# 9.4 Sum-to-product and product-to-sum formulas

 Page 1 / 6
In this section, you will:
• Express products as sums.
• Express sums as products.

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.

## 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:

$\begin{array}{l}\underset{___________________________________}{\begin{array}{ccc}\hfill \mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta +\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta & =& \mathrm{cos}\left(\alpha -\beta \right)\hfill \\ \hfill +\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta -\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta & =& \mathrm{cos}\left(\alpha +\beta \right)\hfill \end{array}}\\ \begin{array}{ccc}\hfill \phantom{\rule{5.7em}{0ex}}2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta & =& \mathrm{cos}\left(\alpha -\beta \right)+\mathrm{cos}\left(\alpha +\beta \right)\hfill \end{array}\end{array}$

Then, we divide by $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ to isolate the product of cosines:

$\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =\frac{1}{2}\left[\mathrm{cos}\left(\alpha -\beta \right)+\mathrm{cos}\left(\alpha +\beta \right)\right]$

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: $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{7x}{2}\right)\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\frac{3x}{2}.$

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

$\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =\frac{1}{2}\left[\mathrm{cos}\left(\alpha -\beta \right)+\mathrm{cos}\left(\alpha +\beta \right)\right]$

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

$\begin{array}{ccc}\hfill 2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{7x}{2}\right)\mathrm{cos}\left(\frac{3x}{2}\right)& =& \left(2\right)\left(\frac{1}{2}\right)\left[\mathrm{cos}\left(\frac{7x}{2}-\frac{3x}{2}\right)\right)+\mathrm{cos}\left(\frac{7x}{2}+\frac{3x}{2}\right)\right]\hfill \\ & =& \left[\mathrm{cos}\left(\frac{4x}{2}\right)+\mathrm{cos}\left(\frac{10x}{2}\right)\right]\hfill \\ & =& \mathrm{cos}\text{\hspace{0.17em}}2x+\mathrm{cos}\text{\hspace{0.17em}}5x\hfill \end{array}$

Use the product-to-sum formula to write the product as a sum or difference: $\text{\hspace{0.17em}}\mathrm{cos}\left(2\theta \right)\mathrm{cos}\left(4\theta \right).$

$\frac{1}{2}\left(\mathrm{cos}6\theta +\mathrm{cos}2\theta \right)$

## 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:

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

$\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =\frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)\right]$

## 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: $\text{\hspace{0.17em}}\mathrm{sin}\left(4\theta \right)\mathrm{cos}\left(2\theta \right).$

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

$\begin{array}{ccc}\hfill \mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta & =& \frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)\right]\hfill \\ \hfill \mathrm{sin}\left(4\theta \right)\mathrm{cos}\left(2\theta \right)& =& \frac{1}{2}\left[\mathrm{sin}\left(4\theta +2\theta \right)+\mathrm{sin}\left(4\theta -2\theta \right)\right]\hfill \\ & =& \frac{1}{2}\left[\mathrm{sin}\left(6\theta \right)+\mathrm{sin}\left(2\theta \right)\right]\hfill \end{array}$

Use the product-to-sum formula to write the product as a sum: $\text{\hspace{0.17em}}\mathrm{sin}\left(x+y\right)\mathrm{cos}\left(x-y\right).$

$\frac{1}{2}\left(\mathrm{sin}2x+\mathrm{sin}2y\right)$

## 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:

#### Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
factoring polynomial
find general solution of the Tanx=-1/root3,secx=2/root3
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
where can I get indices
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Need help with this question please
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has