Question

If `sum_(i=1)^(n) cos(A_i) = n` then find `sum_(i=1)^(n) sin(A_i)`?

Answer 1

0 (assuming all the `A_i`s are real)

Explanation:

Since the maximum value of `cos(A)_i` for real `A_i` is 1, the sum of `n` such numbers can be `n` if and only if each one of them is equal to 1. Thus

`cos(A_i) = 1,qquad i=1,2,...,n implies`

`sin(A_i) = sqrt(1-cos^2(A_i))=0,qquad i=1,2,...,n`

Thus, the required sum is 0.