`sum_(k=1)^6``(sin,(2pik)/7 -icos, (2pik)/7)=?`

1.	`sum_(k=1)^6``(sin,(2pik)/7 -icos, (2pik)/7)=?`
Answer» Let ` S = sum_ (k=1)^6 (sin((2pik)/7) - icos((2pik)/7))` `=>S = -i sum_ (k=1)^6 (cos((2pik)/7) + isin((2pik)/7))` ...(As `i^2 = -1`) `=>S = -i sum_(k=1)^6 (e^((i2pik)/7))` Now, this summation of terms happens to be the sum of the seventh roots of unity; and, the sum of the `n` roots of a number is zero. So, `1+sum_(k=1)^6 (e^((i2pik)/7)) = 0` `=>sum_(k=1)^6 (e^((i2pik)/7)) = -1` `:. S = -i(-1) = i`

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`sum_(k=1)^6``(sin,(2pik)/7 -icos, (2pik)/7)=?`

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