已知a向量=(2,cosx),b向量=(sin(x+30度),-2),函数f(x)=a向量点乘b向量((x属于R),求函数f(x)的单调增区间,若f(x)=6/5,求cos(2x,60度)的值

Question

已知a向量=(2,cosx),b向量=(sin(x+30度),-2),函数f(x)=a向量点乘b向量((x属于R),求函数f(x)的单调增区间,若f(x)=6/5,求cos(2x,60度)的值

Answer 1

f(x)=2sin(x+π/6)-2cosx
=2sinxcosπ/6+2cosxsinπ/6-2cosx
=√3sinx+cosx-2cosx
=√3sinx-cosx
=2(√3/2*sinx-1/2*cosx)
=2(sinxcosπ/6-cosxsinπ/6)
=2sin(x-π/6)
x-π/6∈[－π/2+2kπ,π/2+2kπ]
x∈[－π/3+2kπ,5π/6+2kπ]
2sin(x-π/6)＝6/5
sin(x-π/6)＝3/5
cos(2x-π/3)=1-2sin(x-π/6)^2=7/25