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

Jul 2, 2018

0 (assuming all the ${A}_{i}$s are real)

#### Explanation:

Since the maximum value of $\cos {\left(A\right)}_{i}$ 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

$\cos \left({A}_{i}\right) = 1 , q \quad i = 1 , 2 , \ldots , n \implies$

$\sin \left({A}_{i}\right) = \sqrt{1 - {\cos}^{2} \left({A}_{i}\right)} = 0 , q \quad i = 1 , 2 , \ldots , n$

Thus, the required sum is 0.