De Moivre's theorem — A/L Combined Mathematics

What must I remember about De Moivre's theorem?

• De Moivre's Theorem for Integer Exponents: For any real angle θ and any integer n ∈ ℤ: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ).
• De Moivre's Theorem for Rational Exponents: For rational index p/q (q > 0), one of the values of (cos θ + i sin θ)^p/q is cos[(p/q)θ] + i sin[(p/q)θ]; the complete set of q distinct roots is given by cos[(θ + 2kπ)/q] + i sin[(θ + 2kπ)/q] for k = 0, 1, 2, ..., q - 1.
• Euler's Formula connection: e^iθ = cos θ + i sin θ; then (e^iθ)ⁿ = e^inθ = cos(nθ) + i sin(nθ).
• Expansion of cos(nθ) and sin(nθ) in powers of cos θ and sin θ: Expand (cos θ + i sin θ)ⁿ using Binomial theorem, equate real part to cos(nθ) and imaginary part to sin(nθ).
• Expressing powers cosⁿ θ and sinⁿ θ in terms of multiple angles: Let z = cos θ + i sin θ; then z + 1/z = 2 cos θ and z - 1/z = 2i sin θ; similarly zⁿ + 1/zⁿ = 2 cos(nθ) and zⁿ - 1/zⁿ = 2i sin(nθ); expand (2 cos θ)ⁿ or (2i sin θ)ⁿ using Binomial theorem.

Test yourself (1 minute)

1. n-th roots of unity

   Solutions to zⁿ = 1: z_k = cos(2kπ/n) + i sin(2kπ/n) = ω^k for k = 0, 1, ..., n - 1 where ω = e^{i 2π/n}; sum of all n roots is 0: 1 + ω + ω² + ... + ω^n-1 = 0.

2. Cube roots of unity

   1, ω = (-1 + i√3)/2, and ω² = (-1 - i√3)/2; key identities: 1 + ω + ω² = 0 and ω³ = 1.

3. Factorization of xⁿ - 1

   xⁿ - 1 = (x - 1) ∏_k=1^n-1 (x - ω^k); for odd n, factors into real linear and quadratic factors.

Practise De Moivre's theorem questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 21

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Year 2 · Term 1