:PROPERTIES: :ID: 80dd8196-4453-481d-9a07-c08914a877cb :mtime: 20231024011132 :ctime: 20231024011047 :END: #+title: De Moivre's Theorem #+filetags: :project:public: * De Moivre's Theorem De Moivre's Theorem is a theorem about [[id:2553e0fb-12c1-42cc-8e47-54937a36e2c7][complex number]] which deals with raising a complex number to a power \[(\cos\theta + i \sin \theta)^{n} = \cos(n\theta) + i \sin(n\theta)\] * Examples ** Example: Quadruple Angle Formula *** Question Use De Moivre's Theorem to find an expression for $\cos\4\theta$ and $\sin\4\theta$ in terms of $\cos\theta$ and $\sin\theta$? *** Solution \begin{align} &cos4\theta + i\sin\4\theta \\ &= (\cos^{4}\theta - 6\cos^{2}\theta\sin^{2}\theta + \sin^{4}\theta) +4i (\cos^{3}\theta\sin\theta - \cos\theta\sin^{3}\theta) \end{align} So let's equate the real and imaginary parts ** Example: Exponential forms for sin, cos \[exp(-i\theta) = \cos\theta - i\sin\theta\] so \[\cos\theta = \frac{exp(i\theta) + exp(-i\theta)}{2}\] and \[\cos\theta = \frac{exp(i\theta) - exp(-i\theta)}{2i}\] ** Example: Evaluate a sum *** Question Evaluate \[\sum_{ k = 0}^{N - 1} \cos(k\theta)\] *** Solution \begin{align} &S_n \\ &= Re(\sum_{k = 0}^{N - 1} exp(i k \theta)) \\ &= Re(\sum_{k = 0}^{N - 1} exp(i \theta))^{k} \\ \end{align}