173

Conicoids 175**16 QUADRUPLES IN ASTRONOMY 178**

1. Addition of Perpendicular Triples 178

2. Change of Coordinate System 180

3. Quadruples and Orbits 182

4. Quadruple for given i and A 182

5. Inclination of Orbit 183

6. Quadruple Subtraction 183

7. Quadruple Addition 184

8. Doubling and Halving a Quadruple 185

9. Code Number Addition and Subtraction 185

10. Angle in a Quadruple 187

11. Angular Advance 188

12. Relationship between d and A 188

13. To Obtain a Quadruple with a Given Inclination 191

**PROOFS 193****ANSWERS TO EXERCISES 204****REFERENCES 211****INDEX OF VEDIC FORMULAE 212****INDEX 214**

## Back Cover

Pythagorean triples like 3,4,5 have been a fascination for thousands of years. Now for the first time a simple elegant system, based on these triples, has been developed which

reveals unexpected applications in many areas of pure and applied mathematics.

These include general applications is trigonometry, coordinate geometry (in 2 and 3 dimensions), transformations (in 2 and 3 dimensions), simple harmonic motions, projectile motion,

astronomy etc.

The easy triple method links these areas and replaces large numbers of apparently unconnected formulae with a single device.

This book fully explains the various applications and most of it should be accessible to anyone with the basic understanding of mathematics which a school leaver should have.

Kenneth Williams has been studying, researching and teaching Vedic Mathematics for over 40 years. He has published many articles, DVDs and books and has been invited to many

countries to give seminars and courses. He gives online courses, including teacher training. Research includes left-to-right calculating, Astronomy, applications of Triples, extension of Tirthaji's 'Crowning Gem', Calculus.