在△ABC中,内角A B C所对的边分别是a b c,且a²+b²=c²+ab已知向量m=(sinA,cosA),向量n=(cosB,-sinB),求|M-2N|的取值范围坐等大侠
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![在△ABC中,内角A B C所对的边分别是a b c,且a²+b²=c²+ab已知向量m=(sinA,cosA),向量n=(cosB,-sinB),求|M-2N|的取值范围坐等大侠](/uploads/image/z/1603741-13-1.jpg?t=%E5%9C%A8%E2%96%B3ABC%E4%B8%AD%2C%E5%86%85%E8%A7%92A+B+C%E6%89%80%E5%AF%B9%E7%9A%84%E8%BE%B9%E5%88%86%E5%88%AB%E6%98%AFa+b+c%2C%E4%B8%94a%26%23178%3B%2Bb%26%23178%3B%3Dc%26%23178%3B%2Bab%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fm%3D%EF%BC%88sinA%2CcosA%29%2C%E5%90%91%E9%87%8Fn%3D%EF%BC%88cosB%2C-sinB%EF%BC%89%2C%E6%B1%82%7CM-2N%7C%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4%E5%9D%90%E7%AD%89%E5%A4%A7%E4%BE%A0)
在△ABC中,内角A B C所对的边分别是a b c,且a²+b²=c²+ab已知向量m=(sinA,cosA),向量n=(cosB,-sinB),求|M-2N|的取值范围坐等大侠
在△ABC中,内角A B C所对的边分别是a b c,且a²+b²=c²+ab
已知向量m=(sinA,cosA),向量n=(cosB,-sinB),求|M-2N|的取值范围
坐等大侠
在△ABC中,内角A B C所对的边分别是a b c,且a²+b²=c²+ab已知向量m=(sinA,cosA),向量n=(cosB,-sinB),求|M-2N|的取值范围坐等大侠
∵ a²+b²=c²+ab
由余弦定理知,
cosC=(a^2 + b^2 - c^2)/(2ab)
= ab/(2ab) = 1/2
∵C是三角形内角,∴ C∈(0,π) , ∴C = π/3
∴A + B = 2π/3 ∴ B = 2π/3 - A
|M-2N|^2 = (sinA - 2cosB)^2 + (cosA + 2sinB)^2
= (sinA)^2 - 4sinAcosB + 4(cosB)^2 + (cosA)^2 + 4cosAsinB + 4(sinB)^2
=5 - 4(sinAcosB - cosAsinB)
=5 - 4sin(A - B)
=5 - 4sin(A - (2π/3 - A))
=5 - 4sin(2A - 2π/3)
∵C = π/3 , ∴A∈(0, 2π/3) ∴ (2A - 2π/3)∈( -2π/3,2π/3)
由正弦函数性质知,sin(2A - 2π/3)∈[-1, 1]
∴ |M-2N|^2 = 5 - 4sin(2A - 2π/3) ∈[1, 9]
又∵ |M-2N| > 0 ∴ |M-2N|∈[1, 3]
望采纳!