作业帮若sin^3α-cos^3α≥cosα-sinα,0≤α＜2π,则角α的取值范围是()A.[0,π\4]B.[π\4,π]C.[π\4,5π\4]D.[π\4,3π\2]
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![若sin^3α-cos^3α≥cosα-sinα,0≤α＜2π,则角α的取值范围是()A.[0,π\4]B.[π\4,π]C.[π\4,5π\4]D.[π\4,3π\2]](/uploads/image/z/3000452-68-2.jpg?t=%E8%8B%A5sin%5E3%CE%B1-cos%5E3%CE%B1%E2%89%A5cos%CE%B1-sin%CE%B1%2C0%E2%89%A4%CE%B1%EF%BC%9C2%CF%80%2C%E5%88%99%E8%A7%92%CE%B1%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4%E6%98%AF%28%29A.%5B0%2C%CF%80%5C4%5DB.%5B%CF%80%5C4%2C%CF%80%5DC.%5B%CF%80%5C4%2C5%CF%80%5C4%5DD.%5B%CF%80%5C4%2C3%CF%80%5C2%5D)
若sin^3α-cos^3α≥cosα-sinα,0≤α＜2π,则角α的取值范围是()A.[0,π\4]B.[π\4,π]C.[π\4,5π\4]D.[π\4,3π\2]
若sin^3α-cos^3α≥cosα-sinα,0≤α＜2π,则角α的取值范围是()
A.[0,π\4]
B.[π\4,π]
C.[π\4,5π\4]
D.[π\4,3π\2]
若sin^3α-cos^3α≥cosα-sinα,0≤α＜2π,则角α的取值范围是()A.[0,π\4]B.[π\4,π]C.[π\4,5π\4]D.[π\4,3π\2]
选【C.[π\4,5π\4]】
(sinα)^3-(cosα)^3=(sinα-cosα)((sinα)^2+sinα*cosα+(cosα)^2)
=(sinα-cosα)(1+sinα*cosα)
(sinα)^3-(cosα)^3-(cosα-sinα)
=(sinα-cosα)(2+sinα*cosα)≥0
∵2+sinα*cosα=2+sin2α/2>0
∴sinα-cosα≥0
∴sin(α-π/4)=sinα*√2/2-cosα*√2/2≥0
∴选C.[π\4,5π\4]
若sin³α-cos³α≥cosα-sinα， 0≤α＜2π，则角α的取值范围是()
(sinα-cosα)(sin²α+sinαcosα+cos²α)+(sinα-cosα)
=(sinα-cosα)(sin²α+sinαcosα+cos²α+1)
=(sinα-cosα)(sinαcosα+2)=(sinα...
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若sin³α-cos³α≥cosα-sinα， 0≤α＜2π，则角α的取值范围是()
(sinα-cosα)(sin²α+sinαcosα+cos²α)+(sinα-cosα)
=(sinα-cosα)(sin²α+sinαcosα+cos²α+1)
=(sinα-cosα)(sinαcosα+2)=(sinα-cosα)[(1/2)sin2α+2]≥0
由于-1≦sin2α≦1，-1/2≦(1/2)sin2α≦1/2；故0<3/2≦(1/2)sin2α+2≦5/2；
故必有sinα-cosα≥0，即sinα≥cosα，
故π/4≦x≦5π/4，选C.
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