What is the value of ∑k=16(sin2πk/7−icos2πk/7)=?

Question

https://rankerer.com/question/mathematics/1GL6DbxIntWa/the-value-of-displaystyle-sum-k-1-6-bigg-sin-frac-2-pi-k-7-i-cos-frac-2-pi-k-7-bigg-is

this is the link of the question.please analyse it properly.TIA.

Answer 1

Hint:

That sum is, apparently, just

$$\sum_{k=1}^6\left( e^{-2\pi i/7}\right)^k ....\text{the sum of some elements of a geometric sequence}$$

Answer 2

We have $$\sin(\frac{2k\pi}{7})-i\cos(\frac{2k\pi}{7})$$ Taking out $-i$ we get: $$-i\bigg[\cos(\frac{2k\pi}{7})+i\sin(\frac{2k\pi}{7})\bigg]$$ Thus our sum becomes: $$\sum_{k=1}^{6}{\bigg[-ie^{\frac{2k\pi i}{7}}}\bigg]=-i\bigg[\sum_{k=1}^{6}\big[e^{\frac{2k\pi i}{7}}\big]\bigg]$$ $$=-i\bigg[e^{\frac{2\pi i}{7}}+e^{\frac{4\pi i}{7}}+e^{\frac{6\pi i}{7}}+e^{\frac{8\pi i}{7}}+e^{\frac{10\pi i}{7}}+e^{\frac{12\pi i}{7}}\bigg]$$