**NEW EDITION - 2018**

Pothi paperback (for India only): Rs.400

## Description

The book explains trigonometry in a simple way beginning with squares and square roots and Pythagoras Theorem up to solution of trig equations. It is aimed at the child and forms a complete and self-contained introduction to the subject. Suitable for children around grade 7. This is a new and very easy approach to trigonometry that begins with triples like 3,4,5 which the children make themselves from thin card and then use in various ways. Four new chapters have been added in this new edition - on solution of right-angled triangles. Please see Contents for the full range of topics covered.

## Details

186 pages.

Size: 24.5cm by 17cm.

Paperback. 2018

Author: Kenneth Williams

ISBN 978-1-902517-45-2.

## Introduction

**INTRODUCTION**

Right-angled triangles are a fundamental unit in many areas of mathematics, particularly trigonometry.

The approach of this book is to establish certain simple results using Pythagorean Triples (where the sides are whole numbers, such as 3, 4, 5).

These then serve to introduce techniques and principles which apply to right-angled triangles in general.

Another such triple is 12, 5, 13 and in fact there are infinitely many of these. In this book we are not restricted to whole numbers though.

This is a fascinating area of mathematics and the Triples have many interesting and beautiful properties.

But, in addition to this, the Triples can be extremely useful. Once we have a way of combining the triples, as shown in this book, they give us surprising and easy ways of solving mathematical problems. In many cases the solutions can be obtained with a small fraction of the work needed by the usual method.

In fact solutions become so easy that areas of mathematics normally done at a certain age can be covered much earlier.

Because these triples can be used in many different areas of mathematics they have a unifying influence, which is a very welcome in our study and teaching of mathematics.

These triples, as developed in my book “Triples” (first published back in 1984) have many applications and link many diverse areas of mathematics. That book is not very suitable for use in schools however and so this present book has been created which aims to make the material child-friendly: proceeding in small steps and covering topics normally covered at school level.

This book is self-contained, starting with squaring, and developing the material in a gentle and logical way up to the solution of trigonometrical equations. The book

can be used in different ways; apart from running right through the book sequentially, specific sections may be chosen by a teacher/home-schooler as appropriate. Or the book may be used for general interest by anyone interested in a simple and easy approach to topics generally considered hard.

* Note, exercise questions preceded by an asterisk are harder questions.

**Acknowledgements**

My gratitude and thanks go to my colleagues Nathan Annenberg in the USA, Kuldeep Singh in New Delhi and Vera Stevens in Brisbane, Australia, for reading the book, using it with their students and giving me valuable feedback for improving the content.

**New edition, 2018 **

This edition has four new chapters added at the end. These are on the solution of right-angles triangles without use of a calculator. Though placed at the end of the original 12 chapters this material could be studied at any point after Chapter 9.

## Contents

**1) Squares and Square Roots**1

Squaring Numbers that End in 5

Squaring Numbers near 50

General Squaring

Grouping

Algebraic Squaring

Last Digit of Square Numbers

Square Roots of Perfect Squares

**2) Pythagoras’ Theorem**13

A Short Cut

The Theorem in Reverse

**3) Triples**22

Similar or Equal Triples

Angle in a Triple

**4) Families of Triples**28

Triple Groups

Making Triple Triangles

Extending the Triples List

Finding a Triple’s Sharpness

Complementary Triples

The Group 3 Triples

Finding a Triple Knowing its Sharpness

A Formula

**Prelude 1: The Similar Triangles Pattern**42

**5) Adding Triples**44

Triple Addition

Negative Elements

Double Angle

Triple Subtraction

**Prelude 2: Manipulating Square Roots**57

**6) Triple Geometry**59

Quadrant Angles

Solving Triangles

Angles of 45, 30, 60 etc.

Half Angle

Spirals

**7) Rotations**69

90 Rotations

Other Rotations

A Different Centre of Rotation

**Prelude 3: 45°, 30° and 60° Triples**77

**8) Coordinate Geometry**79

Point Triple and Line Triple

Angle Between Two Lines

Distance of a Point from a Line

Sloping Lines

Line not Passing Through the Origin

**9) Solving Triples**88

Finding a Side

Finding an Angle

Multiple Angles

**10) Sines, Cosines, Tangents**95

Definitions

Sketching Triangles

Connected Angles

Combinations

Angles not both Acute

**11) Proofs with Triples**105

The General Triple

Equation and Identity

Proofs

Double Angle

**12) Equations**112

Answers in Triple Form

Answers in Degrees

More Answers

Quadratic Equations

Another Type of Trig Equation

**13) Estimating Sides and Angles**122

Estimating a Side

Estimating an Angle

**14) Small Angles**129

The Number 57

Finding a Side

Finding a Small Angle

**15) Finding a Side**137

Using 4,3,5

Angle Less Than 37°

Using a Different Triple

Flipping the Triangle

Finding the Hypotenuse

Two Shorter Sides

**16) Finding an Angle**149

Knowing the Hypotenuse

Knowing Base and Height

**Appendix**154

**Glossary**158

**Answers**159

**Index**179

## Back Cover

This self-contained book shows a unified and easy approach to trigonometry using triples like 3,4,5 which can represent the sides of a right-angled triangle.

This method is so powerful and simple that quite advanced topics can be tackled and understood at an earlier age than usual.

Though, as we see here, the usual sines, cosines etc. can be avoided they are included to connect up with the traditional approach.

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, extension of Tirthaji's ‘Crowning Gem', Calculus.