Product to Sum Formulas

Product-to-sum formulas are trigonometric identities that convert the product of sine and cosine functions into a sum (or difference) of trigonometric functions. These formulas are particularly useful in simplifying the integrals and solving trigonometric equations.

In this article, we will learn about, Product to Sum Formulae, related examples and others in detail.

Product-to-sum formulas provide a powerful tool for simplifying trigonometric expressions involving products of sines and cosines, and the product to sum formulas are:

• sin A cos B = (1/2) [sin (A + B) + sin (A – B)]
• cos A sin B = (1/2) [sin (A + B) – sin (A – B)]
• cos A cos B = (1/2) [cos (A + B) + cos (A – B)]
• sin A sin B = (1/2) [cos (A – B) – cos (A + B)]

Product-to-sum identities are used to express the product between sine and/or cosine functions as a sum or difference. The sum and difference formulas of sine and cosine functions are added or subtracted to derive these identities. The product-to-sum identities can be used to simplify the trigonometric expression.

The integrals or derivatives of the trigonometric functions can be solved with ease by using these identities. For all possible sine and cosine product combinations, there are four products to sum or difference formulae in total.

Product to Sum Identities

The product to sum formulae can be derived using the trigonometric sum/difference formulae. 

The Sum/difference formulae are given below:

• sin (A + B) = (½) {sin A cos B + cos A sin B}———— (1)
• sin (A – B) = (½) {sin A cos B – cos A sin B}———— (2)
• cos (A + B) = (½) {cos A cos B – sin A sin B}———— (3)
• cos (A – B) = (½) {cos A cos B + sin A sin B}———— (4)

cos A cos B Formula (Product of Cosines)

To derive the cos A cos B formula add equations (3) and (4)

⇒ cos (A + B) + cos (A – B) = [cos A cos B – sin A sin B] + [cos A cos B + sin A sin B]

⇒ cos (A + B) + cos (A – B)  = cos A cos B + cos A cos B

⇒ cos (A + B) + cos (A – B) = 2 cos A cos B

Hence, 

  cos A cos B =  (½) [cos (A + B) + cos (A – B)]

cos A sin B Formula (Product of Sine and Cosine)

To derive the cos A sin B formula subtract equations (2) from (1)

⇒ sin (A + B) – sin (A – B) = [sin A cos B + cos A sin B] – [sin A cos B – cos A sin B]

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

⇒ sin (A + B) – sin (A – B) = 2 cos A sin B

Hence,

cos A sin B = (½) [sin (A + B) – sin (A – B)]

sin A cos B Formula (Product of Sine and Cosine)

To derive the sin A cos B formula add equations (1) and (2)

⇒ sin (A + B) + sin (A – B) = [sin A cos B + cos A sin B] + [sin A cos B – cos A sin B]

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

⇒ sin (A + B) + sin (A – B) = 2 sin A cos B

Hence,

sin A cos B = (½) [sin (A + B) + sin (A – B)]

sin A sin B Formula (Product of Sines)

To derive the sin A sin B formula subtract equations (4) from (3)

⇒ cos (A – B) – cos (A + B) = [cos A cos B + sin A sin B] – [cos A cos B – sin A sin B]

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

⇒ cos (A – B) – cos (A + B) = 2 sin A sin B

Hence,

sin A sin B = (½) [cos (A – B) – cos (A + B)]

Product to sum formulas are useful tools in trigonometry that make it easier to work with sine and cosine functions. By converting products into sums or differences, these formulas help simplify complex trigonometric problems. Learning and using these formulas can help you solve a wide range of math problems more easily.

Read More,

• Trigonometric in Maths
• Trigonometric Identities
• Inverse Trigonometric Identities

Problem 1: Express 6 cos 8x sin 5x as sum/difference.

Solution:

From one of the product to sum formulas, we have

cos A sin B = (½) [sin (A + B) – sin (A – B)]

So, by substituting A = 8x and B = 5x in the above formula, we get

cos 8x sin 5x = (½) [ sin (8x + 5x) – sin (8x – 5x) ]

cos 8x sin 5x = (½) [sin 13x – sin 3x]

Now, 6 cos 8x sin 5x = 6 × (½) [sin 13x – sin 3x]

Hence, 6 cos 8x sin 5x = 3 [sin 13x – sin 3x]

Problem 2: Determine the value of the integral of cos 4x cos 6x.

Solution:

From one of the product to sum formulas, we have

cos A cos B = (½) [cos (A + B) + cos (A – B)]

cos 4x cos 6x = ½ [cos (4x + 6x) + cos (4x – 6x)]

= (½) [cos 10x + cos (-2x)]

= (½) [cos 10x + cos 2x]       {Since, cos (-θ) = cos θ}

Now, integral of cos 4x cos 6x = ∫ cos 4x cos 6x dx

= ∫(½) [cos 10x + cos 2x] dx

= (½) [1/10 sin 10x + 1/2 sin 2x] + C   {Since, ∫cos(ax) dx = 1/a sin(ax) + c}

= 1/20 sin(10x) + 1/2 sin(2x) + C

Hence, integral of cos 4x cos 6x = 1/20 sin(10x) + 1/2 sin(2x) + C

Problem 3: Determine the value of sin 36° cos 54° without evaluating the sin 36° and cos 54° values. 

Solution:

From one of the product to sum formulas, we have

sin A cos B = (½) [sin (A + B) + sin (A – B)]

So, sin 36° cos 54° = (½) [sin (36° + 54°) + sin (36° – 54°)]

= (½) [sin (90°) + sin (-18°)]

= (½) [sin 90° – sin 18°]       {Since, sin (-θ) = – sin θ}

= (½) [1 – 0.3090]                {Since, sin 90° = 1, sin 18° = 0.3090}

= 0.3455

Hence, sin 36° cos 54° = 0.3455.

Problem 4: Determine the value of the derivative of 4 cos 3x sin 2x.

Solution:

From one of the product to sum formulas, we have

cos A sin B = (½) [sin (A + B) – sin (A – B)]

Now, 4 cos 3x sin 2x = 4 × (½) [sin (3x + 2x) – sin (3x – 2x)]

= 2 [sin 5x – sin x]

Now, derivative of 4 cos 3x sin 2x = d(4 cos 3x sin 2x)/dx

= d/(2 [sin 5x – sin x])/dx

= 2 [ d(sin 5x)/dx – d(sin x)/dx ]

= 2 [5 cos 5x – cos x]                 {Since, d(sin ax)/dx = a cos ax}

Hence, derivative of 4 cos 3x sin 2x = 2 [5 cos 5x – cos x] .

Problem 5: Determine the value of sin 15° sin 45° without evaluating the sin 15° and sin 45° values. 

Solution:

From one of the product to sum formulas, we have

sin A sin B = (½) [cos (A – B) – cos (A + B)]

Now, sin 15° sin 45° = (½)[cos (15° – 45°) – cos (15° + 45°)]

= (½) [cos (-30°) – cos (60° )]

= (½) [cos 30° – cos 60°]   {Since, cos (-θ) = cos θ}

= (½) [√3/2 – 1/2]            {Since, cos 30° = √3/2 and cos 60° = 1/2}

= (½) [(√3 -1)/2]

= (√3 -1)/4

Hence, sin 15° sin 45° = (√3 -1)/4.

Problem 6: Express 2 cos 9x cos 7x as sum/difference.

Solution:

From one of the product to sum formulas, we have

cos A cos B = (½) [cos (A + B) + cos (A – B)]

Now, 2 cos 9x cos 7x = 2 × (½) [cos (9x + 7x) + cos (9x – 7x)]

= [cos (16x) + cos (2x)]

Hence, 2 cos 9x cos 7x = [cos 16x + cos 2x]

What are Product to Sum Formulas?

Product to sum formulas are trigonometric identities that convert product of sine and cosine functions into a sum or difference of trigonometric functions.

What is the formula for sin 3x?

Formula for sin 3x is, sin3x = 3sin x – 4 sin³x.

What is the formula for cos 3x?

Formula for cos 3x is, cos3x = 4 cos³x – 3 cos x.