(1)
設(shè)a/sinA=b/sinB=c/sinC=k
(2c-b)/a=(2ksinC - ksinB)/(ksinA)=(2sinC-sinB)/sinA
∴(2sinC-sinB)/sinA=cosB/cosA
即sinAcosB=(2sinC-sinB)cosA=2sinCcosA-sinBcosA
即sinAcosB+sinBcosA=2sinCcosA
即sin(A+B)=2sinCcosA
即sinC=2sinCcosA
∴cosA=1/2
A=60°
(2)
∵a/sinA=b/sinB=C/sinC=2√5/(√3/2)=4√5/√3
∴(bc)/(sinBsinC)=(4√5/√3)²=80/3
bc=(80/3)sinBsinC
S△ABC
=(1/2)bcsinA
=(1/2)×(80/3)sinBsinC×(√3/2)
=(10/√3)×(2sinBsinC)
=(10/√3)×[cos(B-C)-cos(B+C)]
=(10/√3)×[cos(B-C)+(1/2)]
≤(10/√3)×[1+(1/2)]=5√3
當(dāng)且僅當(dāng)B=C=60°時(shí)等號(hào)成立
∴當(dāng)B=C=60°時(shí),Smax=5√3