ABC中,sinA,cosB,sinC成等比数列,cosA,sinB,cosC成等差数列,求B.

2sinB＝cosA＋cosC →4·sinB/2·cosB/2＝2·cos(A＋C)/2·cos(A－C)/2 →4·sinB/2·cosB/2＝2·cos(π－B)/2·cos(A－C)/2 →4·sinB/2·cosB/2＝2·sinB/2·cos(A－C)/2 →2cosB/2＝cos(A－C)/2 →8(cosB/2)＾2－1＝2[cos(A－C)/2]＾2－1 →cos(A－C)＝4cosB＋3…………………① (cosB)＾2＝sinA·sinB →(cosB)＾2＝－1/2×[cos(A＋C)－cos(A－C)] →(cosB)＾2＝－1/2×[cos(π－B)－cos(A－C)] →(cosB)＾2＝1/2×[cosB＋cos(A－C)]……………② 把①式代入②式得 (cosB)＾2＝1/2×[cosB＋4cosB＋3] →2×(cosB)＾2－5cosB－3＝0 →cosB＝－1/2 (舍弃cosB＝3) 所以,B＝120°