Compound Angle Formulae

The important trigonometrical ratios of compound angle formulae are given below:

sin (A + B) = sin A cos B + cos A sin B

sin (A - B) = sin A cos B - cos A sin B

cos (A + B) = cos A cos B - sin A sin B

cos (A - B) = cos A cos B + sin A sin B

sin ( A + B) sin (A - B) = sin^2 A – sin^2 B = cos^2 B – cos^2 A

cos ( A + B) cos (A - B) = cos^2 A – sin^2 B = cos^2 B – sin^2 A

tan (A + B) = tan A + tan B/1 - tan A tan B

tan (A - B) = tan A - tan B/1 + tan A tan B

cot (A + B) = cot A cot B - 1/cot B + cot A

cot (A - B) = cot A + cot B + 1/cot B - cot A

tan (A + B + C) = tan A + tan B + tan C - tan A tan B tan C/1- tan A tan B - tan B tan C - tan C tan A.

Now we will learn how to use the above formulae for solving different types of trigonometric problems on compound angle.

1. Using the formula of cos (A - B) = cos A cos B + sin A sin B prove that cos (π/2 - x) = sin x, for all real numbers x.

Proof:

cos (π/2 - x) = cos π/2 cos x + sin π/2 sin x, [Applying the formula of cos (A - B) = cos A cos B + sin A sin B]

= 0 × cos x + 1 × sin x, [Since we know that cos π/2 = 0 and sin π/2 = 1]

= 0 + sin x

= sin x                        Proved

Therefore cos (π/2 - x) = sin x.

 

2. Using the formula of cos (A + B) = cos A cos B - sin A sin B prove that cos (π/2 + x) = - sin x, for all real numbers x.

Proof:

cos (π/2 + x) = cos π/2 cos x - sin π/2 sin x, [Applying the formula of cos (A + B) = cos A cos B - sin A sin B]

= 0 × cos x - 1 × sin x, [Since we know that cos π/2 = 0 and sin π/2 = 1]

= 0 - sin x

= - sin x                        Proved

Therefore cos (π/2 + x) = - sin x

 

3. Using the formula of sin (A + B) = sin A cos B + cos A sin B prove that sin (π/2 + x) = cos x, for all real numbers x.

Proof:

sin (π/2 + x) = sin π/2 cos x + cos π/2 sin x, [Applying the formula of sin (A + B) = sin A cos B + cos A sin B]

= 1 × cos x + 0 × sin x, [Since we know that sin π/2 = 1 and cos π/2 = 0]

= cos x + 0

= cos x                        Proved

Therefore sin (π/2 + x) = cos x

 

4. Using the formula of sin (A - B) = sin A cos B - cos A sin B prove that sin (π/2 - x) = cos x, for all real numbers x.

Proof:

sin (π/2 - x) = sin π/2 cos x - cos π/2 sin x, [Applying the formula of sin (A - B) = sin A cos B - cos A sin B]

= 1 × cos x - 0 × sin x, [Since we know that sin π/2 = 1 and cos π/2 = 0]

= cos x - 0

= cos x                        Proved

Therefore sin (π/2 - x) = cos x

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● 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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