Question

△ABC中,abc是三个内角ABC对应的三边,已知(b-c)sinB=asinA-csinC？

Answer 1

1、a/sinA=2R,b/sinB=2R,c/sinC=2R
(b-c)sinB=asinA-csinC
（b-c)×b=a²-c²
b²+c²-a²=bc
∴cosA=(b²+c²-a²)/2bc
=bc/2bc
=1/2
∴A=60°
2、sinC-cos(B+30°)=1/2
sin(180°-60°-B)-cos(B+30°)=1/2
sin(120°-B)-cos(B+30°)=1/2
cos(90°-120°+B)-cos(B+30°)=1/2
cos(B-30°)-cos(B+30°)=1/2
-2sinBsin(-30­°)=1/2
sinB=1/2
B=30°或150°（舍去）
∴B=30°
C=90°
3、b+c=5
b²+c²=25-2bc
a²=b²+c²-2bc×cos60°
7=25-2bc-bc
3bc=18
bc=6
∴b=2或3
c=3或2
(1)a=√7,b=2,c=3时
cosB=(7+9-4)/2√7×3=2√7/7
∴AD²=c²+a²/2-2ca/2cosB
=9+7/4-2×3×√7/2×2√7/7
=19/4
∴AD=√19/2
(2)a=√7,b=3,c=2时
cosB=(7+4-9)/2√7×2=√7/14
AD²=c²+a²/2-2ca/2cosB
=4+7/4-2×2×√7/2×√7/14
=19/4
∴AD=√19/2,1,