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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.