Question

cos(2派/7)+cos(4派/7)+cos(6派/7)=?

Answer 1

Pi表示派。
cos(2pi/7)+cos(4pi/7)+cos(6pi/7)
=1/[2sin(2pi/7)]*[2sin(2pi/7)cos(2pi/7)+2sin(2pi/7)cos(4pi/7)+2sin(2pi/7)cos(6pi/7)]
进行积化和差
=1/[2sin2pi/7)]*{sin(4pi/7)+[sin(6pi/7)-sin(2pi/7)] +[sin(8pi/7)-sin(4pi/7)]}
=1/[2sin(2pi/7)][-sin(2pi/7)+sin(6pi/7)-sin(6pi/7)]
=1/[2sin(2pi/7)]*[-sin(2pi/7)]
=-1/2.

Answer 2

利用和差公式化简后 4cos(pi/7)cos(2pi/7)cos(3pi/7)-1
令C=cos(pi/7)cos(2pi/7)cos(3pi/7)
原式为 4C-1, 假设 S=sin(pi/7)sin(2pi/7)sin(3pi/7)
8*C*S=2sin(pi/7)cos(pi/7)*2sin(2pi/7)c…
=sin(2pi/7)sin(4pi/7)sin(6pi/7)
=sin(2pi/7)sin(4pi/7)sin(pi/7)
=S
所以 C=1/8
则原式 4C-1=-1/2

Answer 3

见下图