已知向量a=(cos3θ/2,sin3θ/2),b=(cosθ/2,-sinθ/2),θ属于[0,π/3] 1) 求a·b/(|a+b|)的最大值已知向量a=(cos3θ/2,sin3θ/2),b=(cosθ/2,-sinθ/2),θ属于[0,π/3]1) 求a·b/(|a+b|)的最大值、最小值;2) 若|ka+b|=(√3)|a-
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![已知向量a=(cos3θ/2,sin3θ/2),b=(cosθ/2,-sinθ/2),θ属于[0,π/3] 1) 求a·b/(|a+b|)的最大值已知向量a=(cos3θ/2,sin3θ/2),b=(cosθ/2,-sinθ/2),θ属于[0,π/3]1) 求a·b/(|a+b|)的最大值、最小值;2) 若|ka+b|=(√3)|a-](/uploads/image/z/2483087-23-7.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28cos3%CE%B8%2F2%2Csin3%CE%B8%2F2%29%2Cb%3D%28cos%CE%B8%2F2%2C-sin%CE%B8%2F2%29%2C%CE%B8%E5%B1%9E%E4%BA%8E%5B0%2C%CF%80%2F3%5D+1%EF%BC%89+%E6%B1%82a%C2%B7b%2F%28%7Ca%2Bb%7C%29%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28cos3%CE%B8%2F2%2Csin3%CE%B8%2F2%29%2Cb%3D%28cos%CE%B8%2F2%2C-sin%CE%B8%2F2%29%2C%CE%B8%E5%B1%9E%E4%BA%8E%5B0%2C%CF%80%2F3%5D1%EF%BC%89+%E6%B1%82a%C2%B7b%2F%28%7Ca%2Bb%7C%29%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E3%80%81%E6%9C%80%E5%B0%8F%E5%80%BC%EF%BC%9B2%EF%BC%89+%E8%8B%A5%7Cka%2Bb%7C%3D%EF%BC%88%E2%88%9A3%EF%BC%89%7Ca-)
已知向量a=(cos3θ/2,sin3θ/2),b=(cosθ/2,-sinθ/2),θ属于[0,π/3] 1) 求a·b/(|a+b|)的最大值已知向量a=(cos3θ/2,sin3θ/2),b=(cosθ/2,-sinθ/2),θ属于[0,π/3]1) 求a·b/(|a+b|)的最大值、最小值;2) 若|ka+b|=(√3)|a-
已知向量a=(cos3θ/2,sin3θ/2),b=(cosθ/2,-sinθ/2),θ属于[0,π/3] 1) 求a·b/(|a+b|)的最大值
已知向量a=(cos3θ/2,sin3θ/2),b=(cosθ/2,-sinθ/2),θ属于[0,π/3]
1) 求a·b/(|a+b|)的最大值、最小值;
2) 若|ka+b|=(√3)|a-kb|(k属于R),求k的取值范围
已知向量a=(cos3θ/2,sin3θ/2),b=(cosθ/2,-sinθ/2),θ属于[0,π/3] 1) 求a·b/(|a+b|)的最大值已知向量a=(cos3θ/2,sin3θ/2),b=(cosθ/2,-sinθ/2),θ属于[0,π/3]1) 求a·b/(|a+b|)的最大值、最小值;2) 若|ka+b|=(√3)|a-
1)a·b=1/4(cos3θcosθ-sin3θsinθ)=cos4θ/4
a+b=1/2(cos3θ+cosθ,sin3θ-sinθ)
|a+b|=1/2(2+2cos3θsin3θ-2cosθsinθ)^(1/2)=1/2(2+2cos4θ)^(1/2)=cos2θ
a·b/|a+b|=cos4θ/4cos2θ=(2(cos2θ)^2-1)/4cos2θ
将cos2θ变作自变量,其取值范围为[1/2,1],a·b/|a+b|是关于cos2θ的单调递增函数,从而a·b/|a+b|的最大值在θ=0时取到,为1/4,最小值在θ=π/3时取到,为-1/4.
2)|ka+b|=(√3)|a-kb|两边平方得,并整理得(a^2-3b^2)k^2+8ka·b+b^2-3a^2=0,再将a^2=b^2=1/4及a·b=cos4θ/4代入,得-k^2/2+4kcos4θ-1/2=0,从而k^2-4kcos4θ+1=0,即k+1/k=4cos4θ∈[-4,2],所以,k的取值范围为[-2-√3,-2+√3]∪\{1}或写作-2-√3≤k≤-2+√3或k=1
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1) a*b=cos(3θ/2)cos(θ/2)-sin(3θ/2)sin(θ/2)=cos2θ
a+b=(cos(3θ/2)+cos(θ/2), sin(3θ/2)-sin(θ/2))
Ia+bI=√[2+2cos2θ]=2IcosθI=2cosθ (θ属于[0,π/3])
a*b/Ia+bI=cos2θ/2cosθ=cosθ-1/(2cosθ)
∴最大值=1-1...
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1) a*b=cos(3θ/2)cos(θ/2)-sin(3θ/2)sin(θ/2)=cos2θ
a+b=(cos(3θ/2)+cos(θ/2), sin(3θ/2)-sin(θ/2))
Ia+bI=√[2+2cos2θ]=2IcosθI=2cosθ (θ属于[0,π/3])
a*b/Ia+bI=cos2θ/2cosθ=cosθ-1/(2cosθ)
∴最大值=1-1/2=1/2, 最小值=1/2-1=-1/2
2) |ka+b|=(√3)|a-kbI
k^2a^2+b^2+2kab=3(a^2-2kab+k^2b^2)
即k^2+1+2kcos2θ=3-6kcos2θ+k^2
8kcos2θ=2
k=1/(4cos2θ)
θ属于[0,2π/3]
k属于[-1/2, +∞)
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