Mathematics Grade 10 Notes PDF

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M ATHEM ATIC ICS S GRADE 10 NOTES

Contents DISCLAIMER

vii viii

Preface 1 The 1.1 1.2 1.3 1.4

usefulness of mathematics What can I learn from math? . Problem solving techniques . . . The ultimate in problem solving Take a break . . . . . . . . . .

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2 Geometric foundations 2.1 What’s special about triangles? 2.2 Some definitions on angles . . . 2.3 Symbols in mathematics . . . . 2.4 Isoceles triangles . . . . . . . . 2.5 Right triangles . . . . . . . . . 2.6 Angle sum in triangles . . . . . 2.7 Supplemental problems . . . . .

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14 14 15 16 18 19 20 20 21

3 The 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Pythagorean theorem The Pythagorean theorem . . . . . . . . . . . . . . The Pythagorean theorem and dissection . . . . . . Scaling . . . . . . . . . . . . . . . . . . . . . . . . . The Pythagorean theorem and scaling . . . . . . . Cavalieri’s principle . . . . . . . . . . . . . . . . . . The Pythagorean theorem and Cavalieri’s principle The beginning of measurement . . . . . . . . . . . . Supplemental problems . . . . . . . . . . . . . . . .

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1 1 2 3 3

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CONTENTS 4 Angle measurement 4.1 The wonderful world of π . . . . . . . . . 4.2 Circumference and area of a circle . . . . 4.3 Gradians and degrees . . . . . . . . . . . 4.4 Minutes and seconds . . . . . . . . . . . 4.5 Radian measurement . . . . . . . . . . . 4.6 Converting between radians and degrees 4.7 Wonderful world of radians . . . . . . . . 4.8 Supplemental problems . . . . . . . . . .

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24 24 25 25 27 28 28 29 30

5 Trigonometry with right triangles 5.1 The trigonometric functions . . . . . . . . . . . . . . 5.2 Using the trigonometric functions . . . . . . . . . . . 5.3 Basic Identities . . . . . . . . . . . . . . . . . . . . . 5.4 The Pythagorean identities . . . . . . . . . . . . . . . 5.5 Trigonometric functions with some familiar triangles . 5.6 A word of warning . . . . . . . . . . . . . . . . . . . 5.7 Supplemental problems . . . . . . . . . . . . . . . . .

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32 32 34 35 35 36 37 38

6 Trigonometry with circles 6.1 The unit circle in its glory . . . . . . . . . . . . . . . . . . . . . . . 6.2 Different, but not that different . . . . . . . . . . . . . . . . . . . . 6.3 The quadrants of our lives . . . . . . . . . . . . . . . . . . . . . . . 6.4 Using reference angles . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Pythagorean identities . . . . . . . . . . . . . . . . . . . . . . . 6.6 A man, a plan, a canal: Panama! . . . . . . . . . . . . . . . . . . . 6.7 More exact values of the trigonometric functions . . . . . . . . . . . 6.8 Extending to the whole plane . . . . . . . . . . . . . . . . . . . . . 6.9 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . .

42 42 43 44 44 46 46 48 48 49

7 Graphing the trigonometric functions 7.1 What is a function? . . . . . . . . . . . . . . 7.2 Graphically representing a function . . . . . 7.3 Over and over and over again . . . . . . . . 7.4 Even and odd functions . . . . . . . . . . . 7.5 Manipulating the sine curve . . . . . . . . . 7.6 The wild and crazy inside terms . . . . . . . 7.7 Graphs of the other trigonometric functions 7.8 Why these functions are useful . . . . . . . . 7.9 Supplemental problems . . . . . . . . . . . .

53 53 54 55 55 56 58 60 60 62

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CONTENTS 8 Inverse trigonometric functions 8.1 Going backwards . . . . . . . . . . . . . . . 8.2 What inverse functions are . . . . . . . . . . 8.3 Problems taking the inverse functions . . . . 8.4 Defining the inverse trigonometric functions 8.5 So in answer to our quandary . . . . . . . . 8.6 The other inverse trigonometric functions . . 8.7 Using the inverse trigonometric functions . . 8.8 Supplemental problems . . . . . . . . . . . .

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64 64 65 65 66 67 68 68 71

9 Working with trigonometric identities 9.1 What the equal sign means . . . . . . . 9.2 Adding fractions . . . . . . . . . . . . 9.3 The conju-what? The conjugate . . . . 9.4 Dealing with square roots . . . . . . . 9.5 Verifying trigonometric identities . . . 9.6 Supplemental problems . . . . . . . . .

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72 72 73 74 75 75 77

10 Solving conditional relationships 10.1 Conditional relationships . . . . 10.2 Combine and conquer . . . . . . 10.3 Use the identities . . . . . . . . 10.4 ‘The’ square root . . . . . . . . 10.5 Squaring both sides . . . . . . . 10.6 Expanding the inside terms . . 10.7 Supplemental problems . . . . .

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79 79 79 81 82 82 83 84

sum and difference formulas Projection . . . . . . . . . . . . . . . . . Sum formulas for sine and cosine . . . . Difference formulas for sine and cosine . Sum and difference formulas for tangent Supplemental problems . . . . . . . . . .

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85 85 86 87 88 89

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91 91 91 92 93 93 94

11 The 11.1 11.2 11.3 11.4 11.5

12 Heron’s formula 12.1 The area of triangles . . . 12.2 The plan . . . . . . . . . . 12.3 Breaking up is easy to do . 12.4 The little ones . . . . . . . 12.5 Rewriting our terms . . . 12.6 All together . . . . . . . .

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CONTENTS

12.7 Heron’s formula, properly stated . . . . . . . . . . . . . . . . . . . . 95 12.8 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 95 13 Double angle identity and such 13.1 Double angle identities . . . . 13.2 Power reduction identities . . 13.3 Half angle identities . . . . . . 13.4 Supplemental problems . . . .

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97 . 97 . 98 . 99 . 100

14 Product to sum and vice versa 14.1 Product to sum identities . . 14.2 Sum to product identities . . 14.3 The identity with no name . . 14.4 Supplemental problems . . . .

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109 . 109 . 109 . 110 . 112 . 113

15 Law of sines and cosines 15.1 Our day of liberty . . . 15.2 The law of sines . . . . 15.3 The law of cosines . . . 15.4 The triangle inequality 15.5 Supplemental problems

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103 103 104 105 107

16 Bubbles and contradiction 16.1 A back door approach to proving . 16.2 Bubbles . . . . . . . . . . . . . . . 16.3 A simpler problem . . . . . . . . . 16.4 A meeting of lines . . . . . . . . . . 16.5 Bees and their mathematical ways . 16.6 Supplemental problems . . . . . . .

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116 . 116 . 117 . 117 . 118 . 121 . 121

17 Solving triangles 17.1 Solving triangles . . . . . . . . . . 17.2 Two angles and a side . . . . . . . 17.3 Two sides and an included angle . . 17.4 The scalene inequality . . . . . . . 17.5 Three sides . . . . . . . . . . . . . 17.6 Two sides and a not included angle 17.7 Surveying . . . . . . . . . . . . . . 17.8 Supplemental problems . . . . . . .

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123 123 123 124 125 126 126 128 129

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CONTENTS 18 Introduction to limits 18.1 One, two, infinity... . . . . . . 18.2 Limits . . . . . . . . . . . . . 18.3 The squeezing principle . . . . 18.4 A limit involving trigonometry 18.5 Supplemental problems . . . .

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133 . 133 . 134 . 134 . 135 . 136

19 Vi`ete’s formula 139 19.1 A remarkable formula . . . . . . . . . . . . . . . . . . . . . . . . . . 139 19.2 Vi`ete’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 20 Introduction to vectors 20.1 The wonderful world of vectors . . . . . . . . . . . 20.2 Working with vectors geometrically . . . . . . . . . 20.3 Working with vectors algebraically . . . . . . . . . 20.4 Finding the magnitude of a vector . . . . . . . . . . 20.5 Working with direction . . . . . . . . . . . . . . . . 20.6 Another way to think of direction . . . . . . . . . . 20.7 Between magnitude-direction and component form . 20.8 Applications to physics . . . . . . . . . . . . . . . . 20.9 Supplemental problems . . . . . . . . . . . . . . . .

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141 141 141 143 144 145 146 146 147 147

21 The 21.1 21.2 21.3 21.4 21.5 21.6 21.7

dot product and its applications A new way to combine vectors . . . . . The dot product and the law of cosines Orthogonal . . . . . . . . . . . . . . . Projection . . . . . . . . . . . . . . . . Projection with vectors . . . . . . . . . The perpendicular part . . . . . . . . . Supplemental problems . . . . . . . . .

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150 150 151 152 153 154 154 155

22 Introduction to complex numbers 22.1 You want me to do what? . . . . . . . . . . . . . 22.2 Complex numbers . . . . . . . . . . . . . . . . . . 22.3 Working with complex numbers . . . . . . . . . . 22.4 Working with numbers geometrically . . . . . . . 22.5 Absolute value . . . . . . . . . . . . . . . . . . . 22.6 Trigonometric representation of complex numbers 22.7 Working with numbers in trigonometric form . . . 22.8 Supplemental problems . . . . . . . . . . . . . . .

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158 158 159 159 160 160 161 162 163

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CONTENTS

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23 De Moivre’s formula and induction 164 23.1 You too can learn to climb a ladder . . . . . . . . . . . . . . . . . . 164 23.2 Before we begin our ladder climbing . . . . . . . . . . . . . . . . . . 164 23.3 The first step: the first step . . . . . . . . . . . . . . . . . . . . . . 165 23.4 The second step: rinse, lather, repeat . . . . . . . . . . . . . . . . . 166 23.5 Enjoying the view . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 23.6 Applying De Moivre’s formula . . . . . . . . . . . . . . . . . . . . . 167 23.7 Finding roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 23.8 Supplemental problems . . . . . . . . . . . . . . . . . . . . . . . . . 170 A Collection of equations

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NOTES ON TRIG IGO ONOM ETRY

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Lecture 1 The usefulness of mathematics

1

LECTURE 1. THE USEFULNESS OF MATHEMATICS

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Your ability to solve problems must be developed, and one of the many ways to develop your problem solving ability is to do mathematics starting with simple problems and working your way up to the more complicated problems. Now let me carry this analogy with bodybuilding a little further. When I played football in high school I would spend just as much time in the weight room as any member of the team. But I never developed huge biceps, a flat stomach or any of a number of features that many of my teammates seemed to gain with ease. Some people have bodies that respond to training and bulk up right away, and then some bodies do not respond to training as quickly. You will notice the same thing when it comes to doing mathematics. Some people pick up the subject quickly and fly through it, while others struggle to understand the basics. If you find yourself in this latter group, don’t give up. Everyone has the ability to understand and enjoy mathematics, be patient, work problems and practice thinking. Anyone who follows this practice will develop an ability to do mathematics.

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Problem solving techniques

There are a number of books written on the subject of mathematical problem solving. One of the best, and most famous, is How to Solve It by George Polya. The following outline for solving problems is adopted from his book. Essentially there are four steps involved in solving a problem. UNDERSTANDING THE PROBLEM—Before beginning to solve any problem you must understand what it is that you are trying to solve. Look at the problem. There are two parts, what you are given and what you are trying to show. Clearly identify these parts. What are you given? What are you trying to show? Is it reasonable that there is a connection between the two? DEVISING A PLAN—Once we understand the problem that we are trying to solve we need to find a way to connect what we are given to what we are trying to show, in other words, we need a plan. Mathematicians are not very original and often use the same ideas over and over, so look for similar proble...