Question

Find the value of $\sum _{k=1}^{10}(sin\frac{2\pi k}{11}-icos\frac{2\pi k}{11})$

• A. $-i$
• B. $i$
• C. $1$
• D. $-1$

Answer 1

The correct option is B $i$

Let $S=\sum _{k=1}^{10}(\sin \frac{2\pi k}{11}-i\cos \frac{2\pi k}{11})$

$={\sum }_{k=1}^{10}(-i^2\sin \frac{2\pi k}{11}-i\cos \frac{2\pi k}{11})$

$=-i\sum _{k=1}^{10}(\cos \frac{2\pi k}{11}+i\sin \frac{2\pi k}{11})=-i\sum _{k=1}^{10}{e}^{i\frac{2\pi k}{11}}$

$=-i[\sum _{k=0}^{10}{e}^{i\frac{2\pi k}{11}}-1]=-i$ (sum of 11th roots of unity=-1)

$=-i(0-1)=i$
Ans: B