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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

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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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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