在三角形ABC中,已知sinAcos^2(C/2)+sinC-cos^2(A/2)=(3/2)sinB求证：sinA+sinC=2_数学解答

在三角形ABC中,已知sinAcos^2(C/2)+sinC-cos^2(A/2)=(3/2)sinB
求证：sinA+sinC=2sinB

人气：447 ℃　时间：2019-09-29 01:35:03

解答

cosC=2(cosC/2)^2-1,
(cosC/2)^2=(1+cosC)/2
(cosA/2)^2=(1+cosA)/2
sin(A+C)=sinB
sin(90°-B/2)=cosB/2
sinAcos^(C/2)+sinCcos^(A/2)=3/2sinB
sinA*[(1+cosC)/2]+sinC*[(1+cosA)/2]=3/2sinB
sinA+sinC+sin(A+C)=3sinB
sinA+sinC+sin(A+C)=3sinB
sinA+sinC=2sinB