<u>NOTES:</u>
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)
<u>PROOF </u><em><u>(from left side)</u></em><u>:</u>
sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
2 sin (A + B) cos (A - B) + sin 2C <em>refer to NOTE 4</em>
2 sin (A + B) cos (A - B) + 2 sin C cos C <em>double angle formula</em>
2 sin C cos (A - B) + 2 sin C cos C <em>refer to NOTE 2</em>
2 sin C [cos (A - B) + cos C] <em>factored out 2 sin C</em>
2 sin C [cos (A - B) - (cos(A + B)<u>]</u> <em>refer to NOTE 3</em>
2 sin C <u>[</u>2 sin A sin B<u>]</u> <em>sum/difference formula</em>
4 sin A sin B sin C <em>multiplied 2 sin C by 2 sin A sin B</em>
<em>Proof completed:</em> 4 sin A sin B sin C = 4 sin A sin B sin C
