Question

Find the value of

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

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Answer 1

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

Since above expression is not a standard polar form. First convert this expression for standard polar from by taking i common

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

We know,

Sum of n roots of Unity = 0

$\sum {}_{k=0}^6$ $(cos\frac{2k\pi }{7}+isin\frac{2k\pi }{7})$ = 0

Given expression -i $\sum {}_{k=1}^6$ $(cos\frac{2k\pi }{7}+isin\frac{2k\pi }{7})$ can be written as -i[ Sum of all the roots - first root]

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

= -i[0 - 1] = i