求sin²20°+cos²80°+(根号3)sin20°cos80°的值

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求sin²20°+cos²80°+(根号3)sin20°cos80°的值
求sin²20°+cos²80°+(根号3)sin20°cos80°的值

求sin²20°+cos²80°+(根号3)sin20°cos80°的值
sin²20°+cos²80° = (1 - cos40°)/2 + (1 + cos160°)/2 = 1 + (cos160°- cos40°)/2 = 1 - sin[(160°+ 40°)/2]*sin[(160°- 40°)/2] = 1 - sin100°sin60° = 1 - √3/2*sin100° √3sin20°cos80° = √3/2*(sin100°-sin60°) = √3/2*sin100°- 3/4 sin²20°+cos²80°+√3sin20°cos80° = 1 - 3/4 = 0.25