![\bf \textit{ roots of complex numbers, DeMoivre's theorem} \\\\ \sqrt[n]{z}=\sqrt[n]{r}\left[ cos\left( \cfrac{\theta+2\pi k}{n} \right) +i\ sin\left( \cfrac{\theta+2\pi k}{n} \right)\right]\quad \begin{array}{llll} k\ roots\\ 0,1,2,3,... \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \left[ 32\left[ cos\left( \frac{\pi }{3} \right) +i~sin\left( \frac{\pi }{3} \right) \right] \right]^{\frac{1}{5}}](/tpl/images/0213/9727/228c7.png)
![\bf 32^{\frac{1}{5}}\left[ cos\left( \frac{\pi +2\pi (0)}{15} \right)+i~sin\left( \frac{\pi +2\pi (0)}{15} \right) \right]\implies \stackrel{\textit{first root, k = 0}}{2\left[ cos\left( \frac{\pi }{15} \right)+isin\left( \frac{\pi }{15} \right) \right]}](/tpl/images/0213/9727/1ff77.png)
![\bf 32^{\frac{1}{5}}\left[ cos\left( \frac{\pi +2\pi (1)}{15} \right)+i~sin\left( \frac{\pi +2\pi (1)}{15} \right) \right]\implies \stackrel{\textit{second root, k = 1}}{2\left[ cos\left( \frac{7\pi }{15} \right)+isin\left( \frac{7\pi }{15} \right) \right]} \\\\\\ \stackrel{\textit{third root, k = 2}}{2\left[ cos\left( \frac{13\pi }{15} \right)+isin\left( \frac{13\pi }{15} \right) \right]}\qquad \bigotimes](/tpl/images/0213/9727/cd451.png)
![\bf \stackrel{\textit{fourth root, k = 3}}{2\left[ cos\left( \frac{19\pi }{15} \right)+isin\left( \frac{19\pi }{15} \right) \right]}\qquad \bigotimes \\\\\\ \stackrel{\textit{fifth root, k =4}}{2\left[ cos\left( \frac{5\pi }{3} \right)+isin\left( \frac{5\pi }{3} \right) \right]}\qquad \bigotimes](/tpl/images/0213/9727/2a8d7.png)
Trigonometric question, 30 points, will give brainliest.
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Mathematics, 31.08.2019 04:10, mildredelizam
Trigonometric question, 30 points, will give brainliest.
Answers: 1
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