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

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Question

(A) We know that $\sum_{k=0}^{6} \cos \frac{2 \pi k}{7} = 0$ and $\sum_{k=0}^{6} \sin \frac{2 \pi k}{7} = 0$. Since $\cos 0 = 1$ and $\sin 0 = 0$,we h

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$i$

Vedclass · Answer

(A) We know that $\sum_{k=0}^{6} \cos \frac{2 \pi k}{7} = 0$ and $\sum_{k=0}^{6} \sin \frac{2 \pi k}{7} = 0$. Since $\cos 0 = 1$ and $\sin 0 = 0$,we have $\sum_{k=1}^{6} \cos \frac{2 \pi k}{7} = -1$ and $\sum_{k=1}^{6} \sin \frac{2 \pi k}{7} = 0$. The given expression is $\sum_{k=1}^{6} (\sin \frac{2 \pi k}{7} - i \cos \frac{2 \pi k}{7}) = \sum_{k=1}^{6} \sin \frac{2 \pi k}{7} - i \sum_{k=1}^{6} \cos \frac{2 \pi k}{7}$. Substituting the values,we get $0 - i(-1) = i$.

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

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