作业帮已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4].1）求a*b及|a+b|的表达式2）若f（x）=a*b-|a+b|.求发（x）最大值和最小值
来源：学生作业帮助网 编辑：作业帮 时间：2024/11/27 22:27:34
![已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4].1）求a*b及|a+b|的表达式2）若f（x）=a*b-|a+b|.求发（x）最大值和最小值](/uploads/image/z/351253-37-3.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28cos3x%2F2%2Csin3x%2F2%29%2Cb%3D%28cosx%2F2%2C-sinx%2F2%29%2C%E4%B8%94x%E2%88%88%5B-%CF%80%2F3%2C%CF%80%2F4%5D.1%EF%BC%89%E6%B1%82a%2Ab%E5%8F%8A%7Ca%2Bb%7C%E7%9A%84%E8%A1%A8%E8%BE%BE%E5%BC%8F2%EF%BC%89%E8%8B%A5f%EF%BC%88x%EF%BC%89%3Da%2Ab-%7Ca%2Bb%7C.%E6%B1%82%E5%8F%91%EF%BC%88x%EF%BC%89%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC)
已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4].1）求a*b及|a+b|的表达式2）若f（x）=a*b-|a+b|.求发（x）最大值和最小值
已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4].
1）求a*b及|a+b|的表达式
2）若f（x）=a*b-|a+b|.求发（x）最大值和最小值
已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4].1）求a*b及|a+b|的表达式2）若f（x）=a*b-|a+b|.求发（x）最大值和最小值
ab=cos(3x/2)*cos(x/2)-sin(3x/2)*sin(x/2)
=cos[(3x+x)/2]
=cos(2x).
a+b=(cos(3x/2)+cos(x/2),sin(3x/2)-sin(x/2)),
|a+b|=√[(cos(3x/2)+cos(x/2))^2+(sin(3x/2)-sin(x/2))^2]
=√[2(1+cos2x)]
=2*|cosx|,
因为,x∈[-π/3,π/4].则有,cosx>0,
即,
|a+b|=2*|cosx|=2cosx.
2.若f（x）=a*b-|a+b|.则有,
f(x)=cos2x-2cosx,
=2cos^2x-1-2cosx
=2(cosx-1/2)^2-3/2.
而,x∈[-π/3,π/4].则有,
1)当X=0时,cos0=1,则f(x)=2(1-1/2)^2-3/2=-1.
2)当X=π/4时,cosπ/4=√2/2,则f(x)=2*(√2/2-1/2)^2-3/2=-√2.
则,f(x)最大值=-1,f(x)最小值=-√2.