De Moivre’s theorem, n-th roots and square roots of complex numbers

Last week you saw that multiplying two complex numbers in trigonometric form multiplies the moduli and adds the amplitudes. Multiply a number by itself n times and the modulus is raised to the power n, while the amplitude is added n times. For a number on the unit circle this is De Moivre’s theorem (in Arabic نظرية ديموافر): (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for n = 1, 2, 3, … It gives powers at once, with no repeated multiplication. If the number is not yet in trigonometric form, find its modulus r and amplitude θ first; then the power is rⁿ(cos nθ + i sin nθ), and any amplitude beyond 360° can be reduced by subtracting full turns. The theorem has a second use. The left side can also be expanded with the binomial theorem of Unit 1, remembering i² = −1, i³ = −i and i⁴ = 1. Both expansions are the same complex number, so their real parts agree and their imaginary parts agree. The real parts give cos nθ and the imaginary parts give sin nθ, written with powers of cos θ and sin θ. To keep lines short, write c for cos θ and s for sin θ. For n = 2: (c + is)² = c² − s² + 2cs·i, so cos 2θ = c² − s² and sin 2θ = 2sc, formulas you already know. At the end use s² + c² = 1 to remove one of the two letters if you want only one.