已知sina+cosa=p,求(1)sina*cosa(2)sin^2a-cos^2a(3)sin^3a+-cos^3a
1个回答
sina+cosa=p
(1)
(sina+cosa)²=sin²a+2sina·cosa+cos²a=1+2sina·cosa=p²
sina·cosa=(p²-1)/2
(2)
(sina-cosa)²=sin²a-2sina·cosa+cos²a=1-2sina·cosa=1-(p²-1)=2-p²
sina-cosa=±√(2-p²)
sin²a-cos²a=(sina+cosa)(sina-cosa)=±p√(2-p²)
(3)
sin³a+cos³a=(sina+cosa)(sin²a-sina·cosa+cos²a)
=p[1-(p²-1)/2]
=p(3-p²)/2
=(3p-p³)/2
sin³a-cos³a=(sina-cosa)(sin²a+sina·cosa+cos²a)
=±√(2-p²)·[1+(p²-1)/2]
=±√(2-p²)·(p²+1)/2
=±(1/2)(p²+1)√(2-p²)
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