Notes_from_Trigonometry-Butler.pdf

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Trig_book.dvi
Notes from Trigonometry
Steven Butler
Brigham Young
University
Fall 2002
Contents
Preface
vii
1 The usefulness of mathematics 1
1.1 WhatcanIlearnfrommath? ..................... 1
1.2 Problemsolvingtechniques....................... 2
1.3 Theultimateinproblemsolving.................... 3
1.4 Takeabreak .............................. 3
1.5 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Geometric foundations 5
2.1 What’s special about triangles? . . . . . . . . . . . . . . . . . . . . 5
2.2 Somedefinitionsonangles....................... 6
2.3 Symbolsinmathematics ........................ 7
2.4 Isocelestriangles ............................ 8
2.5 Righttriangles ............................. 8
2.6 Anglesumintriangles ......................... 9
2.7 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 The Pythagorean theorem 13
3.1 The Pythagorean theorem . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 The Pythagorean theorem and dissection . . . . . . . . . . . . . . . 14
3.3 Scaling.................................. 15
3.4 The Pythagorean theorem and scaling . . . . . . . . . . . . . . . . 17
3.5 Cavalieri’sprinciple........................... 18
3.6 The Pythagorean theorem and Cavalieri’s principle . . . . . . . . . 19
3.7 Thebeginningofmeasurement..................... 19
3.8 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Angle measurement 23
4.1 The wonderful world of π ........................ 23
4.2 Circumferenceandareaofacircle................... 24
i
CONTENTS
ii
4.3 Gradiansanddegrees.......................... 24
4.4 Minutesandseconds .......................... 26
4.5 Radianmeasurement.......................... 26
4.6 Convertingbetweenradiansanddegrees ............... 27
4.7 Wonderfulworldofradians....................... 28
4.8 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Trigonometry with right triangles 30
5.1 Thetrigonometricfunctions ...................... 30
5.2 Usingthetrigonometricfunctions................... 32
5.3 BasicIdentities ............................. 33
5.4 The Pythagorean identities . . . . . . . . . . . . . . . . . . . . . . . 33
5.5 Trigonometric functions with some familiar triangles . . . . . . . . . 34
5.6 Awordofwarning ........................... 35
5.7 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 35
6 Trigonometry with circles 39
6.1 Theunitcircleinitsglory ....................... 39
6.2 Different,butnotthatdifferent .................... 40
6.3 Thequadrantsofourlives....................... 41
6.4 Usingreferenceangles ......................... 41
6.5 The Pythagorean identities . . . . . . . . . . . . . . . . . . . . . . . 43
6.6 A man, a plan, a canal: Panama! . . . . . . . . . . . . . . . . . . . 43
6.7 More exact values of the trigonometric functions . . . . . . . . . . . 45
6.8 Extendingtothewholeplane ..................... 45
6.9 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 46
7 Graphing the trigonometric functions 50
7.1 Whatisafunction?........................... 50
7.2 Graphicallyrepresentingafunction.................. 51
7.3 Over and over and over again . . . . . . . . . . . . . . . . . . . . . 52
7.4 Evenandoddfunctions ........................ 52
7.5 Manipulatingthesinecurve ...................... 53
7.6 Thewildandcrazyinsideterms.................... 55
7.7 Graphs of the other trigonometric functions . . . . . . . . . . . . . 57
7.8 Whythesefunctionsareuseful..................... 58
7.9 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 58
CONTENTS
iii
8 Inverse trigonometric functions 60
8.1 Goingbackwards ............................ 60
8.2 Whatinversefunctionsare....................... 61
8.3 Problemstakingtheinversefunctions................. 61
8.4 Definingtheinversetrigonometricfunctions ............. 62
8.5 Soinanswertoourquandary ..................... 63
8.6 Theotherinversetrigonometricfunctions............... 63
8.7 Usingtheinversetrigonometricfunctions............... 64
8.8 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 66
9Working with trigonometric identities 67
9.1 Whattheequalsignmeans....................... 67
9.2 Addingfractions ............................ 68
9.3 The conju-what? The conjugate . . . . . . . . . . . . . . . . . . . . 69
9.4 Dealingwithsquareroots ....................... 69
9.5 Verifyingtrigonometricidentities ................... 70
9.6 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 72
10 Solving conditional relationships 73
10.1 Conditional relationships . . . . . . . . . . . . . . . . . . . . . . . . 73
10.2Combineandconquer.......................... 73
10.3Usetheidentities............................ 75
10.4‘The’squareroot ............................ 76
10.5Squaringbothsides........................... 76
10.6Expandingtheinsideterms ...................... 77
10.7 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 78
11 The sum and difference formulas 79
11.1Projection................................ 79
11.2Sumformulasforsineandcosine ................... 80
11.3 Difference formulas for sine and cosine . . . . . . . . . . . . . . . . 81
11.4Sumanddifferenceformulasfortangent ............... 82
11.5 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 83
12 Heron’s formula 85
12.1Theareaoftriangles .......................... 85
12.2Theplan................................. 85
12.3Breakingupiseasytodo........................ 86
12.4Thelittleones.............................. 87
12.5Rewritingourterms .......................... 87
12.6Alltogether............................... 88
CONTENTS
iv
12.7Heron’sformula,properlystated.................... 89
12.8 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 90
13 Double angle identity and such 91
13.1Doubleangleidentities......................... 91
13.2Powerreductionidentities ....................... 92
13.3Halfangleidentities........................... 93
13.4 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 94
14 Product to sum and vice versa 97
14.1Producttosumidentities ....................... 97
14.2Sumtoproductidentities ....................... 98
14.3Theidentitywithnoname....................... 99
14.4 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 101
15 Law of sines and cosines 102
15.1Ourdayofliberty............................102
15.2Thelawofsines.............................102
15.3Thelawofcosines............................103
15.4Thetriangleinequality.........................105
15.5 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 106
16 Bubbles and contradiction 108
16.1 A back door approach to proving . . . . . . . . . . . . . . . . . . . 108
16.2 Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
16.3Asimplerproblem ...........................109
16.4Ameetingoflines............................110
16.5 Bees and their mathematical ways . . . . . . . . . . . . . . . . . . . 113
16.6 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 113
17 Solving triangles 115
17.1Solvingtriangles ............................115
17.2Twoanglesandaside .........................115
17.3Twosidesandanincludedangle....................116
17.4Thescaleneinequality .........................117
17.5Threesides ...............................118
17.6Twosidesandanotincludedangle..................118
17.7Surveying ................................120
17.8 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 121

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