Using Complex Numbers - Comprehensive Course on De Moivre's Theorem, Argand Diagrams, and Complex Analysis

Using De Moivre's Theorem to Prove Trigonometric Identity
Understanding & Applying the Conjugate Root Theorem
Argand Diagram / Locus Question
Interesting Complex Polynomial Question (1 of 2: Factorisation)
Interesting Complex Polynomial Question (2 of 2: Trigonometric Result)
Complex Numbers as Points (1 of 4: Geometric Meaning of Addition)
Complex Numbers as Points (2 of 4: Geometric Meaning of Subtraction)
Complex Numbers as Points (3 of 4: Geometric Meaning of Multiplication)
Complex Numbers as Points (4 of 4: Second Multiplication Example)
De Moivre's Theorem
How to graph the locus of |z-1|=1
Complex Numbers as Vectors (1 of 3: Introduction & Addition)
Complex Numbers as Vectors (2 of 3: Subtraction)
Complex Numbers as Vectors (3 of 3: Using Geometric Properties)
Complex Roots (2 of 5: Expanding in Rectangular Form)
The Triangle Inequalities (1 of 3: Sum of Complex Numbers)
Graphs in the Complex Plane (1 of 4: Introductory Examples)
The Triangle Inequalities (2 of 3: Discussing Specific Cases)
The Triangle Inequalities (3 of 3: Difference of Complex Numbers)
Graphs in the Complex Plane (2 of 4: Graphing Complex Inequalities)
Graphs in the Complex Plane (3 of 4 : Shifting the Point of Reference)
Graphs in the Complex Plane (4 of 4: Where is the argument measured from?)
Further Graphs on the Complex Plane (1 of 3: Geometrical Representation of Moduli)
Further Graphs on the Complex Plane (2 of 3: Algebraically verifying Graphs concerning the Moduli)
Further Graphs on the Complex Plane (3 of 3: Geometrical Representation of Arguments)
Graphs on the Complex Plane (3 of 4: Geometry of arg(z)-arg(z-1))
Graphs on the Complex Plane (4 of 4: Exploring how the argument traced the graph)
Graphs on the Complex Plane [Continued] (1 of 4: What's behind the graph?)
Graphs on the Complex Plane [Continued] (2 of 4: Finding Regions of Inequality by Testing Points)
Using Inverse tan to find arguments? (1 of 2: Why it doesn't work... Sometimes)
Complex Roots (1 of 5: Introduction)
Complex Roots (3 of 5: Through Polar Form Using De Moivre's Theorem)
Using Inverse tan to find arguments? (2 of 2: Why it works... Sometimes)
Complex Roots (4 of 5: Through Polar Form Generating Solutions)
Complex Roots (5 of 5: Flowing Example - Solving z^6=64)
Complex Conjugate Root Theorem (1 of 4: Using DMT and Polar Form to solve for Complex Roots)
Complex Conjugate Root Theorem (2 of 4: Introduction to the Conjugate Root Theorem)
Complex Conjugate Root Theorem (3 of 4: Geometrical Shape represented by Conjugate Root Theorem)
Complex Conjugate Root Theorem (4 of 4: Using Factorisation to find patterns with Roots of Unity)
DMT and Trig Identities (1 of 4: Deriving multi-angle identities with compound angles)
DMT and Trig Identities (2 of 4: Using De Moivre's Theorem and Binomial Expansions)
DMT and Trig Identities (3 of 4: Deriving tan expression from cos and sin)
DMT and Trig Identities (4 of 4: Using Multi-angle formula to solve polynomials)
Complex Numbers (1 of 6: Solving Harder Complex Numbers Questions) [Student requested problem]
Complex Numbers (2 of 6: Solving Harder Complex Numbers Questions) [Student Requested Problem]
Complex Numbers (3 of 6: Harder Complex Numbers Question) [Student Requested Problem]
Complex Numbers (4 of 6: Harder Complex Numbers Questions) [Student Requested Problem]
Complex Numbers (5 of 6: Complex Numbers Proofs [Using the Conjugate])
Complex Numbers (6 of 6: Finishing off the Proof)
Roots and Coefficients (1 of 3: Using DMT & Binomial Theorem to find identities)
Roots and Coefficients (2 of 3: Using Trigonometry to solve polynomial problems)
Roots and Coefficients (3 of 3: Using the results to find a relation in cosine)
Extension II Assessment Review (5 of 5: De Moivre's Theorem and Polynomials)
Why Complex Numbers? (1 of 5: Atoms & Strings)
Why Complex Numbers? (2 of 5: Impossible Roots)
Why Complex Numbers? (3 of 5: The Imaginary Unit)
Why Complex Numbers? (4 of 5: Turning the key)
Why Complex Numbers? (5 of 5: Where to now?)
Complex Arithmetic (1 of 2: Addition, Subtraction & Multiplication)
Complex Arithmetic (2 of 2: Division)
Factorisation with Complex Numbers
Vectors (1 of 4: Outline of vectors and their ability to represent complex number)
Vectors (2 of 4: Representing addition & subtraction of complex numbers with vectors)
Vectors (3 of 4: Geometrically representing multiplication of complex numbers with vectors)
Vectors (4 of 4: Outlining the usefulness of vectors in representing geometry)
Curves and Regions on the Complex Plane (1 of 4: Introductory example plotting |z|=5 geometrically)
Curves and Regions on the Complex Plane (2 of 4: Deciphering terminology to plot complex numbers)
Curves and Regions on the Complex Plane (3 of 4: Simplifying expressions to plot on a complex plane)
Curves and Regions on the Complex Plane (4 of 4: Plotting simultaneous shifted complex numbers)
Further Curves and Regions (1 of 5: Why does Sine & the Sine Rule produce an ambiguous case?)
Further Curves and Regions (2 of 5: Finding the Range of |z| )
Further Curves and Regions (3 of 5: Finding the range of arg z)
Further Curves and Regions (4 of 5: Geometrical expression of expressions of arg)
Further Curves and Regions (5 of 5: Using Circle properties to graph an expression of args)
DMT & Complex Roots (1 of 4: Reviewing geometrical expression of arg equations)
DMT & Complex Roots (2 of 4: Using DMT to find roots of a complex polynomial)
DMT & Complex Roots (3 of 4: Using the fundamental theorem of algebra to justify number of roots)
DMT & Complex Roots (4 of 4: Solving for roots of a complex number taking advantage of DMT)
Complex Roots (1 of 5: Observing Complex Conjugate Root Theorem through seventh roots of unity)
Complex Roots (2 of 5: Using Trigonometrical Identities & Conjugates to solve an equation)
Complex Roots (3 of 5: Using DMT to solve an equation of roots of unity)
Complex Roots (4 of 5: Using Polynomial Identities to prove unity identities)
Complex Roots (5 of 5: Using Geometric Progression to find factors of ω^n - 1)
DMT and Trig Identities (1 of 4: Noticing a pattern in natural numbers)
DMT and Trig Identities (2 of 4: Using Trig expansion to find the sine triple angle formula)
DMT and Trig Identities (3 of 4: Using DMT and Polynomials to verify triple angle formula for sine)
DMT and Trig Identities (4 of 4: Using Trig Identities to solve polynomial equations)
HSC Question on Complex Numbers, Vectors & Triangle Area (1 of 2: Thinking geometrically)
HSC Question on Complex Numbers, Vectors & Triangle Area (2 of 2: Manipulating trigonometric terms)
HSC Question on Complex Numbers, Vectors & Polynomials (1 of 2: How to "explain")
HSC Question on Complex Numbers, Vectors & Polynomials (2 of 2: Combining results)
2016 HSC - Complex Numbers on Unit Circle (1 of 2: Considering Re & Im Parts)
2016 HSC - Complex Numbers on Unit Circle (2 of 2: Evaluating the arguments)
2016 HSC - Complex Identity Proof (1 of 3: Convert to polar form)
2016 HSC - Complex Identity Proof (2 of 3: Using binomial theorem)
2016 HSC - Complex Identity Proof (3 of 3: Combining results)
Semi Circles on Argand Diagrams (3 of 3: Oblique example)
Semi Circles on Argand Diagrams (2 of 3: Graphing the locus)
Semi Circles on Argand Diagrams (1 of 3: Relating the angles)
Algebraic Proof for Opposing Rays (3 of 3: Testing cases)
Algebraic Proof for Opposing Rays (2 of 3: Generating the equation)
Algebraic Proof for Opposing Rays (1 of 3: Foundational knowledge)
cos⁴θ Identity
Graphs in the Complex Plane (3 of 3: Opposing rays)
Graphs in the Complex Plane (2 of 3: Algebraic method)
Graphs in the Complex Plane (1 of 3: Perpendicular bisector - visual method)
Complex Polynomial Identity Question (4 of 4: Roots & coefficients)
Complex Polynomial Identity Question (3 of 4: de Moivre's Theorem)
Complex Polynomial Identity Question (2 of 4: Difference of cubes)
Complex Polynomial Identity Question (1 of 4: Quadratic factors)
Sum & Product of Cosines (3 of 3: Drawing ℝ conclusions)
Sum & Product of Cosines (2 of 3: Simplifying with conjugates)
Sum & Product of Cosines (1 of 3: 9th roots of unity)
Nth Roots of a ℂ Number (2 of 2: Example problem)
Nth Roots of a ℂ Number (1 of 2: General form)
Roots of Unity (2 of 2: Insights from polar & exponential forms)
Roots of Unity (1 of 2: Evaluating the cube roots)
Complex Conjugate Root Theorem (2 of 2: Other conjugate properties)
Complex Conjugate Root Theorem (1 of 2: Conjugate of a sum)
Solving Higher Degree Trigonometric Equations (3 of 3: Finding solutions)
Solving Higher Degree Trigonometric Equations (2 of 3: Combining results into proof)
Solving Higher Degree Trigonometric Equations (1 of 3: Initial use of de Moivre's Theorem)
Equations with Complex Solutions (2 of 2: Solving & factorising)
Equations with Complex Solutions (1 of 2: Relation to square roots)
Polynomials with Trigonometric Solutions (3 of 3: Simplifying with identities)
Polynomials with Trigonometric Solutions (2 of 3: Substitute & solve)
Polynomials with Trigonometric Solutions (1 of 3: de Moivre's Theorem)
Trigonometric Expansions from Complex Numbers (3 of 3: General compound angles)
Trigonometric Expansions from Complex Numbers (2 of 3: Double angle results)
Trigonometric Expansions from Complex Numbers (1 of 3: Concept map)
Using de Moivre's Theorem - example question (2 of 2: Purely imaginary)
Using de Moivre's Theorem - example question (1 of 2: Purely real)
Proving de Moivre's Theorem (2 of 2: Derivation & example problem)
Proving de Moivre's Theorem (1 of 2: Prologue)
Arcs on the Complex Plane (1 of 4: Review questions)
Arcs on the Complex Plane (3 of 4: Identifying length and direction)
Arcs on the Complex Plane (2 of 4: Exploring circle properties)
Arcs on the Complex Plane (4 of 4: Cartesian equation)
Angles in the Same Segment
Algebraic Approach for Major Arc (1 of 2: Foundational steps)
Algebraic Approach for Major Arc (2 of 2: Identifying intercept & equation)
Max/Min Value of |z| (1 of 2: Geometric solution)
Max/Min Value of |z| (2 of 2: Triangle inequality)
Investigating de Moivre's Theorem (1 of 3: Why must we be cautious?)
Investigating de Moivre's Theorem (2 of 3: Infinite values for i-th powers?!)
Investigating de Moivre's Theorem (3 of 3: Proof by mathematical induction)
Varying |z| & Argz on a Locus
Maximising Sum of Moduli (1 of 3: Geometric approach)
Maximising Sum of Moduli (2 of 3: Differentiation)
Maximising Sum of Moduli (3 of 3: Interpreting stationary points)
Sketching (z-1)÷(z-i) (1 of 2: When it's real)
Sketching (z-1)÷(z-i) (2 of 2: When it's imaginary)
Complex Numbers Exam Review (1 of 4: Visualising & Manipulating Arithmetic)
Complex Numbers Exam Review (2 of 4: Proving i^i is real, identity proof)
Complex Numbers Exam Review (3 of 4: Cube roots of unity)
Complex Numbers Exam Review (4 of 4: Locus; polynomial identity)
Prove arg(z₁z₂) = arg(z₁+z₂)² (1 of 2: Preliminary thoughts)
Prove arg(z₁z₂) = arg(z₁+z₂)² (2 of 2: Geometric approach)
Complex Geometry - Square Problem (1 of 2: Complex numbers → vectors)
Complex Geometry - Square Problem (2 of 2: Vectors → complex numbers)
Complex Geometry - Equilateral Triangle (3 of 3: Vector proof)
Complex Geometry - Equilateral Triangle (2 of 3: Algebraic method)
Complex Geometry - Equilateral Triangle (1 of 3: Arithmetic proof)
The Basel Problem (9 of 9: Squeeze law)
The Basel Problem (8 of 9: Returning to trigonometric terms)
The Basel Problem (7 of 9: Manipulating the polynomial integral)
The Basel Problem (5 of 9: Telescoping sum)
The Basel Problem (6 of 9: Equations → inequalities)
The Basel Problem (4 of 9: Introducing x² to the integrand)
The Basel Problem (3 of 9: Integration by *different* parts)
The Basel Problem (1 of 9: Prologue)
The Basel Problem (2 of 9: Recurrence relation)
Centre of a Major Arc (5 of 5: Algebraic proof)
Centre of a Major Arc (4 of 5: Inscribed equilateral triangle)
Centre of a Major Arc (3 of 5: Using trigonometric & vectors)
Centre of a Major Arc (2 of 5: Finding centre and radius)
Centre of a Major Arc (1 of 5: Evaluating internal angle)
How to graph a region on the complex plane
Three ways to find a parallelogram's area