Summing sines

We can convert degrees to radians: $S=\displaystyle \sum_{k=1}^{35}=sin(5k)=\displaystyle \sum_{k=1}^{35}=sin(\frac{k\pi}{36})$ And know using complex numbers we can write That: $S=Im(\displaystyle \sum_{k=1}^{35}e^{i\frac{\pi}{36}})$

Now using geometric sum we get: $S=Im(\frac{1+e^{\frac{i\pi}{36}}}{1-e^{\frac{i\pi}{36}}})$ After some manipulations we get(This result is left for the reader to prove): $S=Im(i*tan(\frac{35\pi}{72}))$ Now all we do is convert the above back to degrees to get ou desired result: $\boxed{177}$

Simply multiply and divide by 2*(common difference) and you get telescoping series.