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De Moivre's Theorem

mathematics complex analysis trigonometry
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De Moivre’s Theorem, named after the French mathematician Abraham de Moivre, is a fundamental result in complex numbers and trigonometry. It states that:

$$ (\cos x + i \sin x)^n = \cos(nx) + i\sin(nx) $$

Where:

  • \( \small \small x\ \) is a real number representing the angle in radians,
  • \( \small \small n\ \) is an integer exponent,
  • \( \small \small i\ \) is the imaginary unit, \( \small \small i^2 = -1\ \),
  • \( \small \small \cos\ \) and \( \small \small \sin\ \) are the trigonometric functions cosine and sine, respectively.

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