已知向量m=(cosθ,-sinθ),n=(√2+sinθ,cosθ),θ∈(π,3π/2),且cos(θ/2+π/8),求绝对值m+n的值.且cos(θ/2+π/8)=-4/5,求绝对值m+n的值
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![已知向量m=(cosθ,-sinθ),n=(√2+sinθ,cosθ),θ∈(π,3π/2),且cos(θ/2+π/8),求绝对值m+n的值.且cos(θ/2+π/8)=-4/5,求绝对值m+n的值](/uploads/image/z/8663899-67-9.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fm%3D%28cos%CE%B8%2C-sin%CE%B8%29%2Cn%3D%28%E2%88%9A2%2Bsin%CE%B8%2Ccos%CE%B8%29%2C%CE%B8%E2%88%88%28%CF%80%2C3%CF%80%EF%BC%8F2%29%2C%E4%B8%94cos%28%CE%B8%EF%BC%8F2%2B%CF%80%EF%BC%8F8%EF%BC%89%2C%E6%B1%82%E7%BB%9D%E5%AF%B9%E5%80%BCm%2Bn%E7%9A%84%E5%80%BC.%E4%B8%94cos%28%CE%B8%EF%BC%8F2%2B%CF%80%EF%BC%8F8%29%3D-4%EF%BC%8F5%2C%E6%B1%82%E7%BB%9D%E5%AF%B9%E5%80%BCm%2Bn%E7%9A%84%E5%80%BC)
已知向量m=(cosθ,-sinθ),n=(√2+sinθ,cosθ),θ∈(π,3π/2),且cos(θ/2+π/8),求绝对值m+n的值.且cos(θ/2+π/8)=-4/5,求绝对值m+n的值
已知向量m=(cosθ,-sinθ),n=(√2+sinθ,cosθ),θ∈(π,3π/2),且cos(θ/2+π/8),求绝对值m+n的值.
且cos(θ/2+π/8)=-4/5,求绝对值m+n的值
已知向量m=(cosθ,-sinθ),n=(√2+sinθ,cosθ),θ∈(π,3π/2),且cos(θ/2+π/8),求绝对值m+n的值.且cos(θ/2+π/8)=-4/5,求绝对值m+n的值
mn=cosθ(√2+sinθ)-sinθcosθ
n^2=(√2+sinθ)^2+(cosθ)^2=3+2√2sinθ
|m+n|^2=(m+n)^2=m^2+2mn+n^2
=1+2【cosθ(√2+sinθ)-sinθcosθ】+(3+2√2sinθ)
=4+2√2(cosθ+sinθ)
=4+4sin(θ+π/4)
cos(θ/2+π/8)=-4/5,两边平方,得cos^2 (θ/2+π/8)=16/25
得1/2【cos(θ+π/4)+1】=16/25
得cos(θ+π/4)=7/25
θ∈(π,3π/2),则θ+π/4∈(5π/4,7π/4),
所以sin(θ+π/4)=-24/25
所以|m+n|=2/5
|m+n|^2=m^2+n^2+2m*n=1+2-2√2sinθ+1+2√2sinθcosθ
=4-4sin(θ-45度)
45度<=θ-45度<=225度
-√2/2<=sin(θ-45度)<=1
所以得到|M+N|^2<=4+2√2
|M+N|<=√(4+2√2)
绝对就是小于等于√(4+2√2)咯
(2)由(1),则4-4sin(θ-4...
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|m+n|^2=m^2+n^2+2m*n=1+2-2√2sinθ+1+2√2sinθcosθ
=4-4sin(θ-45度)
45度<=θ-45度<=225度
-√2/2<=sin(θ-45度)<=1
所以得到|M+N|^2<=4+2√2
|M+N|<=√(4+2√2)
绝对就是小于等于√(4+2√2)咯
(2)由(1),则4-4sin(θ-45度)=(4√10/5)^2
所以sin(θ-45度)=-3/5
sin2θ=cos[90度-2θ]=cos[2(θ-45度)]=1-2sin(θ-45度)^2=7/25
希望有所帮助咯。~
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