Hello from MrBillDoesMath!
Answer:
See Discussion below
Discussion:
As a,b,c are interior angles of a triangle
a + b + c = 180 =>
b + c = 180 -a =>
(b +c)/2 = (1/2) (180 -a) =>
(b+c)/2 = (1/2) 180 * (1/2) (-a) =>
(b+c)/2 = 90 - a/2 ( Exhibit A)
In general, for any angle @
sin (90 - @ ) = sin(90) cos(@) - cos(90)sin(@)
= 1 * cos(@) - 0 as cos(90) = 0, sin(90) = 1
= cos(@) (exhibit B)
So
sin( (b+c)/2) = sin(90 - a/2) from Exhibit A
= cos (a/2)
where, in the final step, we let @ = a/2 in Exhibit B
Thank you,
MrB
