Question

FInd the value of $\sum {}_{k=0}^7$ $(cos\frac{2k\pi }{7}+isin\frac{2k\pi }{7})$

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Answer 1

We know,Sum of the n roots of unity = 0

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

$\sum {}_{k=0}^7$ $cos\frac{2k\pi }{7}+isin\frac{2k\pi }{7}$ = $\sum {}_{k=0}^6$ $(cos\frac{2k\pi }{7}+isin\frac{2k\pi }{7})$ + cos2$\pi$ +isin2$\pi$

= 0 + 1 = 1