Question

高一数学观察等式:cos10°^2+cos110°^2+cos130°^2=3/2 及cos15^2+cos105^2+cos135°^2=3/2,

观察等式:cos10°^2+cos110°^2+cos130°^2=3/2 及cos15^2+cos105^2+cos135°^2=3/2,请归纳出一个一般性的结论，使以上两个等式为这个结论的特例，并证明你给出的结论。

Answer 1

一般性的结论是：cos²α+cos²(120°-α)+cos²(120°+α) = 3/2
证明如下：
cos²α+cos²(120°-α)+cos²(120°+α)
= (1/2)[1+cos(2α)]+(1/2)[1+cos(240°-2α)]+(1/2)[1+cos(240°+2α)]
= 3/2+(1/2)[cos(2α)+cos(240°-2α)+cos(240°+2α)]
= 3/2+(1/2){cos(2α)+2cos[(240°-2α)/2+(240°+2α)/2]cos[(240°-2α)/2-(240°+2α)/2]}
= 3/2+(1/2)[cos(2α)+2cos240°cos(-2α)]
= 3/2+(1/2)[cos(2α)-2cos60°cos(2α)]
= 3/2+(1/2)[cos(2α)-cos(2α)]
= 3/2