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7.4 Sum-to-product and product-to-sum formulas  (Page 2/6)

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

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

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

Express the product as a sum or difference

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

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

          cos   α   cos   β = 1 2 [ cos ( α − β ) + cos ( α + β ) ] cos ( 3 θ ) cos ( 5 θ ) = 1 2 [ cos ( 3 θ − 5 θ ) + cos ( 3 θ + 5 θ ) ]                          = 1 2 [ cos ( 2 θ ) + cos ( 8 θ ) ]   Use even-odd identity .
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Use the product-to-sum formula to evaluate   cos   11 π 12   cos   π 12 .

− 2 − 3 4

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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   and   u − v 2 = β .

Then,

α + β = u + v 2 + u − v 2           = 2 u 2           = u α − β = u + v 2 − u − v 2           = 2 v 2           = v

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

                     sin   α   cos   β = 1 2 [ sin ( α + β ) + sin ( α − β ) ]    sin ( u + v 2 ) cos ( u − v 2 ) = 1 2 [ sin   u + sin   v ]   Substitute for ( α + β )  and  ( α − β ) 2   sin ( u + v 2 ) cos ( u − v 2 ) = sin   u + sin   v

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

Writing the difference of sines as a product

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

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

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

Substitute the values into the formula, and simplify.

sin ( 4 θ ) − sin ( 2 θ ) = 2 sin ( 4 θ − 2 θ 2 )   cos ( 4 θ + 2 θ 2 )                             = 2 sin ( 2 θ 2 )   cos ( 6 θ 2 )                             = 2   sin   θ   cos ( 3 θ )
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Use the sum-to-product formula to write the sum as a product:   sin ( 3 θ ) + sin ( θ ) .

2 sin ( 2 θ ) cos ( θ )

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Evaluating using the sum-to-product formula

Evaluate   cos ( 15 ∘ ) − cos ( 75 ∘ ) .

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

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

Then we substitute the given angles and simplify.

cos ( 15 ∘ ) − cos ( 75 ∘ ) = − 2 sin ( 15 ∘ + 75 ∘ 2 )   sin ( 15 ∘ − 75 ∘ 2 )                                 = − 2 sin ( 45 ∘ )   sin ( − 30 ∘ )                                 = − 2 ( 2 2 ) ( − 1 2 )                                 = 2 2
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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
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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
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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
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ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
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ismail
Rectangle coordinate
Asma Reply
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Jhon Reply
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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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