The value of sum_{k=1}^{6}left(sin frac{2 k p | vedclass

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(A) We are given the sum $S = \sum_{k=1}^{6}\left(\sin \frac{2 k \pi}{7}-i \cos \frac{2 k \pi}{7}\right)$.Factor out $-i$ from the expression:$S = -i

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$i$

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(A) We are given the sum $S = \sum_{k=1}^{6}\left(\sin \frac{2 k \pi}{7}-i \cos \frac{2 k \pi}{7}\right)$.Factor out $-i$ from the expression:$S = -i \sum_{k=1}^{6}\left(\cos \frac{2 k \pi}{7} + i \sin \frac{2 k \pi}{7}\right)$.Let $\omega = e^{i \frac{2 \pi}{7}} = \cos \frac{2 \pi}{7} + i \sin \frac{2 \pi}{7}$. Then the sum becomes:$S = -i \sum_{k=1}^{6} \omega^k$.This is a geometric series with $6$ terms,where the first term is $\omega$ and the common ratio is $\omega$:$S = -i \left( \frac{\omega(1 - \omega^6)}{1 - \omega} \right) = -i \left( \frac{\omega - \omega^7}{1 - \omega} \right)$.Since $\omega^7 = e^{i 2 \pi} = 1$,we have:$S = -i \left( \frac{\omega - 1}{1 - \omega} \right) = -i (-1) = i$.

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

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