De Moivre's TheoremJEE Main

Powers of complex numbers spiral — interactive Mathematics simulation for IIT-JEE.

Concept

In polar form, multiplying complex numbers is geometric: moduli multiply, arguments add. De Moivre is the n-fold repeat: $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$ . Powers of z march around the origin in equal angular steps while the modulus follows rⁿ — a perfect logarithmic spiral.

Key formula

z^n = r^n(\cos n\theta + i\sin n\theta), \qquad z = re^{i\theta} \Rightarrow z^n = r^ne^{in\theta}

Derivation

One multiplication: $r_1e^{i\theta_1} \cdot r_2e^{i\theta_2} = r_1r_2\,e^{i(\theta_1+\theta_2)}$ (angle-addition formulas in disguise). Induct n times.

Runs backwards too: n-th roots have modulus $r^{1/n}$ and arguments $(\theta + 2k\pi)/n$ — n of them, equally spaced. Also generates trig identities: expand $(\cos\theta + i\sin\theta)^3$ binomially and match parts to get cos 3θ and sin 3θ.

Scenarios to explore

De Moivre's Theorem — Powers of complex numbers spiral around the Argand plane.

Real-world applications

Fast computation of large powers: (1+i)²⁰ in two lines.
cos nθ / sin nθ expansion identities.
AC phasors & rotations in the plane.

JEE exam tips

(1+i) = √2·cis 45° — memorise; its powers cycle every 8.
iⁿ cycles with period 4; ωⁿ (cube root of unity) with period 3.
z + 1/z = 2cosθ when |z| = 1 ⇒ zⁿ + 1/zⁿ = 2cos nθ — a JEE workhorse.

Common mistakes

Applying De Moivre to (cosθ − i·sinθ)ⁿ without rewriting as cos(−θ) + i·sin(−θ).
Forgetting to reduce nθ mod 2π before converting back.
(cos α + i sin β)ⁿ with α ≠ β — De Moivre does NOT apply.

Exam traps to avoid

De Moivre is proved for integers; for fractional n it gives ONE of several values.
cis(90°)⁴ = cis(360°) = 1: i⁴ = 1 — sanity check every polar answer.