if xi>0 for 1<=i<=n and
x1+x2+x3+x4 -------xn=n
then greatest value of sin x1+sinx2+sinx3+----sin xn=
It looks as though the maximum will occur when $\displaystyle x_k = 1$ for each k = 1,2,...,n. Then the sum will be $\displaystyle n\sin1\approx0.84n$. I don't see that it's possible to do any better than that, but I haven't tried to prove it.