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

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