Question

The value of $\left({\sum }_{k=1}^{10}(sin\frac{2\pi k}{11}-icos\frac{2\pi k}{11})\right)$ is

• A. 1
• B. -1
• C. i
• D. -i

Answer 1

The correct option is C i

${\sum }_{k=1}^{10}(sin\frac{2\pi k}{11}-icos\frac{2\pi k}{11})={\sum }_{k=1}^{10}(i^{2}sin\frac{2\pi k}{11}-icos\frac{2\pi k}{11})=-i{\sum }_{k=1}^{10}(cos\frac{2\pi k}{11}+isin\frac{2\pi k}{11})=-i{\sum }_{k=1}^{10}{e}^{i\frac{2\pi k}{11}}=-i\left[{\sum }_{k=1}^{10}{e}^{i\frac{2\pi k}{11}-1}\right]$
= -i( sum of ${11}^{th}$ roots of unity – 1)
= – i(0 – 1) = i