Question

$ifi=\sqrt{-1},then4+5{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{334}+3{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{365}isequalto$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Given equation is
${(-\frac{1}{2}+i\frac{\sqrt{3}}{2})}^{365}=4+5{(cos\frac{2\pi }{3}+isin\frac{2\pi }{3})}^{334}+3{(cos\frac{2\pi }{3}+isin\frac{2\pi }{3})}^{365}=4+5[cos\frac{668}{3}\pi +isin\frac{668}{3}\pi ]3[cos\frac{730}{3}\pi +isin\frac{730}{3}\pi ]$
$=4+5[cos(222\pi +\frac{2\pi }{3})+isin(222\pi +\frac{2\pi }{3})]+3\left[cos(243\pi +\frac{\pi }{3})isin(243\pi +\frac{\pi }{3})\right]$
$=4+5(cos\frac{2\pi }{3}+isin\frac{2\pi }{3})+3(-cos\frac{\pi }{3}-isin\frac{\pi }{3})=4+5(-\frac{1}{2}+i\frac{\sqrt{3}}{2})+3(-\frac{1}{2}-i\frac{\sqrt{3}}{2})=4-4+2i\frac{\sqrt{3}}{2}=i\sqrt{3}.$