Sample Problems on Product to Sum Formulas

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]

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.