De Moivre's Formula for Integer Powers

On this page

Interactive

A power multiplies the angle on the unit circle

PlayReset

Angle θ

30°

Power n

3

z and zⁿ on the unit circle

Move θ or n to rotate zⁿ.

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

Multiplying complex numbers in polar form multiplies their moduli and adds their arguments. De Moivre’s formula packages that rule for integer powers, making powers of points on the unit circle especially easy to compute.

Statement and domain

Write $z=r(\cos\theta+i\sin\theta)$ with $r\ge0$ and angle $\theta$ in radians. For every positive integer $n$,

$z^n=r^n\bigl(\cos(n\theta)+i\sin(n\theta)\bigr),$

including $z=0$. For $n=0$, both sides are defined and equal to $1$ when $z\ne0$. For negative integers $n$, require $z\ne0$; then the same identity holds, with the usual reciprocal meaning of $z^n$. Thus the formula is valid for all integer $n$ whenever the indicated power is defined.

Proof using Euler’s formula

For nonzero $z$, Euler’s formula gives $z=re^{i\theta}$. For positive integers, exponentiation yields

$$\begin{aligned} z^n &= (re^{i\theta})^n=r^n e^{in\theta}\\ &=r^n\bigl(\cos(n\theta)+i\sin(n\theta)\bigr). \end{aligned}$$

where $e^{i\phi}=\cos\phi+i\sin\phi$. For $n=0$, the identity is $1=1$. If $n<0$, take reciprocals of the positive-power identity; this is permitted precisely when $z\ne0$. When $z=0$ and $n>0$, both sides are zero. This establishes the stated cases.

Example and use

Let $z=\cos(\pi/6)+i\sin(\pi/6)$, a unit-modulus complex number, and take $n=3$. Then

$$\begin{aligned} z^3 &= \cos(3\pi/6)+i\sin(3\pi/6)\\ &=\cos(\pi/2)+i\sin(\pi/2)=i. \end{aligned}$$

The calculation triples the argument while keeping the modulus $1$. More generally, the formula reduces an integer power to a modulus power and an angle multiplication; it is useful when those polar data are easier to work with than repeated Cartesian multiplication.