## Product To Sum Formulas

The product to sum formulas are used to express the product of sine and cosine functions as a sum. These are derived from the sum and difference formulas of trigonometry. These formulas are very helpful while solving the integrals of trigonometric functions. Let us learn the product to sum formulas along with proofs and examples.

## What Are Product To Sum Formulas?

The product-to-sum formulas are a set of formulas from trigonometric formulas and as we discussed in the previous section, they are derived from the sum and difference formulas. Here are the product to sum formulas and you can see their derivation below the formulas.

### Product To Sum Formulas

There are 4 product to sum formulas that are widely used as trigonometric identities.

- 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

We will use the sum/difference formulas of trigonometry to derive the product to sum formulas. Let us recall the sum and difference formulas of sin and cos and give some numbers to each of the sum/difference formulas.

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) ]

ou can see the applications of product to sum formulas in the section below.

## Examples on Product To Sum Formulas

**Example 1: **Find the value of sin 75^{o} sin 15^{o} without actually evaluating the values of sin 75^{o} and sin 15^{o}.

**Solution:**

Using one of the product to sum formulas,

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

Substitute A = 75^{o} and B = 15^{o}, we get

sin 75^{o} sin 15^{o} = (1/2) [ cos (75^{o} – 15^{o}) – cos (75^{o} + 15^{o}) ]

= (1/2) [ cos 60^{o} – cos 90^{o}]

= (1/2) [ (1/2) – 0] (from trigonometry table)

= 1/4

**Answer: **sin 75^{o} sin 15^{o} = 1/4.

**Example 2: **Express 2 cos 5x sin 2x as sum/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 above 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 ∫ 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 on both sides,

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

Now, we will evaluate the given integral using the above 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.

## FAQs on Product To Sum Formulas

### What Are Product To Sum Formulas?

The product to sum formulas in trigonometry are formulas that are used to convert the product of trigonometric functions into the sum of trigonometric functions. There are 4 important product to sum formulas.

- 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) ]

### How To Derive Product To Sum Formulas?

The product to sum formulas are derived using the sum and difference formulas which are:

- 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

By adding or subtracting two of these four formulas, we can derive the product to sum formulas. You can find the detailed proof/derivation of product to sum formulas from the “What Are Product To Sum Formulas?” section of this page.

### What Are the Applications of Product To Sum Formulas?

The product to sum formulas are used to write the product of two trigonometric functions (sin and cos) as the sum. These formulas are hence useful in integration as integrating a sum is pretty easier when compared to integrating a product.

### How To Use Product To Sum Formulas?

Let us recall the product to sum formulas:

- 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) ]

We can use one of these formulas to find the product of sin and (or) cos as the sum. For example if we have to convert cos 15^{o} sin 45^{o} into the sum, we will just apply the formula 2 from the above list. Then we get:

cos 15^{o} sin 45^{o} = (1/2) [ sin (15^{o} + 45^{o}) – sin (15^{o} – 45^{o}) ]

= (1/2) [ sin 60^{o} + sin 30^{o}] [because sin (-x) = – sin x]

= (1/2) [ (√3/2) + 1/2] (from trigonometry table)

= (√3 + 1) / 4

### Sum-to-Product and Product-to-Sum Formulas

### Learning Objectives

In this section, you will:

- Express products as sums.
- Express sums as products.

A band marches down the field creating an amazing sound that bolsters the crowd. That sound travels as a wave that can be interpreted using trigonometric functions. For example, Figure 2 represents a sound wave for the musical note A. In this section, we will investigate trigonometric identities that are the foundation of everyday phenomena such as sound waves.

### Expressing Products as Sums

We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the product-to-sum formulas, which express products of trigonometric functions as sums. Let’s investigate the cosine identity first and then the sine identity.

#### Expressing Products as Sums for Cosine

We can derive the product-to-sum formula from the sum and difference identities for cosine. If we add the two equations, we get:

### How To

**Given a product of cosines, express as a sum.**

- Write the formula for the product of cosines.
- Substitute the given angles into the formula.
- Simplify.

**Example 1**

#### Writing the Product as a Sum Using the Product-to-Sum Formula for Cosine

### Try It #1

Use the product-to-sum formula to write the product as a sum or difference: cos(2*θ*)cos(4*θ*).

#### Expressing the Product of Sine and Cosine as a Sum

Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for sine. If we add the sum and difference identities, we get:

Example 2

#### Writing the Product as a Sum Containing only Sine or Cosine

Express the following product as a sum containing only sine or cosine and no products: sin(4*θ*)cos(2*θ*).

#### Solution

Write the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify.

Try It #2

Use the product-to-sum formula to write the product as a sum: sin(*x*+*y*)cos(*x*−*y*).

#### Expressing Products of Sines in Terms of Cosine

Expressing the product of sines in terms of cosine is also derived from the sum and difference identities for cosine. In this case, we will first subtract the two cosine formulas:

Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.

### The Product-to-Sum Formulas

The product-to-sum formulas are as follows:

Example 3

#### Express the Product as a Sum or Difference

Write cos(3*θ*)cos(5*θ*) as a sum or difference.

#### Solution

We have the product of cosines, so we begin by writing the related formula. Then we substitute the given angles and simplify.

### Expressing Sums as Products

Some problems require the reverse of the process we just used. The sum-to-product formulas allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for sine.

Thus, replacing *α* and *β* in the product-to-sum formula with the substitute expressions, we have

The other sum-to-product identities are derived similarly.

### Sum-to-Product Formulas

The sum-to-product formulas are as follows:

Example 4

#### Writing the Difference of Sines as a Product

Write the following difference of sines expression as a product: sin(4*θ*)−sin(2*θ*).

Try It #4

Use the sum-to-product formula to write the sum as a product: sin(3*θ*)+sin(*θ*).

Example 5

#### Evaluating Using the Sum-to-Product Formula

Evaluate cos(15∘)−cos(75∘).

Example 6

#### Proving an Identity

Prove the identity:

Solution

We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.

#### Analysis

Recall that verifying trigonometric identities has its own set of rules. The procedures for solving an equation are not the same as the procedures for verifying an identity. When we prove an identity, we pick one side to work on and make substitutions until that side is transformed into the other side.

Example 7

#### Verifying the Identity Using Double-Angle Formulas and Reciprocal Identities

Solution

For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.

Try It #5

Verify the identity tan*θ*cot*θ*−cos2*θ*=sin2*θ*.

### Section Exercises

#### Verbal

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

#### Numeric

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.

#### Technology

For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

## Để lại một phản hồi