Question

Trigonometry question

Answer 1

`\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}\\\\ {\Large \begin{array}{llll} z=125[\cos(297^o)+i\sin(297^o)] \end{array}} \\\\\\`

`\boxed{k=0}\qquad \sqrt[3]{125}\left[ \cos\left( \cfrac{297 + 360 ( 0 )}{3} \right) +i\sin\left( \cfrac{297 + 360 ( 0 )}{3} \right)\right] \\\\\\ ~\hspace{12em}\boxed{5[\cos(297^o)+i\sin(297^o)]}}\\\\\\ \boxed{k=1}\qquad \sqrt[3]{125}\left[ \cos\left( \cfrac{297 + 360 ( 1 )}{3} \right) +i\sin\left( \cfrac{297 + 360 ( 1 )}{3} \right)\right] \\\\\\ ~\hfill \boxed{5[\cos(99^o)+i\sin(99^o)]}~\hfill`

`\boxed{k=2}\qquad \sqrt[3]{125}\left[ \cos\left( \cfrac{297 + 360 ( 2 )}{3} \right) +i\sin\left( \cfrac{297 + 360 ( 2 )}{3} \right)\right] \\\\\\ ~\hfill \boxed{5[\cos(339^o)+i\sin(339^o)]}~\hfill`