f(x)=向量a.向量b
=sin(x/2)cos(x/2)+√3cos(x/2)cos(x/2).
=(1/2)*2sin(x/2)cos(x/2)+√3cos^2(x/2).
=(1/2)sinx+√3(1+cosx)/2.
=sinxcos(π/3)+cosxsin(π/3)+√3/2.
∴f(x)=sin(x+π/3)+√3/2.
(1) 求f(x)的零点：
令f(x)=0,则,sin(x+π/3)+√3/2=0.
sin(x+π/3)=-√3/2.
x+π/3=π+π/3,(1)
或,x+π/3=2π-π/3 (2)
由(1),得：x=π,得函数f(x)的零点(π,0) ---即为所求“零点”.
由(2),得：x=2π-2π/3.
x=4π/3.
得函数f(x)的第二个零点(4π/3,0).----即为所求“零点”.
[π.4π/3]在『0,2π]内,符合题设要求.
(2) f(A)=sin(A+π/3)+√3/2=√3.
sin(A+π/3)=√3/2
A+π/3=π/3,(1).或A+π/3=2π/3.(2)
∵ 由(1),得A=0.(不符合题设要求,舍去).
由(2)得：A=π/3
∴A=π/3.
又,sinA=2sinC.2sinC=sinπ/3=√3/2.
sinC=√3/4.
∠C=25.66°
∠C≈26°.
∠B=180-∠A-∠C=180°-60°-26°=94°
由正弦定理得：
c/sinC=b/sinB.
c=bsinC/sinB.
=(2*√3/4)/sin94°
=0.866/0.9976.
∴c≈0.87.---即为所求.