Question

The value of $S={\sum }_{k=1}^6(\sin \frac{2\pi k}{7}-i\cos \frac{2\pi k}{7})$ ?

• A. $-1$
• B. $0$
• C. $-i$
• D. $i$

Answer 1

The correct option is D $i$

$sin\frac{2\pi k}{7}-icos\frac{2\pi k}{7}$

$=-i(cos\frac{2\pi k}{7}+isin\frac{2\pi k}{7})$

$=-i{e}^{2\pi k/7i}$

$\Rightarrow S=\sum _{k=1}^6-i\left(e^{2\pi k/i}\right)$

$=-i\left(\sum _{k=1}^6{e}^{i2\pi k/7}\right)$

$=-i$ [sum of roots of the equation

$x^6+x^5+x^4+x^3+x^2+1=0].$

$=-i(-1)$

$=i$