已知函数f(x)=[sin(x+a/2)+根号3cos(x+a/2)]*cos(x+a/2),若a属于(0,π),且f(x)为为偶函数,求a的值

题目

已知函数f(x)=[sin(x+a/2)+根号3cos(x+a/2)]*cos(x+a/2),若a属于(0,π),且f(x)为为偶函数,求a的值

答案

f(x)=[sin(x+a/2)+根号3cos(x+a/2)]*cos(x+a/2)
=sin(x+a/2)cos(x+a/2)+根号3cos(x+a/2)cos(x+a/2)
=1/2sin(2x+a)+根号3/23cos(2x+a)
=cos(π/3)sin(2x+a)+sin(π/3)cos(2x+a)
=sin(2x+a+π/3)
f(x)为为偶函数 a属于(0,π)
f(-x)=sin(-2x+a+π/3)=f(x) =sin(2x+a+π/3)
-2x+a+π/3=π-(2x+a+π/3) a属于(0,π)
故a=π/6