Using Complex Numbers
179 videos • 21,779 views • by Eddie Woo
1
Using De Moivre's Theorem to Prove Trigonometric Identity
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2
Understanding & Applying the Conjugate Root Theorem
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3
Argand Diagram / Locus Question
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4
Interesting Complex Polynomial Question (1 of 2: Factorisation)
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5
Interesting Complex Polynomial Question (2 of 2: Trigonometric Result)
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6
Complex Numbers as Points (1 of 4: Geometric Meaning of Addition)
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7
Complex Numbers as Points (2 of 4: Geometric Meaning of Subtraction)
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8
Complex Numbers as Points (3 of 4: Geometric Meaning of Multiplication)
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9
Complex Numbers as Points (4 of 4: Second Multiplication Example)
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10
De Moivre's Theorem
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11
How to graph the locus of |z-1|=1
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12
Complex Numbers as Vectors (1 of 3: Introduction & Addition)
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13
Complex Numbers as Vectors (2 of 3: Subtraction)
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14
Complex Numbers as Vectors (3 of 3: Using Geometric Properties)
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15
Complex Roots (2 of 5: Expanding in Rectangular Form)
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16
The Triangle Inequalities (1 of 3: Sum of Complex Numbers)
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17
Graphs in the Complex Plane (1 of 4: Introductory Examples)
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18
The Triangle Inequalities (2 of 3: Discussing Specific Cases)
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19
The Triangle Inequalities (3 of 3: Difference of Complex Numbers)
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20
Graphs in the Complex Plane (2 of 4: Graphing Complex Inequalities)
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21
Graphs in the Complex Plane (3 of 4 : Shifting the Point of Reference)
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22
Graphs in the Complex Plane (4 of 4: Where is the argument measured from?)
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23
Further Graphs on the Complex Plane (1 of 3: Geometrical Representation of Moduli)
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24
Further Graphs on the Complex Plane (2 of 3: Algebraically verifying Graphs concerning the Moduli)
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25
Further Graphs on the Complex Plane (3 of 3: Geometrical Representation of Arguments)
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26
Graphs on the Complex Plane (3 of 4: Geometry of arg(z)-arg(z-1))
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27
Graphs on the Complex Plane (4 of 4: Exploring how the argument traced the graph)
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28
Graphs on the Complex Plane [Continued] (1 of 4: What's behind the graph?)
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29
Graphs on the Complex Plane [Continued] (2 of 4: Finding Regions of Inequality by Testing Points)
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30
Using Inverse tan to find arguments? (1 of 2: Why it doesn't work... Sometimes)
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31
Complex Roots (1 of 5: Introduction)
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32
Complex Roots (3 of 5: Through Polar Form Using De Moivre's Theorem)
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33
Using Inverse tan to find arguments? (2 of 2: Why it works... Sometimes)
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34
Complex Roots (4 of 5: Through Polar Form Generating Solutions)
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35
Complex Roots (5 of 5: Flowing Example - Solving z^6=64)
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36
Complex Conjugate Root Theorem (1 of 4: Using DMT and Polar Form to solve for Complex Roots)
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37
Complex Conjugate Root Theorem (2 of 4: Introduction to the Conjugate Root Theorem)
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38
Complex Conjugate Root Theorem (3 of 4: Geometrical Shape represented by Conjugate Root Theorem)
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39
Complex Conjugate Root Theorem (4 of 4: Using Factorisation to find patterns with Roots of Unity)
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40
DMT and Trig Identities (1 of 4: Deriving multi-angle identities with compound angles)
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41
DMT and Trig Identities (2 of 4: Using De Moivre's Theorem and Binomial Expansions)
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42
DMT and Trig Identities (3 of 4: Deriving tan expression from cos and sin)
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43
DMT and Trig Identities (4 of 4: Using Multi-angle formula to solve polynomials)
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44
Complex Numbers (1 of 6: Solving Harder Complex Numbers Questions) [Student requested problem]
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45
Complex Numbers (2 of 6: Solving Harder Complex Numbers Questions) [Student Requested Problem]
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46
Complex Numbers (3 of 6: Harder Complex Numbers Question) [Student Requested Problem]
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47
Complex Numbers (4 of 6: Harder Complex Numbers Questions) [Student Requested Problem]
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48
Complex Numbers (5 of 6: Complex Numbers Proofs [Using the Conjugate])
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49
Complex Numbers (6 of 6: Finishing off the Proof)
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50
Roots and Coefficients (1 of 3: Using DMT & Binomial Theorem to find identities)
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51
Roots and Coefficients (2 of 3: Using Trigonometry to solve polynomial problems)
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52
Roots and Coefficients (3 of 3: Using the results to find a relation in cosine)
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53
Extension II Assessment Review (5 of 5: De Moivre's Theorem and Polynomials)
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54
Why Complex Numbers? (1 of 5: Atoms & Strings)
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55
Why Complex Numbers? (2 of 5: Impossible Roots)
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56
Why Complex Numbers? (3 of 5: The Imaginary Unit)
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57
Why Complex Numbers? (4 of 5: Turning the key)
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58
Why Complex Numbers? (5 of 5: Where to now?)
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59
Complex Arithmetic (1 of 2: Addition, Subtraction & Multiplication)
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60
Complex Arithmetic (2 of 2: Division)
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61
Factorisation with Complex Numbers
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62
Vectors (1 of 4: Outline of vectors and their ability to represent complex number)
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63
Vectors (2 of 4: Representing addition & subtraction of complex numbers with vectors)
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64
Vectors (3 of 4: Geometrically representing multiplication of complex numbers with vectors)
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65
Vectors (4 of 4: Outlining the usefulness of vectors in representing geometry)
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66
Curves and Regions on the Complex Plane (1 of 4: Introductory example plotting |z|=5 geometrically)
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67
Curves and Regions on the Complex Plane (2 of 4: Deciphering terminology to plot complex numbers)
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68
Curves and Regions on the Complex Plane (3 of 4: Simplifying expressions to plot on a complex plane)
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69
Curves and Regions on the Complex Plane (4 of 4: Plotting simultaneous shifted complex numbers)
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70
Further Curves and Regions (1 of 5: Why does Sine & the Sine Rule produce an ambiguous case?)
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71
Further Curves and Regions (2 of 5: Finding the Range of |z| )
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72
Further Curves and Regions (3 of 5: Finding the range of arg z)
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73
Further Curves and Regions (4 of 5: Geometrical expression of expressions of arg)
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74
Further Curves and Regions (5 of 5: Using Circle properties to graph an expression of args)
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75
DMT & Complex Roots (1 of 4: Reviewing geometrical expression of arg equations)
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76
DMT & Complex Roots (2 of 4: Using DMT to find roots of a complex polynomial)
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77
DMT & Complex Roots (3 of 4: Using the fundamental theorem of algebra to justify number of roots)
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78
DMT & Complex Roots (4 of 4: Solving for roots of a complex number taking advantage of DMT)
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79
Complex Roots (1 of 5: Observing Complex Conjugate Root Theorem through seventh roots of unity)
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80
Complex Roots (2 of 5: Using Trigonometrical Identities & Conjugates to solve an equation)
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81
Complex Roots (3 of 5: Using DMT to solve an equation of roots of unity)
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82
Complex Roots (4 of 5: Using Polynomial Identities to prove unity identities)
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83
Complex Roots (5 of 5: Using Geometric Progression to find factors of ω^n - 1)
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84
DMT and Trig Identities (1 of 4: Noticing a pattern in natural numbers)
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85
DMT and Trig Identities (2 of 4: Using Trig expansion to find the sine triple angle formula)
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86
DMT and Trig Identities (3 of 4: Using DMT and Polynomials to verify triple angle formula for sine)
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87
DMT and Trig Identities (4 of 4: Using Trig Identities to solve polynomial equations)
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88
HSC Question on Complex Numbers, Vectors & Triangle Area (1 of 2: Thinking geometrically)
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89
HSC Question on Complex Numbers, Vectors & Triangle Area (2 of 2: Manipulating trigonometric terms)
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90
HSC Question on Complex Numbers, Vectors & Polynomials (1 of 2: How to "explain")
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91
HSC Question on Complex Numbers, Vectors & Polynomials (2 of 2: Combining results)
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92
2016 HSC - Complex Numbers on Unit Circle (1 of 2: Considering Re & Im Parts)
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93
2016 HSC - Complex Numbers on Unit Circle (2 of 2: Evaluating the arguments)
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94
2016 HSC - Complex Identity Proof (1 of 3: Convert to polar form)
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95
2016 HSC - Complex Identity Proof (2 of 3: Using binomial theorem)
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96
2016 HSC - Complex Identity Proof (3 of 3: Combining results)
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97
Semi Circles on Argand Diagrams (3 of 3: Oblique example)
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98
Semi Circles on Argand Diagrams (2 of 3: Graphing the locus)
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99
Semi Circles on Argand Diagrams (1 of 3: Relating the angles)
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100
Algebraic Proof for Opposing Rays (3 of 3: Testing cases)
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101
Algebraic Proof for Opposing Rays (2 of 3: Generating the equation)
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102
Algebraic Proof for Opposing Rays (1 of 3: Foundational knowledge)
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103
cos⁴θ Identity
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104
Graphs in the Complex Plane (3 of 3: Opposing rays)
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105
Graphs in the Complex Plane (2 of 3: Algebraic method)
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106
Graphs in the Complex Plane (1 of 3: Perpendicular bisector - visual method)
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107
Complex Polynomial Identity Question (4 of 4: Roots & coefficients)
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108
Complex Polynomial Identity Question (3 of 4: de Moivre's Theorem)
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109
Complex Polynomial Identity Question (2 of 4: Difference of cubes)
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110
Complex Polynomial Identity Question (1 of 4: Quadratic factors)
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111
Sum & Product of Cosines (3 of 3: Drawing ℝ conclusions)
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112
Sum & Product of Cosines (2 of 3: Simplifying with conjugates)
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113
Sum & Product of Cosines (1 of 3: 9th roots of unity)
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114
Nth Roots of a ℂ Number (2 of 2: Example problem)
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115
Nth Roots of a ℂ Number (1 of 2: General form)
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116
Roots of Unity (2 of 2: Insights from polar & exponential forms)
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117
Roots of Unity (1 of 2: Evaluating the cube roots)
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118
Complex Conjugate Root Theorem (2 of 2: Other conjugate properties)
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119
Complex Conjugate Root Theorem (1 of 2: Conjugate of a sum)
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120
Solving Higher Degree Trigonometric Equations (3 of 3: Finding solutions)
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121
Solving Higher Degree Trigonometric Equations (2 of 3: Combining results into proof)
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122
Solving Higher Degree Trigonometric Equations (1 of 3: Initial use of de Moivre's Theorem)
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123
Equations with Complex Solutions (2 of 2: Solving & factorising)
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124
Equations with Complex Solutions (1 of 2: Relation to square roots)
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125
Polynomials with Trigonometric Solutions (3 of 3: Simplifying with identities)
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126
Polynomials with Trigonometric Solutions (2 of 3: Substitute & solve)
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127
Polynomials with Trigonometric Solutions (1 of 3: de Moivre's Theorem)
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128
Trigonometric Expansions from Complex Numbers (3 of 3: General compound angles)
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129
Trigonometric Expansions from Complex Numbers (2 of 3: Double angle results)
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130
Trigonometric Expansions from Complex Numbers (1 of 3: Concept map)
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131
Using de Moivre's Theorem - example question (2 of 2: Purely imaginary)
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132
Using de Moivre's Theorem - example question (1 of 2: Purely real)
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133
Proving de Moivre's Theorem (2 of 2: Derivation & example problem)
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134
Proving de Moivre's Theorem (1 of 2: Prologue)
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135
Arcs on the Complex Plane (1 of 4: Review questions)
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136
Arcs on the Complex Plane (3 of 4: Identifying length and direction)
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137
Arcs on the Complex Plane (2 of 4: Exploring circle properties)
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138
Arcs on the Complex Plane (4 of 4: Cartesian equation)
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139
Angles in the Same Segment
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140
Algebraic Approach for Major Arc (1 of 2: Foundational steps)
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141
Algebraic Approach for Major Arc (2 of 2: Identifying intercept & equation)
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142
Max/Min Value of |z| (1 of 2: Geometric solution)
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143
Max/Min Value of |z| (2 of 2: Triangle inequality)
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144
Investigating de Moivre's Theorem (1 of 3: Why must we be cautious?)
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145
Investigating de Moivre's Theorem (2 of 3: Infinite values for i-th powers?!)
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146
Investigating de Moivre's Theorem (3 of 3: Proof by mathematical induction)
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147
Varying |z| & Argz on a Locus
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148
Maximising Sum of Moduli (1 of 3: Geometric approach)
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149
Maximising Sum of Moduli (2 of 3: Differentiation)
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150
Maximising Sum of Moduli (3 of 3: Interpreting stationary points)
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151
Sketching (z-1)÷(z-i) (1 of 2: When it's real)
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152
Sketching (z-1)÷(z-i) (2 of 2: When it's imaginary)
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153
Complex Numbers Exam Review (1 of 4: Visualising & Manipulating Arithmetic)
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154
Complex Numbers Exam Review (2 of 4: Proving i^i is real, identity proof)
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155
Complex Numbers Exam Review (3 of 4: Cube roots of unity)
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156
Complex Numbers Exam Review (4 of 4: Locus; polynomial identity)
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157
Prove arg(z₁z₂) = arg(z₁+z₂)² (1 of 2: Preliminary thoughts)
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158
Prove arg(z₁z₂) = arg(z₁+z₂)² (2 of 2: Geometric approach)
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159
Complex Geometry - Square Problem (1 of 2: Complex numbers → vectors)
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160
Complex Geometry - Square Problem (2 of 2: Vectors → complex numbers)
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161
Complex Geometry - Equilateral Triangle (3 of 3: Vector proof)
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162
Complex Geometry - Equilateral Triangle (2 of 3: Algebraic method)
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163
Complex Geometry - Equilateral Triangle (1 of 3: Arithmetic proof)
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164
The Basel Problem (9 of 9: Squeeze law)
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165
The Basel Problem (8 of 9: Returning to trigonometric terms)
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166
The Basel Problem (7 of 9: Manipulating the polynomial integral)
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167
The Basel Problem (5 of 9: Telescoping sum)
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168
The Basel Problem (6 of 9: Equations → inequalities)
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169
The Basel Problem (4 of 9: Introducing x² to the integrand)
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170
The Basel Problem (3 of 9: Integration by *different* parts)
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171
The Basel Problem (1 of 9: Prologue)
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172
The Basel Problem (2 of 9: Recurrence relation)
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173
Centre of a Major Arc (5 of 5: Algebraic proof)
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174
Centre of a Major Arc (4 of 5: Inscribed equilateral triangle)
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175
Centre of a Major Arc (3 of 5: Using trigonometric & vectors)
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176
Centre of a Major Arc (2 of 5: Finding centre and radius)
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177
Centre of a Major Arc (1 of 5: Evaluating internal angle)
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178
How to graph a region on the complex plane
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179
Three ways to find a parallelogram's area
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