Sum to Product Formulas

[MathDefinition title="Sum to Product Formulas"] Sum to product formulas convert the sum or difference of two trigonometric functions into a product of trigonometric functions. These are useful for simplifying expressions and solving equations. [/MathDefinition]

[MathKeyFormula title="Sum to Product Formulas"]

1. sin C + sin D = 2 sin((C+D)/2) cos((C−D)/2)
2. sin C − sin D = 2 cos((C+D)/2) sin((C−D)/2)
3. cos C + cos D = 2 cos((C+D)/2) cos((C−D)/2)
4. cos C − cos D = −2 sin((C+D)/2) sin((C−D)/2) [/MathKeyFormula]

[MathNote title="Product to Sum Formulas (Reverse)"]

1. 2 sin A cos B = sin(A+B) + sin(A−B)
2. 2 cos A sin B = sin(A+B) − sin(A−B)
3. 2 cos A cos B = cos(A+B) + cos(A−B)
4. 2 sin A sin B = cos(A−B) − cos(A+B) [/MathNote]

[MathExample title="Example 1: Converting Sum to Product"] Problem: Express sin 75° + sin 15° as a product.

Solution: Using sin C + sin D = 2 sin((C+D)/2) cos((C−D)/2): Here C = 75°, D = 15°

sin 75° + sin 15° = 2 sin((75+15)/2) cos((75−15)/2) = 2 sin(90/2) cos(60/2) = 2 sin 45° cos 30° = 2 × (1/√2) × (√3/2) = √3/√2 = √6/2

Therefore, sin 75° + sin 15° = √6/2 [/MathExample]

[MathExample title="Example 2: Proving Identity"] Problem: Prove that (sin 3A + sin A) / (cos 3A + cos A) = tan 2A.

Solution: LHS = (sin 3A + sin A) / (cos 3A + cos A)

Using sum to product formulas: sin 3A + sin A = 2 sin((3A+A)/2) cos((3A−A)/2) = 2 sin 2A cos A cos 3A + cos A = 2 cos((3A+A)/2) cos((3A−A)/2) = 2 cos 2A cos A

LHS = (2 sin 2A cos A) / (2 cos 2A cos A) = sin 2A / cos 2A = tan 2A = RHS

Hence proved. [/MathExample]

[MathExample title="Example 3: Evaluating Expression"] Problem: Find the value of cos 50° − cos 70°.

Solution: Using cos C − cos D = −2 sin((C+D)/2) sin((C−D)/2): Here C = 50°, D = 70°

cos 50° − cos 70° = −2 sin((50+70)/2) sin((50−70)/2) = −2 sin(120/2) sin(−20/2) = −2 sin 60° sin(−10°) = −2 sin 60° (−sin 10°) = 2 sin 60° sin 10° = 2 × (√3/2) × sin 10° = √3 sin 10°

Therefore, cos 50° − cos 70° = √3 sin 10° [/MathExample]

[MathExample title="Example 4: Simplifying Expression"] Problem: Simplify: sin 20° + sin 40° + sin 80°.

Solution: First, combine sin 20° + sin 80°: sin 20° + sin 80° = 2 sin((20+80)/2) cos((20−80)/2) = 2 sin 50° cos(−30°) = 2 sin 50° cos 30° = 2 sin 50° × (√3/2) = √3 sin 50°

Now add sin 40°: sin 20° + sin 40° + sin 80° = √3 sin 50° + sin 40° = √3 cos 40° + sin 40°

This can be written as: = 2((√3/2) cos 40° + (1/2) sin 40°) = 2(sin 60° cos 40° + cos 60° sin 40°) = 2 sin(60° + 40°) = 2 sin 100° = 2 cos 10°

Therefore, sin 20° + sin 40° + sin 80° = 2 cos 10° [/MathExample]