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7.4 Sum-to-product and product-to-sum formulas

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

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:

            cos   α   cos   β + sin   α   sin   β = cos ( α − β ) +     cos   α   cos   β − sin   α   sin   β = cos ( α + β ) ________________________________                                                                       2   cos   α   cos   β = cos ( α − β ) + cos ( α + β )

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

cos   α   cos   β = 1 2 [ cos ( α − β ) + cos ( α + β ) ]

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.

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 ( 7 x 2 )   cos   3 x 2 .

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

cos   α   cos   β = 1 2 [ cos ( α − β ) + cos ( α + β ) ]

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

2   cos ( 7 x 2 ) cos ( 3 x 2 ) = ( 2 ) ( 1 2 ) [ cos ( 7 x 2 − 3 x 2 ) + cos ( 7 x 2 + 3 x 2 ) ]                             = [ cos ( 4 x 2 ) + cos ( 10 x 2 ) ]                             = cos   2 x + cos   5 x
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Use the product-to-sum formula to write the product as a sum or difference:   cos ( 2 θ ) cos ( 4 θ ) .

1 2 ( cos 6 θ + cos 2 θ )

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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 ) = sin   α   cos   β + cos   α   sin   β +                  sin ( α − β ) = sin   α   cos   β − cos   α   sin   β _________________________________________ sin ( α + β ) + sin ( α − β ) = 2   sin   α   cos   β

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

sin   α   cos   β = 1 2 [ sin ( α + β ) + sin ( α − β ) ]

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 ) .

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

                          sin   α   cos   β = 1 2 [ sin ( α + β ) + sin ( α − β ) ]     sin ( 4 θ ) cos ( 2 θ ) = 1 2 [ sin ( 4 θ + 2 θ ) + sin ( 4 θ − 2 θ ) ]                                                                             = 1 2 [ sin ( 6 θ ) + sin ( 2 θ ) ]
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Use the product-to-sum formula to write the product as a sum:   sin ( x + y ) cos ( x − y ) .

1 2 ( sin 2 x + sin 2 y )

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Questions & Answers

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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
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Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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