What kind of theorem is this?

Question

For odd denum:
cos(pi/3) + cos(3*pi/3) + cos(5*pi/3) = 0

for even denum :
cos(pi/4) + cos(3*pi/4) = 0
cos (pi/6) + cos(3*pi/6) + cos(5*pi/6) = 0
etc.

Is there any general formula for this?
I need this to prove sumthin.

Thanks Before.

Answer 1

It's not really clear how you want to generalise this.

However, you should note:
cos(π - x) = - cos x

and so cos (π/2n) + cos ((2n-1)π/2n) = 0 for any n
Also, cos(nπ/2n) = cos (π/2) = 0, so we can say

cos (π/2n) + cos (nπ/2n) + cos ((2n-1)π/2n) = 0

which gives you these statements for even denominators:
cos π/4 + cos 2π/4 + cos 3π/4 = 0
cos π/6 + cos 3π/6 + cos 5π/6 = 0
cos π/8 + cos 4π/8 + cos 7π/8 = 0
cos π/10 + cos 5π/10 + cos 9π/10 = 0
etc.

Your odd denominator one is, well, odd. The middle term is cos π, which of course is -1. The first and last terms have angles x and 2π-x, which will always have equal cosines, but the only reason they match the -1 is because the denominator is 3 so we have two lots of 1/2. I don't think there's going to be much you can do with this.