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Algebra and Trigonometry

9.4 Sum-to-Product and Product-to-Sum Formulas

Algebra and Trigonometry9.4 Sum-to-Product and Product-to-Sum Formulas
  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Learning Objectives

In this section, you will:
  • Express products as sums.
  • Express sums as products.
Photo of the UCLA marching band.
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.

Graph of a sound wave for the musical note A - it is a periodic function much like sin and cos - from 0 to .01
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:

cosαcosβ+sinαsinβ = cos(αβ) +cosαcosβsinαsinβ = cos(α+β) ___________________________________ 2cosαcosβ = cos(αβ)+cos(α+β) cosαcosβ+sinαsinβ = cos(αβ) +cosαcosβsinαsinβ = cos(α+β) ___________________________________ 2cosαcosβ = cos(αβ)+cos(α+β)

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

cosαcosβ= 1 2 [cos(αβ)+cos(α+β)] cosαcosβ= 1 2 [cos(αβ)+cos(α+β)]

How To

Given a product of cosines, express as a sum.

  1. Write the formula for the product of cosines.
  2. Substitute the given angles into the formula.
  3. 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: 2cos( 7x 2 )cos 3x 2 . 2cos( 7x 2 )cos 3x 2 .

Try It #1

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

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:

 sin(α+β) = sinαcosβ+cosαsinβ + sin(αβ) = sinαcosβcosαsinβ _________________________________________ sin(α+β)+sin(αβ) = 2sinαcosβ  sin(α+β) = sinαcosβ+cosαsinβ + sin(αβ) = sinαcosβcosαsinβ _________________________________________ sin(α+β)+sin(αβ) = 2sinαcosβ

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

sinαcosβ= 1 2 [ sin( α+β )+sin( αβ ) ] sinαcosβ= 1 2 [ sin( α+β )+sin( αβ ) ]

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( 4θ )cos( 2θ ). sin( 4θ )cos( 2θ ).

Try It #2

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

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:

                    cos( αβ )=cosαcosβ+sinαsinβ                  cos( α+β )=( cosαcosβsinαsinβ ) ____________________________________________________ cos( αβ )cos( α+β )=2sinαsinβ                     cos( αβ )=cosαcosβ+sinαsinβ                  cos( α+β )=( cosαcosβsinαsinβ ) ____________________________________________________ cos( αβ )cos( α+β )=2sinαsinβ

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

sinαsinβ= 1 2 [ cos( αβ )cos( α+β ) ] sinαsinβ= 1 2 [ cos( αβ )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:

cosαcosβ = 1 2 [cos(αβ)+cos(α+β)] cosαcosβ = 1 2 [cos(αβ)+cos(α+β)]
sinαcosβ = 1 2 [sin(α+β)+sin(αβ)] sinαcosβ = 1 2 [sin(α+β)+sin(αβ)]
sinαsinβ = 1 2 [cos(αβ)cos(α+β)] sinαsinβ = 1 2 [cos(αβ)cos(α+β)]
cosαsinβ = 1 2 [sin(α+β)sin(αβ)] cosαsinβ = 1 2 [sin(α+β)sin(αβ)]

Example 3

Express the Product as a Sum or Difference

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

Try It #3

Use the product-to-sum formula to evaluate cos 11π 12 cos π 12 . cos 11π 12 cos 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 u+v 2 =α u+v 2 =α and uv 2 =β. uv 2 =β.

Then,

α+β = u+v 2 + uv 2 = 2u 2 = u αβ = u+v 2 uv 2 = 2v 2 = v α+β = u+v 2 + uv 2 = 2u 2 = u αβ = u+v 2 uv 2 = 2v 2 = v

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

sinαcosβ = 1 2 [sin(α+β)+sin(αβ)] sin( u+v 2 )cos( uv 2 ) = 1 2 [sinu+sinv] Substitute for(α+β) and (αβ) 2sin( u+v 2 )cos( uv 2 ) = sinu+sinv sinαcosβ = 1 2 [sin(α+β)+sin(αβ)] sin( u+v 2 )cos( uv 2 ) = 1 2 [sinu+sinv] Substitute for(α+β) and (αβ) 2sin( u+v 2 )cos( uv 2 ) = sinu+sinv

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

Sum-to-Product Formulas

The sum-to-product formulas are as follows:

sinα+sinβ = 2sin( α+β 2 )cos( αβ 2 ) sinα+sinβ = 2sin( α+β 2 )cos( αβ 2 )
sinαsinβ = 2sin( αβ 2 )cos( α+β 2 ) sinαsinβ = 2sin( αβ 2 )cos( α+β 2 )
cosαcosβ = −2sin( α+β 2 )sin( αβ 2 ) cosαcosβ = −2sin( α+β 2 )sin( αβ 2 )
cosα+cosβ = 2cos( α+β 2 )cos( αβ 2 ) cosα+cosβ = 2cos( α+β 2 )cos( αβ 2 )

Example 4

Writing the Difference of Sines as a Product

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

Try It #4

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

Example 5

Evaluating Using the Sum-to-Product Formula

Evaluate cos(15°)cos(75°). cos(15°)cos(75°). Check the answer with a graphing calculator.

Example 6

Proving an Identity

Prove the identity:

cos( 4t )cos( 2t ) sin( 4t )+sin( 2t ) =tant cos( 4t )cos( 2t ) sin( 4t )+sin( 2t ) =tant

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 θ2= cos(2θ) sin 2 θ . csc 2 θ2= cos(2θ) sin 2 θ .

Try It #5

Verify the identity tanθcotθ cos 2 θ= sin 2 θ. tanθcotθ 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.

9.4 Section Exercises

Verbal

1.

Starting with the product to sum formula sinαcosβ= 1 2 [sin(α+β)+sin(αβ)], sinαcosβ= 1 2 [sin(α+β)+sin(αβ)], explain how to determine the formula for cosαsinβ. cosαsinβ.

2.

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

3.

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

4.

Describe 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.

5.

16sin(16x)sin(11x) 16sin(16x)sin(11x)

6.

20cos( 36t )cos( 6t ) 20cos( 36t )cos( 6t )

7.

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

8.

10cos( 5x )sin( 10x ) 10cos( 5x )sin( 10x )

9.

sin( x )sin( 5x ) sin( x )sin( 5x )

10.

sin( 3x )cos( 5x ) sin( 3x )cos( 5x )

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

11.

cos( 6t )+cos( 4t ) cos( 6t )+cos( 4t )

12.

sin( 3x )+sin( 7x ) sin( 3x )+sin( 7x )

13.

cos( 7x )+cos( 7x ) cos( 7x )+cos( 7x )

14.

sin( 3x )sin( 3x ) sin( 3x )sin( 3x )

15.

cos( 3x )+cos( 9x ) cos( 3x )+cos( 9x )

16.

sinhsin( 3h ) sinhsin( 3h )

For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

17.

cos( 45° )cos( 15° ) cos( 45° )cos( 15° )

18.

cos( 45° )sin( 15° ) cos( 45° )sin( 15° )

19.

sin( −345° )sin( −15° ) sin( −345° )sin( −15° )

20.

sin( 195° )cos( 15° ) sin( 195° )cos( 15° )

21.

sin( −45° )sin( −15° ) sin( −45° )sin( −15° )

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

22.

cos( 23° )sin( 17° ) cos( 23° )sin( 17° )

23.

2sin( 100° )sin( 20° ) 2sin( 100° )sin( 20° )

24.

2sin(−100°)sin(−20°) 2sin(−100°)sin(−20°)

25.

sin( 213° )cos( ) sin( 213° )cos( )

26.

2cos(56°)cos(47°) 2cos(56°)cos(47°)

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

27.

sin(76°)+sin(14°) sin(76°)+sin(14°)

28.

cos( 58° )cos( 12° ) cos( 58° )cos( 12° )

29.

sin(101°)sin(32°) sin(101°)sin(32°)

30.

cos( 100° )+cos( 200° ) cos( 100° )+cos( 200° )

31.

sin(−1°)+sin(−2°) sin(−1°)+sin(−2°)

For the following exercises, prove the identity.

32.

cos(a+b) cos(ab) = 1tanatanb 1+tanatanb cos(a+b) cos(ab) = 1tanatanb 1+tanatanb

33.

4sin( 3x )cos( 4x )=2sin( 7x )2sinx 4sin( 3x )cos( 4x )=2sin( 7x )2sinx

34.

6cos( 8x )sin( 2x ) sin( 6x ) =−3sin( 10x )csc( 6x )+3 6cos( 8x )sin( 2x ) sin( 6x ) =−3sin( 10x )csc( 6x )+3

35.

sinx+sin( 3x )=4sinx cos 2 x sinx+sin( 3x )=4sinx cos 2 x

36.

2( cos 3 xcosx sin 2 x )=cos( 3x )+cosx 2( cos 3 xcosx sin 2 x )=cos( 3x )+cosx

37.

2tanxcos( 3x )=secx( sin( 4x )sin( 2x ) ) 2tanxcos( 3x )=secx( sin( 4x )sin( 2x ) )

38.

cos( a+b )+cos( ab )=2cosacosb cos( a+b )+cos( ab )=2cosacosb

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°)+cos(12°) cos(58°)+cos(12°)

40.

sin()sin() sin()sin()

41.

cos(44°)cos(22°) cos(44°)cos(22°)

42.

cos(176°)sin() cos(176°)sin()

43.

sin(−14°)sin(85°) sin(−14°)sin(85°)

Technology

For the following exercises, algebraically determine whether each of the given equation is an 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.

2sin(2x)sin(3x)=cosxcos(5x) 2sin(2x)sin(3x)=cosxcos(5x)

45.

cos( 10θ )+cos( 6θ ) cos( 6θ )cos( 10θ ) =cot( 2θ )cot( 8θ ) cos( 10θ )+cos( 6θ ) cos( 6θ )cos( 10θ ) =cot( 2θ )cot( 8θ )

46.

sin( 3x )sin( 5x ) cos( 3x )+cos( 5x ) =tanx sin( 3x )sin( 5x ) cos( 3x )+cos( 5x ) =tanx

47.

2cos(2x)cosx+sin(2x)sinx=2sinx 2cos(2x)cosx+sin(2x)sinx=2sinx

48.

sin( 2x )+sin( 4x ) sin( 2x )sin( 4x ) =tan( 3x )cotx sin( 2x )+sin( 4x ) sin( 2x )sin( 4x ) =tan( 3x )cotx

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.

sin( 9t )sin( 3t ) cos( 9t )+cos( 3t ) sin( 9t )sin( 3t ) cos( 9t )+cos( 3t )

50.

2sin( 8x )cos( 6x )sin( 2x ) 2sin( 8x )cos( 6x )sin( 2x )

51.

sin( 3x )sinx sinx sin( 3x )sinx sinx

52.

cos( 5x )+cos( 3x ) sin( 5x )+sin( 3x ) cos( 5x )+cos( 3x ) sin( 5x )+sin( 3x )

53.

sinxcos( 15x )cosxsin( 15x ) sinxcos( 15x )cosxsin( 15x )

Extensions

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

54.

sinxsiny=2sin( xy 2 )cos( x+y 2 ) sinxsiny=2sin( xy 2 )cos( x+y 2 )

55.

cosx+cosy=2cos( x+y 2 )cos( xy 2 ) cosx+cosy=2cos( x+y 2 )cos( xy 2 )

For the following exercises, prove the identity.

56.

sin(6x)+sin(4x) sin(6x)sin(4x) =tan(5x)cotx sin(6x)+sin(4x) sin(6x)sin(4x) =tan(5x)cotx

57.

cos(3x)+cosx cos(3x)cosx =cot(2x)cotx cos(3x)+cosx cos(3x)cosx =cot(2x)cotx

58.

cos(6y)+cos(8y) sin(6y)sin(4y) =cotycos(7y)sec(5y) cos(6y)+cos(8y) sin(6y)sin(4y) =cotycos(7y)sec(5y)

59.

cos( 2y )cos( 4y ) sin( 2y )+sin( 4y ) =tany cos( 2y )cos( 4y ) sin( 2y )+sin( 4y ) =tany

60.

sin( 10x )sin( 2x ) cos( 10x )+cos( 2x ) =tan( 4x ) sin( 10x )sin( 2x ) cos( 10x )+cos( 2x ) =tan( 4x )

61.

cosxcos(3x)=4 sin 2 xcosx cosxcos(3x)=4 sin 2 xcosx

62.

(cos(2x)cos(4x)) 2 + (sin(4x)+sin(2x)) 2 =4 sin 2 (3x) (cos(2x)cos(4x)) 2 + (sin(4x)+sin(2x)) 2 =4 sin 2 (3x)

63.

tan( π 4 t )= 1tant 1+tant tan( π 4 t )= 1tant 1+tant

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