sum_{k=1}^6left(sin frac{2 pi k}{7}-i cos fra | vedclass

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(C) We have the expression $S = \sum_{k=1}^6 \left(\sin \frac{2 \pi k}{7} - i \cos \frac{2 \pi k}{7}\right)$.Factoring out $-i$,we get $S = -i \sum_{k

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$i$

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(C) We have the expression $S = \sum_{k=1}^6 \left(\sin \frac{2 \pi k}{7} - i \cos \frac{2 \pi k}{7}ight)$.Factoring out $-i$,we get $S = -i \sum_{k=1}^6 \left(\cos \frac{2 \pi k}{7} + i \sin \frac{2 \pi k}{7}ight)$.Using Euler's formula $e^{i	heta} = \cos 	heta + i \sin 	heta$,we have $S = -i \sum_{k=1}^6 e^{i \frac{2 \pi k}{7}}$.Let $\omega = e^{i \frac{2 \pi}{7}}$. Then the sum is $S = -i \sum_{k=1}^6 \omega^k$.Since $\omega$ is a $7^{th}$ root of unity,$1 + \omega + \omega^2 + \omega^3 + \omega^4 + \omega^5 + \omega^6 = 0$.Therefore,$\sum_{k=1}^6 \omega^k = -1$.Substituting this into the expression for $S$,we get $S = -i(-1) = i$.

$\sum_{k=1}^6\left(\sin \frac{2 \pi k}{7}-i \cos \frac{2 \pi k}{7}\right)=$

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