NOTES:
1)⇒ A + B + C = 180°
A + B + C = π
A + B = π - C
2)⇒⇒ sin (A + B) = sin (π - C)
= (sin π)(cos C) - (sin C)(cos π)
= (0)(cos C) - (sin C)(-1)
= 0 - (-sin C)
= sin C
3)⇒⇒⇒cos (A + B) = cos (π - c)
= (cos π)(cos C) + (sin π)(sin C)
= (-1)(cos C) + (0)(sin C)
= - cos C
4)⇒⇒⇒⇒ sin 2A + sin 2B = 2 sin (A + B) cos (A - B)
PROOF (from left side):
sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
2 sin (A + B) cos (A - B) + sin 2C refer to NOTE 4
2 sin (A + B) cos (A - B) + 2 sin C cos C double angle formula
2 sin C cos (A - B) + 2 sin C cos C refer to NOTE 2
2 sin C [cos (A - B) + cos C] factored out 2 sin C
2 sin C [cos (A - B) - (cos(A + B)] refer to NOTE 3
2 sin C [2 sin A sin B] sum/difference formula
4 sin A sin B sin C multiplied 2 sin C by 2 sin A sin B
Proof completed: 4 sin A sin B sin C = 4 sin A sin B sin C
