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What is De-Moivre's Theorem?

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De-Moivre's Theorem

  • If $n$ is a rational number, then $(\cos \theta+i \sin \theta)^n=\cos n \theta+i \sin n \theta$
  • If $z=\left(\cos \theta_1+i \sin \theta_1\right)\left(\cos \theta_2+i \sin \theta_2\right) \ldots$ $\left(\cos \theta_n+i \sin \theta_n\right)$, then

$$z=\cos \left(\theta_1+\theta_2+\ldots+\theta_n\right)+i \sin\left(\theta_1+\theta_2+\ldots+\theta_n\right)$$

  • if $z=r(\cos \theta+i \sin \theta)$ and $n$ is a positive integer. Then,$$

$(z)^{1 / n}=r^{1 / n}\left[\cos \left(\frac{2 k \pi+\theta}{n}\right)+i \sin \left(\frac{2 k \pi+\theta}{n}\right)\right]$

where $k=0,1,2,3, \ldots,(n-1)$

  • $(\cos \theta-i \sin \theta)^n=\cos n \theta-i \sin n \theta$
  • $\frac{1}{\cos \theta+i \sin \theta}=(\cos \theta+i \sin \theta)^{-1}=\cos \theta-i \sin \theta$
  • $(\sin \theta \pm i \cos \theta)^n \neq \sin n \theta \pm i \cos n \theta$
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