已知A+b+C=π 求证 cosA+cosB+cosC=1+4sin(A/2)sin(B/2)sin(C/2)

来源:学生作业帮助网 编辑:作业帮 时间:2024/04/28 16:09:06
已知A+b+C=π 求证 cosA+cosB+cosC=1+4sin(A/2)sin(B/2)sin(C/2)

已知A+b+C=π 求证 cosA+cosB+cosC=1+4sin(A/2)sin(B/2)sin(C/2)
已知A+b+C=π 求证 cosA+cosB+cosC=1+4sin(A/2)sin(B/2)sin(C/2)

已知A+b+C=π 求证 cosA+cosB+cosC=1+4sin(A/2)sin(B/2)sin(C/2)
cosA+cosB+cosC
=2cos[(A+B)/2]cos[(A-B)/2]+cosC
=2cos[(π-C)/2]cos[(A-B)/2]+cosC
=2sin(c/2)cos[(A-B)/2]+1-2[sin(C/2)]^2
=1+2sin(c/2){cos[(A-B)/2]-[sin(C/2)]}
=1+2sin(c/2){cos[(A-B)/2]-[sin(π -A-B)/2]}
=1+2sin(c/2){cos[(A-B)/2]-[cos[(A+B)/2]}
=1+2sin(c/2)[-2sin(A/2)sin(-B/2)]
=1+4sin(A/2)sin(B/2)sin(C/2)