How do you evaluate?
#cos^2((pi)/13)+cos^2((2pi)/13)+cos^2((3pi)/13)+cos^2((4pi)/13)+cos^2((5pi)/13)=?#
1 Answer
Explanation:
The
#cos((2npi)/13)+isin((2npi)/13)# for
#n = -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6#
Noting that:
#{ (cos(-theta) = cos theta), (sin(-theta) = -sin theta) :}#
we can take the roots in pairs to find:
#x^13-1 = (x-1)(x^2-2cos((2pi)/13)x+1)(x^2-2cos((4pi)/13)x+1)(x^2-2cos((6pi)/13)x+1)(x^2-2cos((8pi)/13)x+1)(x^2-2cos((10pi)/13)x+1)(x^2-2cos((12pi)/13)x+1)#
#color(white)(x^13-1) = (x-1)(x^12+x^11+x^10+...+x+1)#
Equating the coefficients of
#-2(cos((2pi)/13)+cos((4pi)/13)+cos((6pi)/13)+cos((8pi)/13)+cos((10pi)/13)+cos((12pi)/13)) = 1#
So:
#cos((2pi)/13)+cos((4pi)/13)+cos((6pi)/13)+cos((8pi)/13)+cos((10pi)/13)+cos((12pi)/13)=-1/2#
Now:
#cos 2 theta = 2cos^2 theta-1#
Hence:
#cos^2 theta = 1/2(cos 2theta+1)#
So we find:
#cos^2 (pi/13) + cos^2 ((2pi)/13) + cos^2 ((3pi)/13) + cos^2 ((4pi)/13) + cos^2((5pi)/13) + cos^2((6pi)/13)#
#= 1/2(cos((2pi)/13)+cos((4pi)/13)+cos((6pi)/13)+cos((8pi)/13)+cos((10pi)/13)+cos((12pi)/13)) + 3#
#=-1/4+3#
#=11/4#