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We will learn how to find the expansion of sin (A + B + C). By using the formula of sin (α + β) and cos (α + β) 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 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]
= cos A cos B cos C (tan A + tan B + tan C - tan A tan B tan C)
Therefore, the expansion of sin (A + B + C) = cos A cos B cos C (tan A + tan B + tan C - tan A tan B tan C).
Let us recall the formula of 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]
= cos A cos B cos C (tan A + tan B + tan C - tan A tan B tan C)
Therefore, the expansion of sin (A + B + C) = cos A cos B cos C (tan A + tan B + tan C - tan A tan B tan C).
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