Question

The value of $\displaystyle \sum_{k=1}^{10}$ $\left(sin\, \frac{2k\pi}{11}+i\,cos\, \frac{2k\pi}{11}\right)$ is :

• A. $1$
• B. $-1$
• C. $-i$
• D. $i$

Answer 1

Answer: (C)

$-i$

Answer 2

$\displaystyle \sum_{k=1}^{10}$ $\left(sin\, \frac{2k\pi}{11}+i\,cos\, \frac{2k\pi}{11}\right)$ $i=\displaystyle \sum_{k=1}^{10}$ $\left(cos\, \frac{2k\pi}{11}+i\,sin\, \frac{2k\pi}{11}\right)$ $=i\displaystyle \sum_{k=1}^{10}$ $\left(e^{- \frac{2k\pi}{11}}\right)$ =i\left\\{\displaystyle \sum_{k=0}^{10}\left(e^{-} \frac{2k\pi}{11}\right)-1\right\\}