De Moivre's Theorem

The theorem

For any real angle θ and any integer n:

(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)

Equivalently, in exponential form:

(e^(iθ))^n = e^(inθ)

The geometric content is simple: raising a unit-modulus complex number to the power n multiplies its argument by n. The point at angle θ on the unit circle, raised to the n-th power, lands at angle nθ. The proof is the only tricky bit, and only if you stay inside the cos/sin formulation.

Proof by induction

The classical proof, before Euler's formula was available, uses induction on n.

Base case (n = 1). (cos θ + i sin θ)^1 = cos(1·θ) + i sin(1·θ). Trivially true.

Inductive step. Assume the theorem holds for n = k:

(cos θ + i sin θ)^k = cos(kθ) + i sin(kθ)

Multiply both sides by (cos θ + i sin θ):

(cos θ + i sin θ)^(k+1)
   = (cos(kθ) + i sin(kθ))(cos θ + i sin θ)
   = cos(kθ)cos θ − sin(kθ)sin θ + i(sin(kθ)cos θ + cos(kθ)sin θ)
   = cos((k+1)θ) + i sin((k+1)θ)

The last step uses the angle-addition identities for cos and sin. By induction the theorem holds for all positive integers n. For n = 0, both sides equal 1. For negative n, take reciprocals — (cos θ + i sin θ)^(−1) = cos θ − i sin θ = cos(−θ) + i sin(−θ) — which extends the result to all integers.

Proof via Euler's formula (one line)

Euler's formula e^(iθ) = cos θ + i sin θ reduces De Moivre to the product-of-powers law for the exponential function:

(cos θ + i sin θ)^n = (e^(iθ))^n = e^(inθ) = cos(nθ) + i sin(nθ)

Done. Once you have Euler, De Moivre is a corollary. Historically the order is reversed — De Moivre (1707) preceded Euler's formula (1748) — so the induction proof is the original.

Multiple-angle formulas in one expansion

Expanding the left side with the binomial theorem and matching real and imaginary parts to the right side gives multiple-angle formulas in one shot.

Double angle (n = 2).

(cos θ + i sin θ)² = cos²θ − sin²θ + 2i sin θ cos θ
                   = cos(2θ) + i sin(2θ)

⇒  cos(2θ) = cos²θ − sin²θ
   sin(2θ) = 2 sin θ cos θ

Triple angle (n = 3).

(cos θ + i sin θ)³ = cos³θ + 3i cos²θ sin θ − 3 cos θ sin²θ − i sin³θ
                   = (cos³θ − 3 cos θ sin²θ) + i(3 cos²θ sin θ − sin³θ)
                   = cos(3θ) + i sin(3θ)

⇒  cos(3θ) = cos³θ − 3 cos θ sin²θ = 4 cos³θ − 3 cos θ
   sin(3θ) = 3 cos²θ sin θ − sin³θ = 3 sin θ − 4 sin³θ

For any n, the binomial coefficients give the multiple-angle formulas in one go:

cos(nθ) = Σ_{k even} C(n,k) (−1)^(k/2) cos^(n−k)θ · sin^k θ
sin(nθ) = Σ_{k odd}  C(n,k) (−1)^((k−1)/2) cos^(n−k)θ · sin^k θ

De Moivre vs related results

	De Moivre	Euler's formula	Binomial theorem
Statement	(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)	e^(iθ) = cos θ + i sin θ	(a + b)^n = Σ C(n,k) a^(n−k) b^k
Year	1707	1748	~1665 (Newton's general form)
Domain of n	Integer (all reals if branched)	Real (or complex) θ	Non-negative integer (Newton: any real)
Geometric content	Powers = repeated rotation	Imaginary exp = unit circle	Polynomial expansion
Used to derive	Multiple-angle formulas, n-th roots	De Moivre, Fourier, ODEs	Combinatorial identities
Special case n = 1	Trivial identity	e^(iθ) = e^(iθ)	a + b = a + b
Implies the other	Follows from Euler	Derives De Moivre directly	Used inside De Moivre's expansion

Application — n-th roots of a complex number

De Moivre's most powerful corollary: explicit formulas for n-th roots. To solve z^n = w for any complex w, write w = R·e^(iφ) = R(cos φ + i sin φ). Then:

z_k = R^(1/n) · (cos((φ + 2πk)/n) + i sin((φ + 2πk)/n))
    = R^(1/n) · e^(i(φ + 2πk)/n)         k = 0, 1, ..., n−1

The n roots lie on a circle of radius R^(1/n), evenly spaced 2π/n apart. The +2πk term covers the n distinct branches; missing it gives only the principal root.

Example. Cube roots of 8 (so w = 8, R = 8, φ = 0):

z_0 = 8^(1/3) · e^(i·0)       = 2
z_1 = 8^(1/3) · e^(i·2π/3)    = 2(cos 120° + i sin 120°) = −1 + i√3
z_2 = 8^(1/3) · e^(i·4π/3)    = 2(cos 240° + i sin 240°) = −1 − i√3

Three distinct cube roots. The real cube root 2 is just one of them; the other two are complex conjugates of each other.

Where De Moivre shows up

Fourier analysis

The Discrete Fourier Transform's twiddle factors are powers of ω = e^(2πi/n). De Moivre's theorem ensures these powers cycle through the n-th roots of unity cleanly, which is what makes FFT recursion work.

Solving polynomial equations

Cubic and quartic formulas use De Moivre to extract complex roots from a polynomial's depressed form. Cardano's casus irreducibilis — three real roots that require complex intermediate values — relies on De Moivre to produce the real answers from cosines of one-third the original argument.

Trig identity derivation

Sum-to-product, product-to-sum, half-angle, and many other trigonometric identities derive from De Moivre by setting n equal to small integers. Calculators of trig identities embedded in symbolic systems (Mathematica, SymPy) often go through this representation internally.

Signal modulation

In digital communications, modulating a signal by a carrier corresponds to multiplying its complex envelope by e^(iωt). Discrete-time sample sequences become powers of e^(iω·Δt), and De Moivre describes how successive samples rotate through phase space.

Counterexamples and cautions

• De Moivre is multivalued for non-integer n. (cos θ + i sin θ)^(1/2) has two values: cos(θ/2) + i sin(θ/2) and cos(θ/2 + π) + i sin(θ/2 + π) = −(cos(θ/2) + i sin(θ/2)). The naive single-valued reading misses half the roots.
• The +2πk term is mandatory for roots. A common error: z^3 = 8 ⇒ z = 2. There are three cube roots, not one. The +2πk in (φ + 2πk)/n generates all of them.
• De Moivre fails for arbitrary complex base. The theorem applies to cos θ + i sin θ — a unit-modulus number. For z = re^(iθ) with r ≠ 1, the correct generalization is z^n = r^n · (cos(nθ) + i sin(nθ)); the modulus must be raised to n separately.
• Fractional powers of negative reals. (−1)^(1/2) in real notation gives ±i, two values. Trying to apply De Moivre with θ = π and n = 1/2 to get one answer is wrong; both ±i are valid square roots of −1.
• Branch cuts for irrational exponents. (cos θ + i sin θ)^π is not single-valued; it depends on a branch choice (which spiral arm of the multivalued log to pick). Be explicit about branch when working with non-integer exponents.

Common mistakes

• Forgetting to raise the modulus. (re^(iθ))^n = r^n · e^(inθ), not r · e^(inθ). De Moivre's clean form assumes r = 1.
• Stopping at the principal root. Solving z^n = w gives n roots, not one. Always include the +2πk term and let k run from 0 to n−1.
• Mishandling negative n. (cos θ + i sin θ)^(−1) = cos(−θ) + i sin(−θ) = cos θ − i sin θ. The reciprocal of a unit-modulus complex number is its conjugate.
• Mixing degrees and radians. Multiple-angle formulas derived from De Moivre use the same units throughout. cos(3θ) = 4cos³θ − 3cos θ works whether θ is in degrees or radians, but be consistent.
• Confusing De Moivre with binomial theorem. The expansion of (cos θ + i sin θ)^n via binomial theorem is the bridge to multiple-angle formulas — not the theorem itself. De Moivre is the equality between the expansion and cos(nθ) + i sin(nθ).
• Treating (z^a)^b = z^(ab) as universal. For non-integer a, b this fails on principal branches. ((−1)^2)^(1/2) = 1^(1/2) = 1, but (−1)^(2·1/2) = (−1)^1 = −1. De Moivre obeys the law for integers; outside that, branch issues bite.