又來M2三角學證明題

問：不用計算機，求 cos
π/11 + cos
3π/11 +
cos 5π/11 +
cos 7π/11 +
cos 9π/11 的值。

解：

(sin π/11)(cos π/11 + cos 3π/11 + cos 5π/11 + cos 7π/11
+ cos 9π/11) ÷

(sin π/11)

= [(0.5)(sin 2π/11 + sin 0 + sin 4π/11 - sin 2π/11
+ sin 6π/11 - sin 4π/11 +

sin 8π/11 - sin 6π/11 + sin 10π/11 - sin 8π/11)] ÷ (sin π/11)

= (0.5)(sin 10π/11) ÷ (sin π/11)

= (0.5)(sin π/11) ÷ (sinπ/11) [Note: sin 10π/11 = sin (π - 10π/11) = sin π/11]

= 0.5

(這條想了近三個小時才想到~~~)