Question

formulla of sin(A+B+C)​

Answer 1

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).