Question

在△ABC中，若sinB=sin（A+C）/2 ，则sinB=？

Answer 1

由于
sin[(A+C)/2]
=sin[(π-B)/2]
=sin(π/2-B/2)
=cos(B/2)
所以 sinB=cos(B/2)
而 sinB=2sin(B/2)cos(B/2)
于是 2sin(B/2)=1
sin(B/2)=1/2
B/2=π/6
B=π/3=60°
sinB=√3/2
其中，π的单位为弧度，π=180°

Answer 2

sinB=sin（A+C）/2
∵B=180º-(A+C) ∴B/2=90º-(A+C)/2
∴sin（A+C）/2=cosB/2
∵sinB=2sinB/2cosB/2
∴2sinB/2cosB/2=cosB/2
∴cosB/2(sinB/2-1/2)=0
∵0<B<180º,0º<B/2<90º
∴cosB/2.>0
∴sinB/2-1/2=0,sinB/2=1/2
∴B/2=30º,B=60º
∴sinB=√3/2