设a向量=(1+cosa,sina),b向量=(1-cosb,sinb),c向量=(1,0),a属于(0,π)b属于(π,2π)a向量与c向量夹角为X1,b向量与c向量夹角为X2,且X1-X2=π/6,求sin(a-b)/4的值
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/13 01:34:44
![设a向量=(1+cosa,sina),b向量=(1-cosb,sinb),c向量=(1,0),a属于(0,π)b属于(π,2π)a向量与c向量夹角为X1,b向量与c向量夹角为X2,且X1-X2=π/6,求sin(a-b)/4的值](/uploads/image/z/8895425-41-5.jpg?t=%E8%AE%BEa%E5%90%91%E9%87%8F%3D%EF%BC%881%2Bcosa%2Csina%29%2Cb%E5%90%91%E9%87%8F%3D%281-cosb%2Csinb%29%2Cc%E5%90%91%E9%87%8F%3D%281%2C0%29%2Ca%E5%B1%9E%E4%BA%8E%EF%BC%880%2C%CF%80%EF%BC%89b%E5%B1%9E%E4%BA%8E%EF%BC%88%CF%80%2C2%CF%80%EF%BC%89a%E5%90%91%E9%87%8F%E4%B8%8Ec%E5%90%91%E9%87%8F%E5%A4%B9%E8%A7%92%E4%B8%BAX1%2Cb%E5%90%91%E9%87%8F%E4%B8%8Ec%E5%90%91%E9%87%8F%E5%A4%B9%E8%A7%92%E4%B8%BAX2%2C%E4%B8%94X1-X2%3D%CF%80%2F6%2C%E6%B1%82sin%28a-b%29%2F4%E7%9A%84%E5%80%BC)
设a向量=(1+cosa,sina),b向量=(1-cosb,sinb),c向量=(1,0),a属于(0,π)b属于(π,2π)a向量与c向量夹角为X1,b向量与c向量夹角为X2,且X1-X2=π/6,求sin(a-b)/4的值
设a向量=(1+cosa,sina),b向量=(1-cosb,sinb),c向量=(1,0),a属于(0,π)b属于(π,2π)
a向量与c向量夹角为X1,b向量与c向量夹角为X2,且X1-X2=π/6,求sin(a-b)/4的值
设a向量=(1+cosa,sina),b向量=(1-cosb,sinb),c向量=(1,0),a属于(0,π)b属于(π,2π)a向量与c向量夹角为X1,b向量与c向量夹角为X2,且X1-X2=π/6,求sin(a-b)/4的值
a·c=(1+cosα,sinα)·(1,0)
=1+cosα=2cos(α/2)^2
|a|^2=(1+cosα)^2+sinα^2
=2+2cosα
=4cos(α/2)^2
α∈(0,π),即:α/2∈(0,π/2)
故:|a|=2cos(α/2)
故:cos=a·c/(|a|*|c|)=2cos(α/2)^2/(2cos(α/2))
=cos(α/2)
故:=α/2
-----------------------------
b·c=(1-cosβ,sinβ)·(1,0)
=1-cosβ=2sin(β/2)^2
|b|^2=(1-cosβ)^2+sinβ^2
=2-2cosβ
=4sin(β/2)^2
β∈(π,2π),即:β/2∈(π/2,π)
故:|b|=2sin(β/2)
故:cos=b·c/(|b|*|c|)=2sin(β/2)^2/(2sin(β/2))
=sin(β/2)=cos(π/2-β/2)=cos(β/2-π/2)
β/2∈(π/2,π),即:β/2-π/2∈(0,π/2)
故:=β/2-π/2
故:α/2-β/2+π/2=π/6
即:(α-β)/2=-π/3
即:(α-β)/4=-π/6
故:sin((α-β)/4)=-1/2