De Moivre's formula

De Moivre's formula states that for any real number x and any integer n,

Related Topics:
Real number - Integer

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:(cos x+isin x)^n=cos(nx)+isin(nx).

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The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. The expression "cos x + i sin x" is sometimes abbreviated to "cis x".

Related Topics:
Complex numbers - Imaginary unit - Trigonometry

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By expanding the left hand side and then comparing the real and imaginary parts, it is possible to derive

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useful expressions for cos(nx) and sin(nx) in terms of cos(x) and sin(x). Furthermore, one can use this formula to find explicit expressions for the n-th roots of unity, that is, complex numbers z such that zn = 1.

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Abraham de Moivre was a good friend of Newton; in 1698 he wrote that the formula had been known to Newton as early as 1676. It can be derived from (but historically preceded) Euler's formula eix = cos x + i sin x

Related Topics:
Abraham de Moivre - Newton - Euler's formula

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and the exponential law

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(eix)n = einx (see exponential function).

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De Moivre's formula is actually true in a more general setting than stated above: if z and w are complex numbers, then (cos z + i sin z)w is a multivalued function while cos (wz) + i sin (wz) is not, and one can state that

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:cos (wz) + i sin (wz) is one value of (cos z + i sin z)w.

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