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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

what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
Practice Key Terms 2

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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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