向量OZ1=(3,2sinA),向量OZ2=(sinA,1+cosA)
∵OZ1//OZ2,∴3(1+cosA)=2sinA*sinA,即3(1+cosA)=2(1-(cosA)^2)即2(cosA)^2+3cosA+1=0
即cosA=-1/2或cosA=-1(角A为三角形内角,舍去),∴cosA=-1/2即角A=120°
a^2=b^2+c^2-2bccosA=b^2+c^2+bc,又a^2=(√7)^2(c-b)^2=7(b-c)^2=7b^2-14bc+7c^2,
∴b^2+c^2+bc=7b^2-14bc+7c^2即2b^2-5bc+2c^2=0,即2(b/c)^2-5b/c+2=0即b/c=2或1/2
即b=2c或c=2b
若b=2c,作BD垂直CA于D,角BAC=120°,AB=c,AC=b=2c,BC=√7 c,∴AD=c/2,BD=√3 c/2,
∴CD=5c/2,
∴sinC=√3 / 2√7,cosC=5 / 2√7,
∴cos(C-π/6)=cosCcos(π/6)+sinCsin(π/6)=(5 / 2√7)*(√3 / 2)+(√3 / 2√7)*(1/2)=3√21 / 14
若c=2b,作BD垂直CA于D,角BAC=120°,AC=b,AB=c=2b,BC=√7 b,∴AD=b,BD=√3 b,
∴CD=2b,
∴sinC=√3 / √7,cosC=2 / √7,
∴cos(C-π/6)=cosCcos(π/6)+sinCsin(π/6)=(2 / √7)*(√3 / 2)+(√3 / √7)*(1/2)=3√21 / 14
综上,cos(C-π/6)=3√21 / 14