The value of ∑ k = 1 10 sin ⁡ 2 π k 11 - i cos ⁡ 2 π k 11 is (where i = - 1 )

Step-by-step solution

We have, ∑ k = 1 10 sin ⁡ 2 π k 11 - i cos ⁡ 2 π k 11 = ∑ k = 1 10 - i 2 sin ⁡ 2 π k 11 - i cos ⁡ 2 π k 11 = - i ∑ k = 1 10 cos ⁡ 2 π k 11 + i sin ⁡ 2 π k 11 = - i ∑ k = 1 10 e i 2 π k 11 = - i ∑ k = 0 10 e i 2 π k 11 - 1 = - i s u m o f 11 t h r o o t s o f u n i t y - 1 = - i 0 - 1 = i .