De Moivre's Theorem
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De Moivre’s Theorem is a fundamental statement of complex analysis, where i represents the square root of (-1):
- <math>\left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right)\,</math>
Extension of Euler's formula
De Moivre's formula is a trivial extension of Euler's formula:
- <math>e^{ix} = \cos x + i\sin x\,</math>
Because
- <math>\left( e^{ix} \right)^n = e^{inx} \,</math>
Therefore from Euler's formula:
- <math>e^{i(nx)} = \cos(nx) + i\sin(nx)\,</math>