Question

sin²(π+α)cos(3π/2-α)

已知sin(π-α)-cos(π+α)=根号2/3 求下列各式的值 (1)sinα-cosα (2
)sin^3(π/2-α)-cos^3(π/2+α)

Answer 1

(1) 由题得：sinα+cosα=根号2/3
所以,(sinα+cosα)²=2/9
所以,sin²α+2sinα*cosα+cos²α=2/9
因为,sin²α+cos²α=1
所以,sinα*cosα=-7/18
又因为,(sinα-cosα)²=(sinα+cosα)²-4sinα*cosα=2/9+14/9=16/9
所以,sinα-cosα=4/3 或 sinα-cosα=-4/3
(2) sin^2(π/2-α)-cos^2(π/2+α)=cos²α-sin²α=(cosα-sinα)(cosα+sinα)=-(sinα-cosα)(sinα+cosα)
1、当sinα-cosα=4/3时,sin^2(π/2-α)-cos^2(π/2+α)=-(4/3)*[(根号2)/3]=-4(根号2)/9
2、当sinα-cosα=-4/3时,sin^2(π/2-α)-cos^2(π/2+α)=(4/3)*[(根号2)/3]=4(根号2)/9
所以,sin^2(π/2-α)-cos^2(π/2+α)=-4(根号2)/9 或 4(根号2)/9