Complex numbers and de Moivre's theorem

Learning objectives

State de Moivre's theorem, (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, prove it by induction for positive integer n, and extend its use to negative and rational exponents

Express cos nθ and sin nθ as polynomials in cos θ and sin θ by expanding (cos θ + i sin θ)ⁿ with the binomial theorem and comparing real and imaginary parts

Use z + 1/z = 2 cos θ and z − 1/z = 2i sin θ with the expansion of (z ± 1/z)ⁿ to write cosⁿθ and sinⁿθ as sums of cosines or sines of multiple angles, ready for integration

Find the n distinct nth roots of a complex number in modulus–argument form, show that they lie equally spaced on a circle of radius |z|^(1/n) at intervals of 2π/n, and use the nth roots of unity to factorise expressions such as zⁿ − 1

Sum a trigonometric series such as Σcos rθ by treating it as the real part of a geometric series with complex common ratio, applying the geometric sum formula and simplifying the result

Use the exponential form z = re^(iθ) together with Euler's relation e^(iθ) = cos θ + i sin θ to multiply, divide and take powers of complex numbers efficiently