已知向量m=(cosθ,sinθ),向量n=(√2-sinθ,)cosθ,θ∈【派,1.5派】,若 ︳向量m+向量n ︳=(4√10)/5,求sin2θ的值我已经算出 ︳向量m+向量n ︳max=根号(4+2√2)了向量n=(√2-sinθ,cosθ)
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![已知向量m=(cosθ,sinθ),向量n=(√2-sinθ,)cosθ,θ∈【派,1.5派】,若 ︳向量m+向量n ︳=(4√10)/5,求sin2θ的值我已经算出 ︳向量m+向量n ︳max=根号(4+2√2)了向量n=(√2-sinθ,cosθ)](/uploads/image/z/10129503-39-3.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fm%3D%28cos%CE%B8%2Csin%CE%B8%29%2C%E5%90%91%E9%87%8Fn%3D%EF%BC%88%E2%88%9A2-sin%CE%B8%2C%29cos%CE%B8%2C%CE%B8%E2%88%88%E3%80%90%E6%B4%BE%2C1.5%E6%B4%BE%E3%80%91%2C%E8%8B%A5+%EF%B8%B3%E5%90%91%E9%87%8Fm%2B%E5%90%91%E9%87%8Fn+%EF%B8%B3%3D%284%E2%88%9A10%29%2F5%2C%E6%B1%82sin2%CE%B8%E7%9A%84%E5%80%BC%E6%88%91%E5%B7%B2%E7%BB%8F%E7%AE%97%E5%87%BA+%EF%B8%B3%E5%90%91%E9%87%8Fm%2B%E5%90%91%E9%87%8Fn+%EF%B8%B3max%3D%E6%A0%B9%E5%8F%B7%EF%BC%884%2B2%E2%88%9A2%EF%BC%89%E4%BA%86%E5%90%91%E9%87%8Fn%3D%EF%BC%88%E2%88%9A2-sin%CE%B8%2Ccos%CE%B8%29)
已知向量m=(cosθ,sinθ),向量n=(√2-sinθ,)cosθ,θ∈【派,1.5派】,若 ︳向量m+向量n ︳=(4√10)/5,求sin2θ的值我已经算出 ︳向量m+向量n ︳max=根号(4+2√2)了向量n=(√2-sinθ,cosθ)
已知向量m=(cosθ,sinθ),向量n=(√2-sinθ,)cosθ,θ∈【派,1.5派】,
若 ︳向量m+向量n ︳=(4√10)/5,求sin2θ的值
我已经算出 ︳向量m+向量n ︳max=根号(4+2√2)了
向量n=(√2-sinθ,cosθ)
已知向量m=(cosθ,sinθ),向量n=(√2-sinθ,)cosθ,θ∈【派,1.5派】,若 ︳向量m+向量n ︳=(4√10)/5,求sin2θ的值我已经算出 ︳向量m+向量n ︳max=根号(4+2√2)了向量n=(√2-sinθ,cosθ)
1、m+n=(√2+cosx-sinx,sinx+cosx),则:
|m+n|²=4-2√2[sinx-cosx]=[(4√10)/5]²=32/5
2√2[sinx-cosx]=-12/5 两边平方,得:1-sin2x=18/25
sin2x=7/25
向量m=(cosθ,sinθ),向量n=(√2-sinθ, cosθ ),θ∈【派,1.5派】,
︳向量m+向量n ︳=(4√10)/5
==> | 向量m+向量n |^2 = 16/5
向量m+向量n = (cos +√2-sinθ, sinθ + cosθ )
︳向量m+向量n ︳^2 = (cosθ+√2-sinθ)^2 + (sinθ + cosθ)^2...
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向量m=(cosθ,sinθ),向量n=(√2-sinθ, cosθ ),θ∈【派,1.5派】,
︳向量m+向量n ︳=(4√10)/5
==> | 向量m+向量n |^2 = 16/5
向量m+向量n = (cos +√2-sinθ, sinθ + cosθ )
︳向量m+向量n ︳^2 = (cosθ+√2-sinθ)^2 + (sinθ + cosθ)^2
= 1 + 2 + 2√2cosθ - 2√2*sinθ - 2cosθsinθ + 1+2sinθcosθ
= 4 + 2√2(cosθ - sinθ)
= 16/5
(cosθ - sinθ) = -4/5/(2√2) = -√2/5
(cosθ - sinθ)^2 = 2/25
1 - 2sinθcosθ = 2/25
2sinθcosθ =2 1 - 2/25 = 23/25
sin2θ = 23/25
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m=(cosθ,sinθ), n=(√2-sinθ,cosθ),θ∈[π, 3π/2]
|m+n| =(4√10)/5
m+n = (cosθ+√2-sinθ, sinθ+cosθ)
|m+n|^2
= (cosθ+√2-sinθ)^2+(sinθ+cosθ)^2
= [(cosθ+√2)^2 - 2sinθ(cosθ+√2) + (sinθ)^2 ] +...
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m=(cosθ,sinθ), n=(√2-sinθ,cosθ),θ∈[π, 3π/2]
|m+n| =(4√10)/5
m+n = (cosθ+√2-sinθ, sinθ+cosθ)
|m+n|^2
= (cosθ+√2-sinθ)^2+(sinθ+cosθ)^2
= [(cosθ+√2)^2 - 2sinθ(cosθ+√2) + (sinθ)^2 ] + 1+ 2sinθcosθ
= ( 1+2√2cosθ + 2 -2sinθcosθ - 2√2sinθ ) + 1+ 2sinθcosθ
= 4 + 2√2(cosθ-sinθ) (1)
|m+n|^2
= [(4√10)/5]^2
= 32/5 (2)
(1) =(2)
=>4 + 2√2(cosθ-sinθ) = 35/2
4√2(cosθ-sinθ) = 27
32(cosθ-sinθ)^2 = 729
1- sin2θ = 729/32
sin2θ = -697/32
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向量m+向量n=(cosθ+√2-sinθ,sinθ+cosθ)
︳向量m+向量n ︳^2
=(cosθ+√2-sinθ)^2+(sinθ+cosθ)^2
=1+2cosθ(√2-sinθ)+(√2-sinθ)^2+1+2sinθcosθ
=2+2√2cosθ+2-2√2sinθ=[(4√10)/5]^2=16/5
√2(cosθ-sinθ)=-2/5
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向量m+向量n=(cosθ+√2-sinθ,sinθ+cosθ)
︳向量m+向量n ︳^2
=(cosθ+√2-sinθ)^2+(sinθ+cosθ)^2
=1+2cosθ(√2-sinθ)+(√2-sinθ)^2+1+2sinθcosθ
=2+2√2cosθ+2-2√2sinθ=[(4√10)/5]^2=16/5
√2(cosθ-sinθ)=-2/5
cosθ-sinθ=-√2/5
平方得
(cosθ-sinθ)^2=(-√2/5)^2
1-sin2θ=2/25
sin2θ=23/25
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