Question

在三角形ABC中,设a+c=2b,A-C=60度,求sinB的值

Answer 1

sin(A + B) + sin(A - B) = sin(A)cos(B) + cos(A)sin(B) + sin(A)cos(B)

- cos(A)sin(B)

= 2sin(A)cos(B) ……………………………… ①

令A = (a + c)/2,B = (a - c)/2 则:

sin(A + B) + sin(A - B) = sin[(a + c)/2 + (a - c)/2]

+ sin[(a + c)/2 - (a - c)/2]

= sin(a) + sin(c);

2sin(A)cos(B) = 2sin[(a + c)/2] * cos[(a - c)/2]

由①得 sin(a) + sin(c) = 2sin[(a + c)/2] * cos[(a - c)/2]