Question

The value of $\sum _{k=1}^6\left(\sin \right(2\pi k/7)-i\cos (2\pi k/7\left)\right)$ is

• A. -1
• B. 0
• C. -i
• D. i
• E. none

Answer 1

Answer: (D)

i

$\sum _{k=1}^6\left(sin\right(\frac{2\pi k}{7})-icos(\frac{2\pi k}{7}\left)\right)$

$=(-i)\sum _{k=1}^6\left(cos\right(\frac{2\pi k}{7})-isin(\frac{2\pi k}{7}\left)\right)$

$=(-i)[\sum _{k=0}^6\left(cos\right(\frac{2\pi k}{7})-isin(\frac{2\pi k}{7}\left)\right)-cos0+isin0]$

$=(-i)[\sum _{k=0}^6\left(cos\right(\frac{2\pi k}{7})-isin(\frac{2\pi k}{7}\left)\right)-1]$

$=(-i)[0-1]=i$ (Sum of $n^{th}$ roots of unity is zero)