Trig or treat: an encyclopedia of trigonometric identity proofs (TIPs), intellectually challenging games

415 Pages·2007·1.77 MB·English

by  Ning Hong Yeo

T Intellectually challenging games
Trigonometric Identity Proofs
(TIPs)
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An encyclopedia of
Trigonometric Identity Proofs
(TIPs)
Intellectually challenging games
Y E O Adrian
M.A., Ph.D. (Cambridge University)
Honorary Fellow, Christ’s College, Cambridge University
World Scientific
N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G
S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
TRIG OR TREAT
An Encyclopedia of Trigonometric Identity Proofs (TIPs) with Intellectually
Challenging Games
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN-13 978-981-277-618-1
ISBN-10 981-277-618-4
ISBN-13 978-981-277-619-8 (pbk)
ISBN-10 981-277-619-2 (pbk)
Printed in Singapore.
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September12,2007 19:16 Book:-TrigorTreat (9inx6in) 01˙dedi
dedicatedtothememory
ofmymother
tanpeckhiah
tanA+tanB+tanC(cid:17)tanAtanBtanC
(cid:14)
(A+B+C=180 )
(cid:14) (cid:14) (cid:14)
tanA+tan(A+120 )+tan(A+240 )=3tan(3A+360 )
1 1 1
tan(cid:0)1 =tan(cid:0)1 +tan(cid:0)1
(cid:18)1(cid:19) (cid:18)2(cid:19) (cid:18)3(cid:19)
1 1 p 1 p 1 p 1 p
= tan + tan + tan + tan +(cid:1)(cid:1)(cid:1)
p 22 22 23 23 24 24 25 25
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September17,2007 19:18 Book:-TrigorTreat (9inx6in) 02˙contents
Contents
Preface ix
Introduction xiii
Trig—LevelOne 1
TheBasicsofTrigonometry 1
Pythagoras’Theorem 11
Trig—LevelTwo 15
CompoundAngles,DoubleAnglesandHalfAngles
Trig—LevelThree 25
AnglesinaTriangle
SumandDifferenceofsinandcos
PracticalTrig 33
NumericalValuesofSpecialAngles—
(cid:14) (cid:14)
Graphsofsin,cosandtanfor0 –360
Appendix 45
TheConcordanceofTrigonometricIdentities 47
TheEncyclopediaofTrigonometricGamesor
TrigonometricIdentityProofs(TIPs) 71
vii
September17,2007 19:18 Book:-TrigorTreat (9inx6in) 02˙contents
viii TrigorTreat
Level-One-Games: EasyProofs 73
Level-Two-Games: Less-EasyProofs 251
Level-Three-Games: Not-So-EasyProofs 349
Addenda 393
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Preface
Many students find “Trigonometry” to be a difficult topic in a diffi-
cult subject “Math”. Yet most students have no difficulty with com-
puter games, and enjoy playing them even though many of these
games have lots of pieces to manipulate, and are subjected to com-
plex rules. For example, a simple game like “Tetris” has seven differ-
ent pieces; and the player has to orientate and manipulate each piece
in turn, as it falls. The objective is to construct a solid wall with all the
randompieces —andtodoit,racing against theclock.
“Trigonometry”, or “Trig” for short, can be thought of as an intellec-
tualequivalent of“Tetris”. Therearesixmainpiecestomanipulate. Three
of them, sine, cosine, and tangent, are most important — that is why this
branch of Math is called Trigonometry; the “tri” refers to three functions,
threeanglesandthreesidesofatriangle. Andthereisonlyonesimplerule
—Logic. Trigcanbethoughtofasagamethatinvolvesthelogicalmanip-
ulation of various trig pieces to achieve different identities and equations,
andtosolve numerical problems.
Trig can also be viewed as a non-numerical equivalent of the number
game“Sudoku”. Thelogicandthearrangement ofthedigits 1to9,isnow
applied to the six trig pieces —sine, cosine, tangent, cosecant, secant and
cotangent.
Tetris and Sudoku are both simple games that give lots of fun and
pleasure. Trig is also a simple game, but with a vital difference —
knowledge of it has invaluable applications in Math, surveying, building,
ix