已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/2,π/2].已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/2,π/2](1)求证:(a-b)⊥(a+b)(2)|a+b|=1/3,求cosx的值
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![已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/2,π/2].已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/2,π/2](1)求证:(a-b)⊥(a+b)(2)|a+b|=1/3,求cosx的值](/uploads/image/z/351258-42-8.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28cos3x%2F2%2Csin3x%2F2%29%2Cb%3D%28cosx%2F2%2C-sinx%2F2%29%2C%E4%B8%94x%E2%88%88%5B-%CF%80%2F2%2C%CF%80%2F2%5D.%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28cos3x%2F2%2Csin3x%2F2%29%2Cb%3D%28cosx%2F2%2C-sinx%2F2%29%2C%E4%B8%94x%E2%88%88%5B-%CF%80%2F2%2C%CF%80%2F2%5D%EF%BC%881%EF%BC%89%E6%B1%82%E8%AF%81%EF%BC%9A%EF%BC%88a-b%EF%BC%89%E2%8A%A5%EF%BC%88a%2Bb%EF%BC%89%EF%BC%882%EF%BC%89%7Ca%2Bb%7C%3D1%2F3%2C%E6%B1%82cosx%E7%9A%84%E5%80%BC)
已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/2,π/2].已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/2,π/2](1)求证:(a-b)⊥(a+b)(2)|a+b|=1/3,求cosx的值
已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/2,π/2].
已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/2,π/2]
(1)求证:(a-b)⊥(a+b)
(2)|a+b|=1/3,求cosx的值
已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/2,π/2].已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/2,π/2](1)求证:(a-b)⊥(a+b)(2)|a+b|=1/3,求cosx的值
楼上那种做法是错的,不能那样拆
(1)a-b=(cos3x/2-cosx/2,sin3x/2+sinx/2)
a+b=(cos3x/2+cosx/2,sin3x/2-sinx/2)
(a-b)·(a+b)=(cos3x/2-cosx/2)(cos3x/2+cosx/2)+(sin3x/2+sinx/2)(sin3x/2-sinx/2)=cos^2(3x/2)-cos^2(x/2)+sin^2(3x/2)-cos^2(x/2)=1-1=0
∴(a-b)⊥(a+b)
(2)√[(cos3x/2+cosx/2)^2+(sin3x/2-sinx/2)^2]=1/3
cos^2(3x/2)+2cos3x/2cosx/2+cos^2(x/2)+sin^2(3x/2)-2sin3x/2sinx/2+sin^2(x/2)=1/9(两边平方,左边拆开)
2+2cos3x/2cosx/2-2sin3x/2sinx/2=1/9
cos3x/2cosx/2-sin3x/2sinx/2=-17/18
cos(3x/2+x/2)=-17/18
cos2x=-17/18
cos2x=2cos^2(x)-1=-17/18
cosx=±1/6
∵x∈[-π/2,π/2]
∴cosx=1/6