De Moivre’s theorem: zⁿ = rⁿ (cos nθ + i sin nθ).

z = 1 + i ⇒ r = √2, θ = π/4. z⁴ = r⁴ (cos 4θ + i sin 4θ) = (√2)⁴ (cos π + i sin π) = 4 (cos π + i sin π).

De Moivre’s theorem is primarily used to calculate powers and roots of complex numbers.

(cos θ + i sin θ)⁵ = cos 5θ + i sin 5θ by De Moivre’s theorem.

Cube roots of unity are the three complex solutions to z³ = 1.

General formula for n-th roots of a complex number using De Moivre’s theorem.

Square roots of z = r(cos θ + i sin θ) are w₀ = √r [cos (θ/2) + i sin (θ/2)] and w₁ = √r [cos (θ/2 + π) + i sin (θ/2 + π)].

We expand (cos θ + i sin θ)⁵ to express higher powers of sine and cosine.

Using Euler’s form: (r e^(iθ))ⁿ = rⁿ e^(i n θ), consistent with De Moivre’s theorem.

De Moivre’s theorem helps in deriving multiple-angle formulas in trigonometry.