已知O是锐角△ABC的外接圆的圆心,且∠A=θ,若cosB/sinC*向量ABcosC/sinB*向量AC=2m*向量AO,则m=已知O是锐角△ABC的外接圆的圆心,且∠Aθ=,若cosB/sinC*向量ABcosC/sinB*向量AC=2m*向量AO,则m=（用θ表示)

Question

已知O是锐角△ABC的外接圆的圆心,且∠A=θ,若cosB/sinC*向量AB cosC/sinB*向量AC=2m*向量AO,则m=
已知O是锐角△ABC的外接圆的圆心,且∠Aθ=,若cosB/sinC*向量AB cosC/sinB*向量AC=2m*向量AO,则m=（用θ表示)

Answer 1

取AB中点D,则有 AO = AD + DO ,
代入cosB /sinC AB +cosC/ sinB AC =2m AO 得：
cosB/ sinC AB +cosC/ sinB AC =2m( AD + DO ),
由 OD ⊥ AB ,得 DO • AB =0,
∴两边同乘 AB ,化简得：
cosB /sinC AB • AB +cosC/ sinB AC • AB =2m( AD + DO )• AB =m AB • AB ,
即cosB /sinC c2+cosC/ sinB bc•cosA=mc2,
由正弦定理a/ sinA =b/ sinB =c/ sinC 化简得：
cosB/ sinC* sin^2C+cosC /sinB* sinBsinCcosA=msin^2C,
由sinC≠0,两边同时除以sinC得：cosB+cosAcosC=msinC,
∴m=[cosB+cosAcosC]/ sinC =[-cos(A+C)+cosAcosC]/ sinC=[-cosAcosC+sinAsinC+cosAcosC ]/sinC =sinA,
又∠A=θ,
则m=sinθ．
故答案为：sinθ