已知sinx+cosx=1/3,求sin^3x+cos^3x

由 (sinx+cosx)^3 = (sinx)^3+(cosx)^3 + 3sinxcosx(sinx+cosx)
得 (sinx)^3+(cosx)^3 = (sinx+cosx)^3 - 3sinxcosx(sinx+cosx)
= (1/3)^3-3*(1/3)sinxcosx
= 1/27-sinxcosx (1)
又由 sinx+cosx=1/3 得 (sinx+cosx)^2=(1/3)^2
(sinx)^2+(cosx)^2 + 2sinxcosx =1/9
1+2sinxcosx=1/9
所以 sinxcosx = -4/9
代入 (1) 式,有
(sinx)^3+(cosx)^3 = 1/27+4/9
= 13/27