De Moivre & applications
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Question
State De Moivre's theorem.
Answer
(r cis θ)ⁿ = rⁿ cis(nθ) = rⁿ e^(inθ): power the modulus, multiply the argument by n.
Question
Evaluate (1 + i)⁸ using De Moivre.
Answer
r = √2, θ = π/4; (√2)⁸ cis(8·π/4) = 16 cis(2π) = 16.
Question
When is zⁿ a (positive) real number?
Answer
Real when nθ is a multiple of π; positive real when nθ is a multiple of 2π.
Question
What is the impedance of an AC circuit in complex form?
Answer
Z = R + iX, where R is resistance and X is reactance; |Z| is the total opposition and arg Z is the phase angle.
Question
How do impedances combine in series?
Answer
They add as complex numbers: Z_total = Z₁ + Z₂ + …
Question
How do you add two sinusoids of the same frequency?
Answer
Represent each as a phasor A e^(iφ), add the phasors, then read off the resultant amplitude (modulus) and phase (argument).
Question
Find |Z| for Z = 3 + 4i Ω.
Answer
|Z| = √(3² + 4²) = √25 = 5 Ω.
Question
Find z⁵ for z = 1 − √3 i.
Answer
r = 2, θ = −π/3; z⁵ = 32 cis(−5π/3) = 32 cis(π/3) = 16 + 16√3 i.
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Topic 1.13 hub
Complex numbers: continued (HL only)
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