求cosπ/13+cos3π/13+cos5π/13+cos7π/13+cos9π/13+cos11π/14的值12小时内回答的我会提高悬赏、、

Question

求cosπ/13+cos3π/13+cos5π/13+cos7π/13+cos9π/13+cos11π/14的值
12小时内回答的我会提高悬赏、、

Answer 1

这种题目利用积化和差公式裂项求和
先求 2sin(π/13)*[cosπ/13+cos3π/13+cos5π/13+cos7π/13+cos9π/13+cos11π/13] (你的输入有点小错)
=[sin(2π/13)-sin(0)]+[sin(4π/13)-sin(2π/13)]+[sin(6π/13)-sin(4π/13)]+[sin(8π/13)-sin(6π/13)]
+[sin(10π/13)-sin(8π/13)]+[sin(12π/13)-sin(10π/13)]
=sin(12π/13)-0
=sin(12π/13)
=sin(π/13)
∴ cosπ/13+cos3π/13+cos5π/13+cos7π/13+cos9π/13+cos11π/14
=sin(π/13)/[2sin(π/13)]
=1/2
附：积化和差公式
sinαsinβ=[-cos（α+β）+cos（α-β）]/2
cosαcosβ=[cos（α+β）+cos（α-β）]/2
sinαcosβ=[sin（α+β）+sin（α-β）]/2
cosαsinβ=[sin（α+β）-sin（α-β）]/2=sinβcosα=[sin（β+α）+sin（β-α）]/2