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The value of $S = \sum_{k = 1}^{6} (\sin \frac{2 π k}{7} - i \cos \frac{2 π k}{7})$

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detailed solution

Correct option is D

sin⁡2πk7−icos⁡2πk7 =−icos⁡2πk7+isin⁡2πk7=−iωk[ De Moivre's Theorem ]where ω=cos⁡2π7+isin⁡2π7Note that ω7=1∴ S=−i∑k=16 ωk=−iω1−ω61−ω=−iω−ω71−ω=−i(ω−1)1−ω=i

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The value of S=∑k=16 sin⁡2πk7−icos⁡2πk7

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The value of $S = \sum_{k = 1}^{6} (\sin \frac{2 π k}{7} - i \cos \frac{2 π k}{7})$