Question

sin(A)+sin(2A)+sin(3A)+........+sin(nA)=

Answer 1

I think it would be zero because if you think about on the unit circle for every sine that is positive there is one thats 180 degrees opposite of it and negative.

for example say that A=45 degrees.

then
sin(A)=sqrt(2)/2
sin(2A)=1
sin(3A)=sqrt(2)/2
sin(4A)=0
sin(5A)=-sqrt(2)/2
sin(6A)=-1
sin(7A)=-sqrt(2)/2
sin(8A)=0

So that is one time around the circle for a finite number but as you can see the pattern would be the same and each value would cancel out with another.

Answer 2

\[\mathbb{Im}(e^{iA}+e^{2iA}+...+e^{niA})=\mathbb{Im}(\frac{e^{iA}(e^{niA}-1)}{e^{iA}-1})\]is a good start if u are familiar with complex numbers :)