# Wikipedia Authors - De Moivre's Formula ## Metadata **Author**:: [[Wikipedia Authors]] **Created**:: [[2025-04-24]] **Host**:: [[en.wikipedia.org]] **Source**:: #from/clipper **Status**:: #x **Title**:: De Moivre's formula **URL**:: [en.wikipedia.org](https://en.wikipedia.org/wiki/De_Moivre's_formula) **Zettel**:: #zettel/fleeting ## Synopsis In mathematics, **de Moivre's formula** (also known as **de Moivre's theorem** and **de Moivre's identity**) states that for any real number $x$ and integer $n$ it is the case that $\displaystyle {\big (}\cos x+i\sin x{\big )}^{n}=\cos nx+i\sin nx~,$ where $i$ is the imaginary unit ($i^2 = −1$). The formula is named after Abraham de Moivre, although he never stated it in his works. The expression $\cos x + i \sin x$ is sometimes abbreviated to $\textrm{cis } x$. The formula is important because it connects *complex numbers* and *trigonometry*. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for $\cos nx$ and $\sin nx$ in terms of $\cos x$ and $\sin x$. As written, the formula is not valid for non-integer powers $n$. However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the nth roots of unity, that is, complex numbers $z$ such that $z^n = 1$. Using the standard extensions of the sine and cosine functions to complex numbers, the formula is valid even when $x$ is an arbitrary complex number. ## Example For $\displaystyle x=30^{\circ }$ and $n = 2$, de Moivre's formula asserts that $\displaystyle \left(\cos(30^{\circ })+i\sin(30^{\circ })\right)^{2}=\cos(2\cdot 30^{\circ })+i\sin(2\cdot 30^{\circ })~,$ or equivalently that $\displaystyle \left({\frac {\sqrt {3}}{2}}+{\frac {i}{2}}\right)^{2}={\frac {1}{2}}+{\frac {i{\sqrt {3}}}{2}}~.$ In this example, it is easy to check the validity of the equation by multiplying out the left side.