What is the Sum of a Finite Sine Series?

[shehry1]

Can anyone tell me what is the sum of a finite series of Sines.

$$\sum_{n=1}^N \sin^2 (n \phi)$$. I am going through a text and it gives it as (N+1)/2. I tried to derive it. The N comes out ok when you use the half angle identity but I can't figure out a general rule for the Cosines that appear

 

[HallsofIvy]

What is $\phi$?

 

[shehry1]

What is $\phi$?

$$\phi = \pi s /(N+1)$$ where s = 1,2...N

 

[tim_lou]

resolve sin^2 into 1-cos something. Put it in terms of complex exponentials. you'll get a geometric series.