Question

在△ABC中,∠B=120°,∠BAC=15°，BC=2，则AB

Answer 1

∠C=180°-120°-15°=45°
方法一：根据正弦定理
BC
AB
——
=
——
sinA
sinC
则：AB
=BC·sinC/sinA
=2sin45°/sin15°
=√2/[(√6-√2)/4]
=4√2/(√6-√2)
=√2(√6+√2)
=2√3+2
方法二：作辅助线
过A作AD⊥BC交CB延长线于D
∠ABD=180°-∠ABC=180°-120°=60°
∠DAC=180°-∠D-∠C=180°-90°-45°=45°
∴∠DAC=∠C=45°
AD=CD=BC+BD=2+BD
AD=BD·tan∠ABD=BD·tan60°=BD·√3
即：2+BD=BD·√3
2=BD·√3-BD
(√3-1)BD=2
BD=2/(√3-1)
BD=√3+1
AB=BD/cos∠ABD
=(√3+1)/cos60°
=2(√3+1)
=2√3+2