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 (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).

● Compound Angle

• Proof of Compound Angle Formula sin (α + β)
• Proof of Compound Angle Formula sin (α - β)
• Proof of Compound Angle Formula cos (α + β)
• Proof of Compound Angle Formula cos (α - β)
• Proof of Compound Angle Formula sin 22 α - sin 22 β
• Proof of Compound Angle Formula cos 22 α - sin 22 β
• Proof of Tangent Formula tan (α + β)
• Proof of Tangent Formula tan (α - β)
• Proof of Cotangent Formula cot (α + β)
• Proof of Cotangent Formula cot (α - β)
• Expansion of sin (A + B + C)
• Expansion of sin (A - B + C)
• Expansion of cos (A + B + C)
• Expansion of tan (A + B + C)
• Compound Angle Formulae
• Problems using Compound Angle Formulae
• Problems on Compound Angles

11 and 12 Grade Math

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