What Is De Moivre's Theorem and How Do You Use It?

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De Moivre's theorem gives

for integer $n$. Here, $r$ is the modulus, $\theta$ is an argument, and $\operatorname{cis}\theta=\cos\theta+i\sin\theta$.

The mechanism follows from multiplication in polar form: moduli multiply while arguments add. Multiplying the same number $n$ times therefore produces $r^n$ and the argument $n\theta$. This algebraic rule corresponds graphically to repeated scaling by $r$ and rotation by $\theta$ on the Argand diagram.

For example, calculate $(1+i)^4$. First convert $1+i$ to polar form:

Therefore,

To find all $n$th roots of $z=r\operatorname{cis}\theta$, use

The values of $k$ generate exactly $n$ roots, equally spaced by $\frac{2\pi}{n}$ on a circle of radius $\sqrt[n]{r}$ in the Argand diagram.

For example, the cube roots of $8$ have modulus $2$ and arguments $0$, $\frac{2\pi}{3}$, and $\frac{4\pi}{3}$. They are $2$, $-1+i\sqrt{3}$, and $-1-i\sqrt{3}$. Substituting any one of these into $w^3$ returns $8$, confirming the result.

Before applying the theorem, choose an argument in the correct quadrant; using a calculator's inverse tangent without checking signs can give the wrong angle.

A common misconception is to divide only the principal argument by $n$, producing one root. You must include $2k\pi$ to obtain every distinct root.

Exam technique: This is IB Math AA HL content from AHL 1.14. For a command term such as find, show the Cartesian-to-polar conversion, apply De Moivre's theorem, list all required roots, and convert back to Cartesian form if requested. Paper 1 answers should normally remain exact.