Product to Sum Formula, Derivation and Solved Examples

Product to Sum Formulas, also known as trigonometric identities, are mathematical equations used to express the product of trigonometric functions as sums or differences of trigonometric functions. These formulas are often used to simplify trigonometric expressions or equations. Here are some common product-to-sum formulas:

What Are Product To Sum Formulas?

The product-to-sum formulas are a group of trigonometric identities that are derived from the sum and difference formulas for trigonometric functions. Below, you will find these product-to-sum formulas, along with their derivations explained.

Product To Sum Formulas

There are four widely-used product-to-sum formulas in trigonometry that help simplify trigonometric expressions.

• 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 Formulas Derivation

The sum and difference formulas for sine and cosine and assign some numbers to each of them:

sin (A + B) = sin A cos B + cos A sin B ... (1)

sin (A - B) = sin A cos B - cos A sin B ... (2)

cos (A + B) = cos A cos B - sin A sin B ... (3)

cos (A - B) = cos A cos B + sin A sin B ... (4)

Deriving the formula sin A cos B = (1/2) [ sin (A + B) + sin (A - B) ]:

Adding the equations (1) and (2), we get

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

Dividing both sides by 2,

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

Deriving the formula cos A sin B = (1/2) [ sin (A + B) - sin (A - B) ]:

Subtracting (2) from (1),

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

Dividing both sides by 2,

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

Deriving the formula cos A cos B = (1/2) [ cos (A + B) + cos (A - B) ]

Adding the equations (3) and (4), we get

cos (A + B) + cos (A - B) = 2 cos A cos B

Dividing both sides by 2,

cos A cos B = (1/2) [ cos (A + B) + cos (A - B) ]

Deriving the formula sin A sin B = (1/2) [ cos (A - B) - cos (A + B) ]

Subtracting (3) from (4),

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

Dividing both sides by 2,

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

You can see the applications of products to sum formulas in the section below.

Product To Sum Formulas Examples

Example 1: Find the value of sin 75 degrees sin 15 degrees without directly calculating sin 75 degrees and sin 15 degrees.

Solution:

Using one of the product-to-sum formulas:

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

Let A = 75 degrees and B = 15 degrees:

sin 75 degrees sin 15 degrees = (1/2) [cos(75 degrees - 15 degrees) - cos(75 degrees + 15 degrees)]

= (1/2) [cos 60 degrees - cos 90 degrees]

= (1/2) [(1/2) - 0] (from trigonometry tables)

= 1/4

Answer: sin 75 degrees sin 15 degrees = 1/4.

Example 2: Express 2 cos 5x sin 2x as a sum or difference.

Solution:

Using one of the product-to-sum formulas:

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

Substitute A = 5x and B = 2x in the formula:

cos 5x sin 2x = (1/2) [sin(5x + 2x) - sin(5x - 2x)]

cos 5x sin 2x = (1/2) [sin 7x - sin 3x]

Multiply both sides by 2:

2 cos 5x sin 2x = sin 7x - sin 3x

Answer: 2 cos 5x sin 2x = sin 7x - sin 3x.

Example 3: Find the value of the integral of sin 3x cos 4x dx.

Solution:

Using one of the product-to-sum formulas:

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

Substitute A = 3x and B = 4x:

sin 3x cos 4x = (1/2) [sin(3x + 4x) + sin(3x - 4x)]

sin 3x cos 4x = (1/2) [sin 7x - sin x] (because sin(-x) = -sin x)

Now, evaluate the given integral using this value:

∫ sin 3x cos 4x dx = ∫ (1/2) [sin 7x - sin x] dx

= (1/2) [-cos(7x) / 7 + cos x] + C (using integration by substitution)

Answer: ∫ sin 3x cos 4x dx = (1/2) [-cos(7x) / 7 + cos x] + C.

Product to Sum Formula Applications

Product-to-sum formulas in trigonometry are used to transform products of trigonometric functions into sums or differences of trigonometric functions. They have various applications in mathematics, physics, engineering, and other fields. Here are some common applications:

• Trigonometric Simplification: Product-to-sum formulas are frequently used to simplify complex trigonometric expressions. By converting products into sums or differences, you can make equations or expressions more manageable and easier to work with.
• Integration and Differentiation: When solving integrals or differentiating trigonometric functions, product-to-sum formulas can be applied to simplify the expressions, making it easier to evaluate integrals or find derivatives.
• Trigonometric Identities: Product-to-sum formulas are essential for deriving and proving trigonometric identities. They help establish relationships between different trigonometric functions and are a fundamental tool in trigonometric proofs.
• Waveform Analysis: In physics and electrical engineering, product-to-sum formulas are used to analyze waveforms. Converting products of sinusoidal functions into sums or differences simplifies the analysis of wave interference, resonance, and other wave-related phenomena.
• Signal Processing: In signal processing, product-to-sum formulas help manipulate and analyze signals more effectively. They are often used in applications such as filtering, modulation, and spectral analysis.
• Fourier Series: In mathematics and engineering, Fourier series decomposition of periodic functions involves converting complex trigonometric products into sums of simpler trigonometric functions. Product-to-sum formulas are used in this context.
• Navigation and Astronomy: Navigational calculations and celestial observations frequently involve spherical trigonometry. Product-to-sum formulas help simplify calculations related to the angles and positions of celestial objects.