m-1-10-sin-2pim-11-i-cos-2pim-11-Solve-it-

Question Number 17775 by gourav~ last updated on 10/Jul/17

\underset{m = 1}{\sum^{10}} (\frac{\sin 2 π m}{11} - i \frac{\cos 2 π m}{11}) = ? ? ? ? ?

S o l v e . . i t

Answered by alex041103 last updated on 10/Jul/17

\underset{m = 1}{\sum^{10}} (\frac{\sin 2 π m}{11} - i \frac{\cos 2 π m}{11}) =

= \frac{1}{11} \underset{m = 1}{\sum^{10}} (\sin 2 π m) - \frac{i}{11} \underset{m = 1}{\sum^{10}} (\cos 2 π m)

Because m \in Z

\Rightarrow \sin 2 π m = 0 and \cos 2 π m = 1

\underset{m = 1}{\sum^{10}} (\frac{\sin 2 π m}{11} - i \frac{\cos 2 π m}{11}) =

= \frac{1}{11} \underset{m = 1}{\sum^{10}} (0) - \frac{i}{11} \underset{m = 1}{\sum^{10}} (1)

= 0 \times \frac{1}{11} - \frac{i}{11} \times (1) \times 10

= - \frac{10}{11} i

\underset{m = 1}{\sum^{10}} (\frac{\sin 2 π m}{11} - i \frac{\cos 2 π m}{11}) = - \frac{10}{11} i

Answered by ajfour last updated on 10/Jul/17

= - \frac{i}{11} \underset{m = 0}{\sum^{10}} e^{i (2 π m)} = - \frac{i}{11} (10) = - \frac{10 i}{11} .

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