複素数のｎ乗を求める（ド・モアブルの定理）

ド・モアブルの定理

n

が整数のとき

(\cos\theta + i \sin\theta)^n = \cos n \theta + i \sin n \theta

\left(\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}\right)^4

\begin{align*} & \textcolor{orange}{\small\phase{　ド・モアブルの定理！}}\\ &\textcolor{white}{=} \left(\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}\right)^{\textcolor{red}{4}}\\\\ & \textcolor{orange}{\small\phase{　\theta に 4 をかける}}\\ &= \cos \textcolor{red}{4 \times} \dfrac{\pi}{3}+i \sin \textcolor{red}{4 \times} \dfrac{\pi}{3}\\\\ &= \cos\dfrac{4}{3}\pi +i \sin\dfrac{4}{3}\pi\\\\ &= \textcolor{red}{\underline{\underline{\textcolor{black}{\ -\dfrac12 - \dfrac{\sqrt{3}}{2}i\ }}}} \end{align*}

\left(\cos\dfrac{\pi}{6}+i\sin\dfrac{\pi}{6}\right)^{-6}

\begin{align*} & \textcolor{orange}{\small\phase{　ド・モアブルの定理！}}\\ &\textcolor{white}{=} \left(\cos\dfrac{\pi}{6}+i\sin\dfrac{\pi}{6}\right)^{\textcolor{red}{-6}}\\\\ & \textcolor{orange}{\small\phase{　\theta に -6 をかける}}\\ &= \cos \textcolor{red}{(-6) \times} \dfrac{\pi}{6}+i \sin \textcolor{red}{(-6) \times} \dfrac{\pi}{6}\\\\ &= \cos (-\pi) +i \sin (-\pi)\\\\ & \textcolor{orange}{\small\phase{　\cos(-\theta)=\cos \theta}}　\textcolor{orange}{\small\phase{　\sin(-\theta)=-\sin \theta}}\\ &= \cos \pi -i \sin \pi\\\\ &= -1 - i \cdot 0\\ &= \textcolor{red}{\underline{\underline{\textcolor{black}{\ -1\ }}}} \end{align*}

\left(\cos\dfrac{\pi}{30}+i\sin\dfrac{\pi}{30}\right)^{5}

\begin{align*} & \textcolor{orange}{\small\phase{　ド・モアブルの定理！}}\\ &\textcolor{white}{=} \left(\cos\dfrac{\pi}{30}+i\sin\dfrac{\pi}{30}\right)^{\textcolor{red}{5}}\\\\ & \textcolor{orange}{\small\phase{　\theta に 5 をかける}}\\ &= \cos \textcolor{red}{5 \times} \dfrac{\pi}{30}+i \sin \textcolor{red}{5 \times} \dfrac{\pi}{30}\\\\ &= \cos \dfrac{\pi}{6} +i \sin \dfrac{\pi}{6}\\\\ &= \textcolor{red}{\underline{\underline{\textcolor{black}{\ \dfrac{\sqrt{3}}{2} + \dfrac12 i\ }}}} \end{align*}

\left(\cos\dfrac{2}{3}\pi+i\sin\dfrac{2}{3}\pi\right)^{-5}

\begin{align*} & \textcolor{orange}{\small\phase{　ド・モアブルの定理！}}\\ &\textcolor{white}{=} \left(\cos\dfrac{2}{3}\pi+i\sin\dfrac{2}{3}\pi\right)^{\textcolor{red}{-5}}\\\\ & \textcolor{orange}{\small\phase{　\theta に -5 をかける}}\\ &= \cos \textcolor{red}{(-5) \times} \dfrac{2}{3}\pi+i \sin \textcolor{red}{(-5) \times} \dfrac{2}{3}\pi\\\\ &= \cos \left(-\dfrac{10}{3}\pi\right) +i \sin \left(-\dfrac{10}{3}\pi\right)\\\\ & \textcolor{orange}{\small\phase{　\cos(-\theta)=\cos \theta}}　\textcolor{orange}{\small\phase{　\sin(-\theta)=-\sin \theta}}\\ &= \cos \dfrac{10}{3}\pi -i \sin \dfrac{10}{3}\pi\\\\ & \textcolor{orange}{\scriptsize\phase{　三角関数は\ 2\pi \times n\ 足しても引いても良い！}}\\&= \cos \left(\dfrac{10}{3}\pi-\dfrac{6}{3}\pi\right) -i \sin \left(\dfrac{10}{3}\pi-\dfrac{6}{3}\pi\right)\\\\ &= \cos \dfrac{4}{3}\pi -i \sin \dfrac{4}{3}\pi\\\\ &= \textcolor{red}{\underline{\underline{\textcolor{black}{\ -\dfrac12 +\dfrac{\sqrt{3}}{2} i\ }}}} \end{align*}

\left\{2\left(\cos\dfrac{\pi}{18}+i\sin\dfrac{\pi}{18}\right)\right\}^{6}

\begin{align*} & \textcolor{orange}{\small\phase{　ド・モアブルの定理！}}\\ &\textcolor{white}{=} \left\{2\left(\cos\dfrac{\pi}{18}+i\sin\dfrac{\pi}{18}\right)\right\}^{6}\\\\ & \textcolor{orange}{\small\phase{　(ab)^n = a^nb^n}}\\ &= 2^6\left(\cos\dfrac{\pi}{18}+i\sin\dfrac{\pi}{18}\right)^{\textcolor{red}{6}}\\\\ & \textcolor{orange}{\small\phase{　\theta に 6 をかける}}\\ &= 64\left(\cos \textcolor{red}{6 \times} \dfrac{\pi}{18}+i \sin \textcolor{red}{6 \times} \dfrac{\pi}{18}\right)\\\\ &= 64\left(\cos \dfrac{\pi}{3} +i \sin \dfrac{\pi}{3}\right)\\\\ &= 64\left(\dfrac12 +i \cdot \dfrac{\sqrt{3}}{2}\right)\\\\ &= \textcolor{red}{\underline{\underline{\textcolor{black}{\ 32+32\sqrt{3}\,i\ }}}} \end{align*}

\left(\cos\dfrac{\pi}{6}-i\sin\dfrac{\pi}{6}\pi\right)^{3}

\begin{align*} & \textcolor{orange}{\small\phase{　極形式ではない⇒修正}}\\ &\textcolor{white}{=} \left(\cos\dfrac{\pi}{6}\pi-i\sin\dfrac{\pi}{6}\right)^{3}\\\\ & \textcolor{orange}{\small\phase{　\cos(-\theta)=\cos \theta}}　\textcolor{orange}{\small\phase{　\sin(-\theta) = -\sin\theta}}\\ &= \left\{\cos\left(-\dfrac{\pi}{6}\right)+i\sin\left(-\dfrac{\pi}{6}\right)\right\}^{\textcolor{red}{3}}\\\\ & \textcolor{orange}{\small\phase{　ド・モアブルの定理！}}\\ & \textcolor{orange}{\small\phase{　\theta に 3 をかける}}\\ &= \cos \textcolor{red}{3 \times} \left(-\dfrac{\pi}{6}\right)+i \sin \textcolor{red}{3 \times} \left(-\dfrac{\pi}{6}\right)\\\\ &= \cos \left(-\dfrac{\pi}{2}\right) +i \sin \left(-\dfrac{\pi}{2}\pi\right)\\\\ & \textcolor{orange}{\small\phase{　\cos(-\theta)=\cos \theta}}　\textcolor{orange}{\small\phase{　\sin(-\theta) = -\sin\theta}}\\ &= \cos \dfrac{\pi}{2} -i \sin \dfrac{\pi}{2}\\\\ &= 0-i \cdot 1\\\\ &= \textcolor{red}{\underline{\underline{\textcolor{black}{\ -i\ }}}} \end{align*}

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