高一数学已知向量AC=5,向量AB=8,向量AD=5/11向量DB,
向量CD点向量AB=0,且∠BAC=θ,cos(θ+x)=4/5,
-π<x<-π/4,求sinx的值。
需要具体过程和答案
已知向量|AC|=5,|AB|=8,向量AD=(5/11)DB,CD•AB=0,
且∠BAC=θ,cos(θ+x)=4/5, -π<x<-π/4,求sinx的值
AD=(5/11)DB=(5/11)(AB-AD)--->AD=(5/16)AB
CD=AD-AC=(5/16)AB-AC
0 = CD•AB = [(5/16)AB-AC]•AB = (5/16)AB•AB - AC•AB
--->(5/16)|AB|² = |AB||AC|cosθ
--->cosθ = (5/16)|AB|/|AC| = 1/2--->θ=...全部
已知向量|AC|=5,|AB|=8,向量AD=(5/11)DB,CD•AB=0,
且∠BAC=θ,cos(θ+x)=4/5, -π<x<-π/4,求sinx的值
AD=(5/11)DB=(5/11)(AB-AD)--->AD=(5/16)AB
CD=AD-AC=(5/16)AB-AC
0 = CD•AB = [(5/16)AB-AC]•AB = (5/16)AB•AB - AC•AB
--->(5/16)|AB|² = |AB||AC|cosθ
--->cosθ = (5/16)|AB|/|AC| = 1/2--->θ=π/3
-π<x<-π/4--->-2π/3<θ+x<π/12<arccos(4/5)
cos(θ+x)=4/5--->sin(θ+x)=-3/5
--->sinx = sin[(θ+x)-θ]
= sin(θ+x)cosθ - cos(θ+x)sinθ
= (-3/5)(1/2) - (4/5)(√3/2)
= -(3+4√3)/10。
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