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sum(k=1)^6 (sin,(2pik)/7 -icos, (2pik)/...

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

Text Solution

Verified by Experts

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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