Question

不查表求sin²20°+cos²80°+cos20°cos80°的值.

Answer 1

解析:

解法一: sin²20°+cos²80°+sin²20°cos80°

= (1－cos40°)+ (1+cos160°)+ sin20°cos80°

=1－cos40°+cos160°+sin20°cos(60°+20°)

=1－cos40°+ (cos120°cos40°－sin120°sin40°)

+sin20°(cos60°cos20°－sin60°sin20°)

=1－cos40°－cos40°－sin40°+sin40°－sin²20°

=1－cos40°－(1－cos40°)=

解法二: 設x=sin²20°+cos²80°+sin20°cos80°

y=cos²20°+sin²80°－cos20°sin80°，

則x+y=1+1－sin60°=，

x－y=－cos40°+cos160°+sin100°

=－2sin100°sin60°+sin100°=0

∴x=y=，即x=sin²20°+cos²80°+sin20°cos80°=.