De Moivre's formula

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 ${\big (}\cos x+i\sin x{\big )}^{n}=\cos nx+i\sin nx,$ 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.^[1] The expression $cos x + i sin x$ is sometimes abbreviated to $cis x$ .

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 $n$ th 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.

^ Lial, Margaret L.; Hornsby, John; Schneider, David I.; Callie J., Daniels (2008). College Algebra and Trigonometry (4th ed.). Boston: Pearson/Addison Wesley. p. 792. ISBN 9780321497444.

[1] Lial, Margaret L.; Hornsby, John; Schneider, David I.; Callie J., Daniels (2008). College Algebra and Trigonometry (4th ed.). Boston: Pearson/Addison Wesley. p. 792. ISBN 9780321497444.

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