7.4 Sum-to-Product and Product-to-Sum Formulas - Precalculus 2e

Learning Objectives

In this section, you will:

Express products as sums.
Express sums as products.

7.4 Sum-to-Product and Product-to-Sum Formulas - Precalculus 2e | OpenStax (1)

Figure 1 The UCLA marching band (credit: Eric Chan, Flickr).

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.

7.4 Sum-to-Product and Product-to-Sum Formulas - Precalculus 2e | OpenStax (2)

Figure 2

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:

$\begin{array}{l} \underset{________________________________}{\begin{matrix} \cos α \cos β + \sin α \sin β = \cos (α - β) \\ + \cos α \cos β - \sin α \sin β = \cos (α + β) \end{matrix}} \\ 2 \cos α \cos β = \cos (α - β) + \cos (α + β) \end{array}$

Then, we divide by $2$ to isolate the product of cosines:

$\cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)]$

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

Write the following product of cosines as a sum: $2 \cos (\frac{7 x}{2}) \cos \frac{3 x}{2} .$

Solution

We begin by writing the formula for the product of cosines:

$\cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)]$

We can then substitute the given angles into the formula and simplify.

$\begin{array}{l} 2 \cos (\frac{7 x}{2}) \cos (\frac{3 x}{2}) = (2) (\frac{1}{2}) [\cos (\frac{7 x}{2} - \frac{3 x}{2}) + \cos (\frac{7 x}{2} + \frac{3 x}{2})] \\ = [\cos (\frac{4 x}{2}) + \cos (\frac{10 x}{2})] \\ = \cos 2 x + \cos 5 x \end{array}$

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:

$\begin{array}{l} \underset{_________________________________________}{\begin{array}{l} \begin{array}{l} \sin (α + β) = \sin α \cos β + \cos α \sin β \end{array} \\ + \sin (α - β) = \sin α \cos β - \cos α \sin β \end{array}} \\ \sin (α + β) + \sin (α - β) = 2 \sin α \cos β \end{array}$

Then, we divide by 2 to isolate the product of cosine and sine:

$\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)]$

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.

$\begin{array}{l} \sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin (4 θ) \cos (2 θ) = \frac{1}{2} [\sin (4 θ + 2 θ) + \sin (4 θ - 2 θ)] \\ = \frac{1}{2} [\sin (6 θ) + \sin (2 θ)] \end{array}$

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:

$\begin{array}{l} \underset{____________________________________________________}{\begin{array}{l} \begin{array}{l} \cos (α - β) = \cos α \cos β + \sin α \sin β \end{array} \\ - \cos (α + β) = - (\cos α \cos β - \sin α \sin β) \end{array}} \\ \cos (α - β) - \cos (α + β) = 2 \sin α \sin β \end{array}$

Then, we divide by 2 to isolate the product of sines:

$\sin α \sin β = \frac{1}{2} [\cos (α - β) - \cos (α + β)]$

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:

$\cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)]$

$\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)]$

$\sin α \sin β = \frac{1}{2} [\cos (α - β) - \cos (α + β)]$

$\cos α \sin β = \frac{1}{2} [\sin (α + β) - \sin (α - β)]$

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.

$\begin{array}{l} \cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)] \\ \cos (3 θ) \cos (5 θ) = \frac{1}{2} [\cos (3 θ - 5 θ) + \cos (3 θ + 5 θ)] \\ = \frac{1}{2} [\cos (2 θ) + \cos (8 θ)] & Useeven-oddidentity . \end{array}$

Try It #3

Use the product-to-sum formula to evaluate $\cos \frac{11 π}{12} \cos \frac{π}{12} .$

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. Let $\frac{u + v}{2} = α$ and $\frac{u - v}{2} = β .$

Then,

$\begin{array}{l} α + β = \frac{u + v}{2} + \frac{u - v}{2} \\ = \frac{2 u}{2} \\ = u \\ α - β = \frac{u + v}{2} - \frac{u - v}{2} \\ = \frac{2 v}{2} \\ = v \end{array}$

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

$\begin{array}{l} \sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin (\frac{u + v}{2}) \cos (\frac{u - v}{2}) = \frac{1}{2} [\sin u + \sin v] & Substitutefor (α + β) and (α - β) \\ 2 \sin (\frac{u + v}{2}) \cos (\frac{u - v}{2}) = \sin u + \sin v \end{array}$

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

Sum-to-Product Formulas

The sum-to-product formulas are as follows:

$\sin α + \sin β = 2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2})$

$\sin α - \sin β = 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2})$

$\cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2})$

$\cos α + \cos β = 2 \cos (\frac{α + β}{2}) \cos (\frac{α - β}{2})$

Example 4

Writing the Difference of Sines as a Product

Write the following difference of sines expression as a product: $\sin (4 θ) - \sin (2 θ) .$

Solution

We begin by writing the formula for the difference of sines.

$\sin α - \sin β = 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2})$

Substitute the values into the formula, and simplify.

$\begin{array}{l} \sin (4 θ) - \sin (2 θ) = 2 \sin (\frac{4 θ - 2 θ}{2}) \cos (\frac{4 θ + 2 θ}{2}) \\ = 2 \sin (\frac{2 θ}{2}) \cos (\frac{6 θ}{2}) \\ = 2 \sin θ \cos (3 θ) \end{array}$

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 $\cos (15^{\circ}) - \cos (75^{\circ}) .$

Solution

We begin by writing the formula for the difference of cosines.

$\cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2})$

Then we substitute the given angles and simplify.

$\begin{array}{l} \cos (15^{\circ}) - \cos (75^{\circ}) = - 2 \sin (\frac{15^{\circ} + 75^{\circ}}{2}) \sin (\frac{15^{\circ} - 75^{\circ}}{2}) \\ = - 2 \sin (45^{\circ}) \sin (- 30^{\circ}) \\ = - 2 (\frac{\sqrt{2}}{2}) (- \frac{1}{2}) \\ = \frac{\sqrt{2}}{2} \end{array}$

Example 6

Proving an Identity

Prove the identity:

$\frac{\cos (4 t) - \cos (2 t)}{\sin (4 t) + \sin (2 t)} = - \tan t$

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.

$\begin{array}{l} \frac{\cos (4 t) - \cos (2 t)}{\sin (4 t) + \sin (2 t)} = \frac{- 2 \sin (\frac{4 t + 2 t}{2}) \sin (\frac{4 t - 2 t}{2})}{2 \sin (\frac{4 t + 2 t}{2}) \cos (\frac{4 t - 2 t}{2})} \\ = \frac{- 2 \sin (3 t) \sin t}{2 \sin (3 t) \cos t} \\ = \frac{- 2 \sin (3 t) \sin t}{2 \sin (3 t) \cos t} \\ = - \frac{\sin t}{\cos t} \\ = - \tan t \end{array}$

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

Verify the identity $\csc^{2} θ - 2 = \frac{\cos (2 θ)}{\sin^{2} θ} .$

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.

$\begin{array}{l} \frac{\cos (2 θ)}{\sin^{2} θ} = \frac{1 - 2 \sin^{2} θ}{\sin^{2} θ} \\ = \frac{1}{\sin^{2} θ} - \frac{2 \sin^{2} θ}{\sin^{2} θ} \\ = \csc^{2} θ - 2 \end{array}$

Try It #5

Verify the identity $\tan θ \cot θ - \cos^{2} θ = \sin^{2} θ .$

Media

Access these online resources for additional instruction and practice with the product-to-sum and sum-to-product identities.

Sum to Product Identities
Sum to Product and Product to Sum Identities

7.4 Section Exercises

Verbal

Starting with the product to sum formula $\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)],$ explain how to determine the formula for $\cos α \sin β .$

Explain two different methods of calculating $\cos (195°) \cos (105°),$ one of which uses the product to sum. Which method is easier?

Explain a situation where we would convert an equation from a sum to a product and give an example.

Explain a situation where we would convert an equation from a product to a sum, and give an example.

Algebraic

For the following exercises, rewrite the product as a sum or difference.

$16 \sin (16 x) \sin (11 x)$

$20 \cos (36 t) \cos (6 t)$

$2 \sin (5 x) \cos (3 x)$

$10 \cos (5 x) \sin (10 x)$

$\sin (- x) \sin (5 x)$

10.

$\sin (3 x) \cos (5 x)$

For the following exercises, rewrite the sum or difference as a product.

11.

$\cos (6 t) + \cos (4 t)$

12.

$\sin (3 x) + \sin (7 x)$

13.

$\cos (7 x) + \cos (- 7 x)$

14.

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.

39.

$\cos (58^{\circ}) + \cos (12^{\circ})$

40.

$\sin (2^{\circ}) - \sin (3^{\circ})$

41.

$\cos (44^{\circ}) - \cos (22^{\circ})$

42.

$\cos (176^{\circ}) \sin (9^{\circ})$

43.

$\sin (- 14^{\circ}) \sin (85^{\circ})$

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.

44.

$2 \sin (2 x) \sin (3 x) = \cos x - \cos (5 x)$

45.

$\frac{\cos (10 θ) + \cos (6 θ)}{\cos (6 θ) - \cos (10 θ)} = \cot (2 θ) \cot (8 θ)$

46.

$\frac{\sin (3 x) - \sin (5 x)}{\cos (3 x) + \cos (5 x)} = \tan x$

47.

$2 \cos (2 x) \cos x + \sin (2 x) \sin x = 2 \sin x$

48.

$\frac{\sin (2 x) + \sin (4 x)}{\sin (2 x) - \sin (4 x)} = - \tan (3 x) \cot x$

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

49.

$\frac{\sin (9 t) - \sin (3 t)}{\cos (9 t) + \cos (3 t)}$

50.

$2 \sin (8 x) \cos (6 x) - \sin (2 x)$

51.

$\frac{\sin (3 x) - \sin x}{\sin x}$

52.

$\frac{\cos (5 x) + \cos (3 x)}{\sin (5 x) + \sin (3 x)}$

53.

$\sin x \cos (15 x) - \cos x \sin (15 x)$

Extensions

For the following exercises, prove the following sum-to-product formulas.

54.

$\sin x - \sin y = 2 \sin (\frac{x - y}{2}) \cos (\frac{x + y}{2})$

55.

$\cos x + \cos y = 2 \cos (\frac{x + y}{2}) \cos (\frac{x - y}{2})$

For the following exercises, prove the identity.

56.

$\frac{\sin (6 x) + \sin (4 x)}{\sin (6 x) - \sin (4 x)} = \tan (5 x) \cot x$

57.

$\frac{\cos (3 x) + \cos x}{\cos (3 x) - \cos x} = - \cot (2 x) \cot x$

58.

$\frac{\cos (6 y) + \cos (8 y)}{\sin (6 y) - \sin (4 y)} = \cot y \cos (7 y) \sec (5 y)$

59.

$\frac{\cos (2 y) - \cos (4 y)}{\sin (2 y) + \sin (4 y)} = \tan y$

60.

$\frac{\sin (10 x) - \sin (2 x)}{\cos (10 x) + \cos (2 x)} = \tan (4 x)$

61.

$\cos x - \cos (3 x) = 4 \sin^{2} x \cos x$

62.

${(\cos (2 x) - \cos (4 x))}^{2} + {(\sin (4 x) + \sin (2 x))}^{2} = 4 \sin^{2} (3 x)$

63.

$\tan (\frac{π}{4} - t) = \frac{1 - \tan t}{1 + \tan t}$

7.4 Sum-to-Product and Product-to-Sum Formulas - Precalculus 2e | OpenStax (2024)

FAQs

How do you remember sum to product and product to sum formulas? ›

For example, by changing the signs, you get cos(a−b)=cos(a)cos(b)+sin(a)sin(b). By summing, you have cos(a+b)+cos(a−b)=2cos(a)cos(b), which is your first formula. Similarly, by solving p=a+b and q=a−b, you get the formula cos(p)+cos(q)=2cos(p+q2)cos(p−q2).

Where do the product to sum and sum to product formulas come from? ›

From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine. We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines.

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What is the formula for sum and product? ›

Sum to product formulas is used to find expression for sum and difference of sines and cosines functions as products of sine and cosine functions. Sum to product formulas in trigonometry are: sin A + sin B = 2 sin [(A + B)/2] cos [(A – B)/2] sin A – sin B = 2 sin [(A – B)/2] cos [(A + B)/2]

Get More Info Here ›

What are the 45 formulas of trigonometry? ›

List of Trigonometry Formulas

sin²θ + cos²θ = 1.
tan²θ + 1 = sec²θ
cot²θ + 1 = cosec²θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ – sin²θ
tan 2θ = 2 tan θ / (1 – tan²θ)
cot 2θ = (cot²θ – 1) / 2 cot θ

How do you convert product of sum to sum of product? ›

If the boolean expression is AB+BC. Then the product of sum expression is (A+B)(B+C). To find this, the complement of sum product expression is the product of sum expression. (AB+BC)'=A'B'+B'C' By applying DEMORGANS LAW =(A'+B')(B'+C') Now, (A'+B')'(B'+C')'=(A+B)(B+C).

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How to memorize sum and difference formulas? ›

We can memorize all six sum and difference formulas by remembering only the sum formulas of the trigonometric functions. For difference formulas, we just need to interchange the signs '+' and '-'.

What is the product formula? ›

In mathematics, a product is a number or a quantity obtained by multiplying two or more numbers together. For example: 4 × 7 = 28 Here, the number 28 is called the product of 4 and 7. As another example, the product of 6 and 4 is 24, because 6 times 4 is 24.

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How to rewrite product as sum? ›

Answer and Explanation:

To rewrite a product, a × b, as a sum, we use repeated addition. Repeated addition is a technique used for multiplying two numbers together. It states the following: Repeated addition gives that the multiplication problem, a × b, is equivalent to adding a to itself b times.

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How do you use the SUMPRODUCT formula? ›

To create the formula using our sample list above, type =SUMPRODUCT(C2:C5,D2:D5) and press Enter. Each cell in column C is multiplied by its corresponding cell in the same row in column D, and the results are added up.

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How do you use the sum AND product method? ›

Find two numbers m and n whose product equals the value of a multiplied by c , and whose sum equals the value of b . Product =ac Sum =b.
Decompose the term bx in the trinomial using the two numbers found.
Factor by grouping.

Know More ›

What is the rule of sum AND product in problem solving? ›

If two events are not mutually exclusive (that is, we do them separately), then we apply the product rule. n1 · n2 ways of doing the overall procedure. If two events are mutually exclusive, that is, they cannot be done at the same time, then we must apply the sum rule.

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What are the 7 formulas of trigonometry? ›

By using a right-angled triangle as a reference, the trigonometric functions and identities are derived:

sin θ = Opposite Side/Hypotenuse.
cos θ = Adjacent Side/Hypotenuse.
tan θ = Opposite Side/Adjacent Side.
sec θ = Hypotenuse/Adjacent Side.
cosec θ = Hypotenuse/Opposite Side.
cot θ = Adjacent Side/Opposite Side.

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How hard is trigonometry? ›

The difficulty of college trigonometry can vary from person to person, depending on your previous experience with math and your general math aptitude. However, for most people, it tends to be manageable. Trigonometry primarily focuses on the relationships between angles and side lengths of triangles.

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How to remember trig formulas? ›

S from Some is for Sin, P from People is for Perpendicular and H from Have is for Hypotenuse, C from Curly is for Cos, B from Brown is for Base and H from Hair is for Hypotenuse and T from Turned is for Tan, P from Permanently is for Perpendicular and B from Black is for Base (reference image below).

Learn More Now ›

How to derive product to sum formula? ›

The product to sum identities can be derived from the sum and difference identities. To derive the first identity, add the sine sum identity and the sine difference identity. To derive the second identity, subtract the sine difference identity from the sine sum identity.

Get More Info Here ›

Is product of sum equal to sum of product? ›

Not so much the first from the second, but one can always write a product of sums as the sum of products just by distributing.

Read On ›

7.4 Sum-to-Product and Product-to-Sum Formulas - Precalculus 2e | OpenStax (2024)

Expressing Products as Sums

Expressing Products as Sums for Cosine

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

Solution

Expressing the Product of Sine and Cosine as a Sum

Writing the Product as a Sum Containing only Sine or Cosine

Solution

Expressing Products of Sines in Terms of Cosine

Express the Product as a Sum or Difference

Solution

Expressing Sums as Products

Writing the Difference of Sines as a Product

Solution

Evaluating Using the Sum-to-Product Formula

Solution

Proving an Identity

Solution

Analysis

Verifying the Identity Using Double-Angle Formulas and Reciprocal Identities

Solution

7.4 Section Exercises

Verbal

Algebraic

Numeric

Technology

Extensions

FAQs

How do you remember sum to product and product to sum formulas? ›

What is the rule of sum AND product in problem solving? ›

References