如果sin^3-cos^3>cosθ-sinθ,且θ（0,2π),那么角θ的取值范围是A.(0,π/4) B.(π/2,3π) C.(π/4,5π/4) D.（5π/4,2π)

如果sin^3-cos^3>cosθ-sinθ,且θ（0,2π),那么角θ的取值范围是A.(0,π/4) B.(π/2,3π) C.(π/4,5π/4) D.（5π/4,2π)
如果sin^3-cos^3>cosθ-sinθ,且θ（0,2π),那么角θ的取值范围是
A.(0,π/4) B.(π/2,3π) C.(π/4,5π/4) D.（5π/4,2π)

如果sin^3-cos^3>cosθ-sinθ,且θ（0,2π),那么角θ的取值范围是A.(0,π/4) B.(π/2,3π) C.(π/4,5π/4) D.（5π/4,2π)
(sinθ)^3-(cosθ)^3=(sinθ-cosθ)((sinθ)^2+sinθ*cosθ+(cosθ)^2)
=(sinθ-cosθ)(1+sinθ*cosθ)
(sinθ)^3-(cosθ)^3-(cosθ-sinθ)
=(sinθ-cosθ)(2+sinθ*cosθ)>0
因为2+sinθ*cosθ=2+sin2θ/2>0
所以sinθ-cosθ>0
所以sin(θ-π/4)=sinθ*√2/2-cosθ*√2/2>0
所以C.(π/4,5π/4)