Question

已知sin X+cos X=m,|m|小于等于根号2且|m|不等于1，求sin X^3+cos X^3，sin X^4+cos X^4

Answer 1

sinx+cosx=m
平方
sin²x+cos²x+2sinxcosx=m²
1+2sinxcosx=m²
sinxcosx=(m²-1)/2

所以sin³x+cos³x
=(sinx+cosx)(sin²x-sinxcosx+cos²x)
=m[1-(m²-1)/2]
=(3m-m³)/2

sin^4 x+cos^4 x
=(sin²x+cos²x)²-2sin²xcos²x
=1-2[(m²-1)/2]²
=(-m^4+2m²+1)/2

Answer 2

sinx+cosx=m
两边平方得
1+2sinxcosx=m^2
sinxcosx=(m^2-1)/2

(sinx)^3+(cosx)^3
=(sinx+cosx)[(sinx)^2-sinxcosx+(cosx)^2]
=m*[(sinx+cosx)^2-3sinxcosx]
=m*(m^2-3(m^2-1)/2)
=m*(-m^2/2+3/2)
=-m^3/2+3m/2

(sinx)^4+(cosx)^4
=[(sinx)^2]^2+[(cosx)^2]^2
=[(sinx)^2+(cosx)^2]^2-2(sinxcosx)^2
=1-2[(m^2-1)/2]^2
=1-(1/2)(m^4-2m^2+1)
=-(1/2)(m^4-2m^2+1)