已知cos(75°+α)=1/3,其中α是第三象限角,求cos(15°-α)+sin(a-15°)的值

题目

已知cos(75°+α)=1/3,其中α是第三象限角,求cos(15°-α)+sin(a-15°)的值

答案

α是第三象限角
cos(α+75°)>0
∴ α+75°是第四象限角
∵ cos(α+75°)=1/3
∴ sin(α+75°)=-√[1-cos²(α+75°)]=-√(1-1/9)=-2√2/3
∴ cos(15°-α)+sin(α-15°)
=cos(15°-α)-sin(15°-α)
=cos[90°-(75°+α)]-sin[90°-(75°+α)]
=sin(75°+α)-cos(75°+α)
=-2√2/3-1/3
=-(2√2+1)/3