(1)由(a+b)/(cosA+cosB)=c/cosC,得
(a+b)/c=(cosA+cosB)/cosC
又已知在三角形中有
(a+b)/(sinA+sinB)=c/sinC,得
(a+b)/c=(sinA+sinB)/sinC
则
(sinA+sinB)/sinC=(cosA+cosB)/cosC
(sinA+sinB) cosC =(cosA+cosB) sinC
sinAcosC+sinBcosC=cosAsinC+cosBsinC
sinAcosC-cosAsinC=cosBsinC-sinBcosC
sin(A-C)=sin(C-B)
∵⊿ABC为锐角三角形
∴-90°∵f(x)=sinx在(-90°,90°）内是单调的
∴A-C=C-B
2C=A+B
3C=A+B+C=180°
∴∠C=60°
(2) cos(B+C)+√(3) sinA
=cos(180°-A)+√3sinA
=-cosA+√3sinA
=2((√3/2)sinA-(1/2)cosA)
=2(sinAcos30°-cosAsin30°)
=2sin(A-30°)
∵在锐角三角形ABC中,A最大,C=60°
∴60°≤A<90°
∴30°≤A-30°<60°
∴2sin30°≤2sin(A-30°)1≤2sin(A-30°)<√3
即1≤cos(B+C)+√(3) sinA <√3
(3)由S⊿ABC=1/2absinC,得
ab=2S⊿ABC/sinC
=2S⊿ABC/sin60°
=2√3/(√3/2)
=4
由c^2=a^2+b^2-2abcosC,得
a^2+b^2=c^2+2abcosC
(a+b)^2=a^2+b^2+2ab
=c^2+2abcosC+2ab
=6+2*4*(1/2)+2*4
=18
∴a+b=√18=3√2
∴a+b+c=3√2+√6