Question

1 + cos (2pi/5) + cos (4pi/5) + cos ( 6pi/5 )+ cos( 8pi/5) = ?


Mulțumesc !!!

Answer 1

E = 1 + cos(2pi/5) + cos(4pi/5) +cos(6pi/5 )+cos(8pi/5)

E = cos(0pi/5)+cos(2pi/5) + cos(4pi/5) +cos(6pi/5 )+cos(8pi/5) |×sin(pi/5)

E·sin(pi/5) = sin(pi/5)cos(0pi/5)+ sin(pi/5)cos(2pi/5) + sin(pi/5)cos(4pi/5) +sin(pi/5)cos(6pi/5 )+sin(pi/5)cos(8pi/5)

2E·sin(pi/5) = [sin(pi/5+0pi/5) + sin(pi/5-0pi/5)] + [sin(pi/5+2pi/5) + sin(pi/5-2pi/5)] + [sin(pi/5+4pi/5) + sin(pi/5-4pi/5)] + [sin(pi/5+6pi/5) + sin(pi/5-6pi/5)] + [sin(pi/5+8pi/5) + sin(pi/5-8pi/5)]

2E·sin(pi/5) = sin(pi/5)+sin(pi/5)+sin(3pi/5)-sin(pi/5)+sin(5pi/5)-sin(3pi/5)+sin(7pi/5)-sin(5pi/5)+sin(9pi/5)-sin(7pi/5)

2E·sin(pi/5) = sin(pi/5)+sin(9pi/5)

2E·sin(pi/5) = 2sin(10pi/10)×cos(8pi/10)

2E·sin(pi/5) = 0

=> E = 0