Question

题目如图。

Answer 1

1)
b+c=2acosB
sinB+sinC=2sinAcosB
sinB+sin(A+B)=2sinAcosB
sinB+sinAcosB+cosAsinB=2sinAcosB
sinB+cosAsinB=sinAcosB
sinB=sinAcosB-cosAsinB
sinB=sin(A-B)
B=A-B 或 B+(A-B )=π
A=2B 或 A=π(舍去)
A=2B
2) cosB=2/3 ,sinB=√5/3, π/3<B<π/2 ,2π/3<A<π
所以0<C< π/6
A+B+C=π ,A=2B

C=π-3B
cosC=cos(π-3B)=-cos3B=-4(cosB)^3+3cosB
=-4(2/3)^3+3*2/3
=2-(32/27)
=22/27
cosC=22/27
3)a/b=sinA/sin2B
a/b=sin2B/sin2B
a/b=2cosB
b=a/2cosB
因S=a^2/4
S=1/2absinC=a^2/4
a^2sinC/4cosB=a^2/4
cosB=sinC