# Expansion of sin (A - B + C)

We will learn how to find the expansion of sin (A - B + C). By using the formula of sin (A + B), sin (A - B) and cos (A - B) we can easily expand sin (A - B + C).

Let us recall the formula of sin (α + β) = sin α cos β + cos α sin β, sin (α - β) = sin α cos β - cos α sin β and cos (α - β) = cos α cos β + sin α sin β.

sin (A - B + C) = sin [( A - B) + C]

= sin (A - B) cos C + cos (A - B) sin C, [applying the formula of sin (α + β)]

= (sin A cos B - cos A sin B) cos C + (cos A cos B + sin A sin B) sin C, [applying the formula of sin (α - β) and cos (α - β)]

= sin A cos B cos C - sin B cos C cos A + sin C cos A cos B + sin A sin B sin C, [applying distributive property]

= sin A cos B cos C - cos A sin B cos C + cos A cos B sin C + sin A sin B sin C

Therefore, the expansion of sin (A - B + C) = sin A cos B cos C - cos A sin B cos C + cos A cos B sin C + sin A sin B sin C.

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