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pshichka [43]
3 years ago
5

Trigonometric question, 30 points, will give brainliest.

Mathematics
1 answer:
Elis [28]3 years ago
6 0

\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}}

\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]}

\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

\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

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