If $\sum_{i=1}^n \cos \theta_i$=n, then find the value of $\sum_{i=1}^n \sin \theta_i$

Question

If $\displaystyle\sum_{i=1}^{n} \cos(\theta_{i}) = n$, then find the value of $\displaystyle\sum_{i=1}^n \sin(\theta_i)$

Answer 1

Since $\cos\theta_i\leq1$, the only way a sum of $n$ cosines could equal $n$ is if they were all equal to $1$. So the angles $\theta_i$ are all congruent to $0$. So all of the sines equal $0$, and the sum of sines is $0$.