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7.5: Sum-to-Product and Product-to-Sum Formulas

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    114044
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    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 .

    Answer

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

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

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

    2cos( 7x 2 )cos( 3x 2 )=(2)( 1 2 )[ cos( 7x 2 3x 2 )+cos( 7x 2 + 3x 2 ) ] =[ cos( 4x 2 )+cos( 10x 2 ) ] =cos2x+cos5x 2cos( 7x 2 )cos( 3x 2 )=(2)( 1 2 )[ cos( 7x 2 3x 2 )+cos( 7x 2 + 3x 2 ) ] =[ cos( 4x 2 )+cos( 10x 2 ) ] =cos2x+cos5x

    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 ).

    Answer

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

    sinαcosβ= 1 2 [ sin( α+β )+sin( αβ ) ] sin( 4θ )cos( 2θ )= 1 2 [ sin( 4θ+2θ )+sin( 4θ2θ ) ] = 1 2 [ sin( 6θ )+sin( 2θ ) ] sinαcosβ= 1 2 [ sin( α+β )+sin( αβ ) ] sin( 4θ )cos( 2θ )= 1 2 [ sin( 4θ+2θ )+sin( 4θ2θ ) ] = 1 2 [ sin( 6θ )+sin( 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.

    Answer

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

    cosαcosβ= 1 2 [cos(αβ)+cos(α+β)] cos(3θ)cos(5θ)= 1 2 [cos(3θ5θ)+cos(3θ+5θ)] = 1 2 [cos(2θ)+cos(8θ)] Use even-odd identity. cosαcosβ= 1 2 [cos(αβ)+cos(α+β)] cos(3θ)cos(5θ)= 1 2 [cos(3θ5θ)+cos(3θ+5θ)] = 1 2 [cos(2θ)+cos(8θ)] Use even-odd identity.

    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 ).

    Answer

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

    sinαsinβ=2sin( αβ 2 )cos( α+β 2 ) sinαsinβ=2sin( αβ 2 )cos( α+β 2 )

    Substitute the values into the formula, and simplify.

    sin(4θ)sin(2θ)=2sin( 4θ2θ 2 )cos( 4θ+2θ 2 ) =2sin( 2θ 2 )cos( 6θ 2 ) =2sinθcos(3θ) sin(4θ)sin(2θ)=2sin( 4θ2θ 2 )cos( 4θ+2θ 2 ) =2sin( 2θ 2 )cos( 6θ 2 ) =2sinθcos(3θ)

    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 ).

    Answer

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

    cosαcosβ=2sin( α+β 2 )sin( αβ 2 ) cosαcosβ=2sin( α+β 2 )sin( αβ 2 )

    Then we substitute the given angles and simplify.

    cos( 15 )cos( 75 )=2sin( 15 + 75 2 )sin( 15 75 2 ) =2sin( 45 )sin( 30 ) =2( 2 2 )( 1 2 ) = 2 2 cos( 15 )cos( 75 )=2sin( 15 + 75 2 )sin( 15 75 2 ) =2sin( 45 )sin( 30 ) =2( 2 2 )( 1 2 ) = 2 2

    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

    Answer

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

    cos(4t)cos(2t) sin(4t)+sin(2t) = 2sin( 4t+2t 2 )sin( 4t2t 2 ) 2sin( 4t+2t 2 )cos( 4t2t 2 ) = 2sin(3t)sint 2sin(3t)cost = 2 sin(3t) sint 2 sin(3t) cost = sint cost =tant cos(4t)cos(2t) sin(4t)+sin(2t) = 2sin( 4t+2t 2 )sin( 4t2t 2 ) 2sin( 4t+2t 2 )cos( 4t2t 2 ) = 2sin(3t)sint 2sin(3t)cost = 2 sin(3t) sint 2 sin(3t) cost = sint cost =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 .

    Answer

    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.

    cos(2θ) sin 2 θ = 12 sin 2 θ sin 2 θ = 1 sin 2 θ 2 sin 2 θ sin 2 θ = csc 2 θ2 cos(2θ) sin 2 θ = 12 sin 2 θ sin 2 θ = 1 sin 2 θ 2 sin 2 θ sin 2 θ = csc 2 θ2

    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.

    7.4 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.

    Explain 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.

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

    4.

    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.

    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.

    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( 2 )sin( 3 ) sin( 2 )sin( 3 )

    41.

    cos( 44 )cos( 22 ) cos( 44 )cos( 22 )

    42.

    cos( 176 )sin( 9 ) cos( 176 )sin( 9 )

    43.

    sin( 14 )sin( 85 ) sin( 14 )sin( 85 )

    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.

    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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