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

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(D) We have,$S = \sum_{k=1}^6 \left[ \sin \frac{2 k \pi}{7} - i \cos \frac{2 k \pi}{7} \right]$ Factor out $-i$: $S = \sum_{k=1}^6 (-i) \left( \cos \f

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$i$

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(D) We have,$S = \sum_{k=1}^6 \left[ \sin \frac{2 k \pi}{7} - i \cos \frac{2 k \pi}{7} \right]$ Factor out $-i$: $S = \sum_{k=1}^6 (-i) \left( \cos \frac{2 k \pi}{7} + i \sin \frac{2 k \pi}{7} \right)$ Let $\omega = e^{i \frac{2 \pi}{7}} = \cos \frac{2 \pi}{7} + i \sin \frac{2 \pi}{7}$. Then the expression becomes: $S = -i \sum_{k=1}^6 \omega^k$ This is a geometric progression sum: $S = -i \left( \omega + \omega^2 + \dots + \omega^6 \right)$ Since $\omega$ is a $7^{th}$ root of unity,$1 + \omega + \omega^2 + \dots + \omega^6 = 0$. Therefore,$\omega + \omega^2 + \dots + \omega^6 = -1$. Substituting this into the expression for $S$: $S = -i (-1) = i$.

$\sum_{k=1}^6 \left[ \sin \frac{2 k \pi}{7} - i \cos \frac{2 k \pi}{7} \right]$ is equal to

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