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De Moivre's Formula — Definition, Formula & Examples

De Moivre's Formula states that to raise a complex number in polar form to the nnth power, you raise the modulus to the nnth power and multiply the argument by nn.

For any complex number z=r(cos⁡θ+isin⁡θ)z = r(\cos\theta + i\sin\theta) and any integer nn, De Moivre's Formula gives zn=rn(cos⁡(nθ)+isin⁡(nθ))z^n = r^n(\cos(n\theta) + i\sin(n\theta)).

Key Formula

[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡(nθ)+isin⁡(nθ))\bigl[r(\cos\theta + i\sin\theta)\bigr]^n = r^n\bigl(\cos(n\theta) + i\sin(n\theta)\bigr)
Where:
  • rr = Modulus (absolute value) of the complex number
  • θ\theta = Argument (angle) of the complex number in radians
  • nn = Integer exponent

How It Works

First, convert your complex number to polar (trigonometric) form r(cos⁡θ+isin⁡θ)r(\cos\theta + i\sin\theta). Then apply the formula: raise rr to the power nn and multiply the angle θ\theta by nn. This avoids the tedious algebra of multiplying (a+bi)(a + bi) by itself repeatedly. The formula also extends to finding nnth roots by using n=1nn = \frac{1}{n} and considering all nn equally spaced angles.

Worked Example

Problem: Compute (1+i)6(1 + i)^6 using De Moivre's Formula.
Convert to polar form: Find the modulus and argument of 1+i1 + i.
r=12+12=2,θ=π4r = \sqrt{1^2 + 1^2} = \sqrt{2}, \quad \theta = \frac{\pi}{4}
Apply De Moivre's Formula: Raise the modulus to the 6th power and multiply the angle by 6.
(2)6(cos⁡6π4+isin⁡6π4)=8(cos⁡3π2+isin⁡3π2)(\sqrt{2})^6\left(\cos\frac{6\pi}{4} + i\sin\frac{6\pi}{4}\right) = 8\left(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2}\right)
Evaluate: Use known trig values: cos⁡3π2=0\cos\frac{3\pi}{2} = 0 and sin⁡3π2=−1\sin\frac{3\pi}{2} = -1.
8(0+i(−1))=−8i8(0 + i(-1)) = -8i
Answer: (1+i)6=−8i(1 + i)^6 = -8i

Why It Matters

De Moivre's Formula is essential in precalculus and engineering courses whenever you need to compute powers or roots of complex numbers. It also provides an elegant way to derive trigonometric identities for cos⁡(nθ)\cos(n\theta) and sin⁡(nθ)\sin(n\theta) by expanding the left side with the binomial theorem.

Common Mistakes

Mistake: Multiplying the modulus by nn instead of raising it to the nnth power.
Correction: The modulus gets exponentiated: rnr^n, not r⋅nr \cdot n. Only the angle is multiplied by nn.