Question

求证：a*sinA-b*sinB=c*sin(A-B).

Answer 1

证明：

利用正弦定理

a/sinA=b/sinB=c/sinC=2r

则 左=2rsin²A-2rsin²B

=r(1-cos2A)-r(1-cos2B)

=r(cos2B-cos2A)

=r{[cos[(A+B)-(A-B)]-[cos[(A+B)+(A-B)]}

=r [cos(A+B)cos[(A-B)+sin(A+B)sin(A-B)-cos(A+B)cos(A-B)+sin(A+B)sin(A-B)]

=2rsin(A+B)sin(A-B)

=2rsin(π-c)sin(A-B)

=2rsinCsin(A-B)

=csin(A-B)

得证.