De Moivre's theorem — A/L Combined Mathematics

De Moivre's Theorem states that for any real number θ and any integer n ∈ ℤ: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). The theorem is established for positive integers n ∈ ℤ⁺ by mathematical induction, extended to n = 0 trivially, and to negative integers by writing n = -m (m ∈ ℤ⁺). Key applications in the A/L curriculum include: (1) Expressing cos(nθ) and sin(nθ) as polynomial powers of cos θ and sin θ by expanding (cos θ + i sin θ)ⁿ via the Binomial Theorem and equating real/imaginary parts; (2) Expressing powers cosⁿ θ and sinⁿ θ as sums of sines/cosines of multiple angles using z + 1/z = 2 cos θ and z - 1/z = 2i sin θ; (3) Solving polynomial equations and finding the n distinct n-th roots of unity.

Check before you answer (A/L Pure Maths Paper 1)

1. Proof by Mathematical Induction: Presenting the complete three-step induction proof of De Moivre's theorem for positive integers.
2. Multiple Angle Expansions: Setting (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ), expanding with binomial coefficients, and equating real parts for cos(nθ) and imaginary parts for sin(nθ).
3. Roots of Complex Numbers: Finding all n distinct roots using z_k = r^1/n [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] for k = 0, 1, ..., n - 1.