1. Home
  2. Algebra and trigonometry
  3. Trigonometric identities and
  4. Sum-to-product and product-to

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

<< Chapter < Page
  Algebra and trigonometry     Page 2 / 6
Chapter >> Page >
                       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 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the product-to-sum formula to evaluate   cos   11 π 12   cos   π 12 .

− 2 − 3 4

Got questions? Get instant answers now!

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 )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the sum-to-product formula to write the sum as a product:   sin ( 3 θ ) + sin ( θ ) .

2 sin ( 2 θ ) cos ( θ )

Got questions? Get instant answers now!

Evaluating using the sum-to-product formula

Evaluate   cos ( 15° ) − cos ( 75° ) .   Check the answer with a graphing calculator.

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Proving an identity

Prove the identity:

cos ( 4 t ) − cos ( 2 t ) sin ( 4 t ) + sin ( 2 t ) = − tan   t

We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.

cos ( 4 t ) − cos ( 2 t ) sin ( 4 t ) + sin ( 2 t ) = − 2   sin ( 4 t + 2 t 2 )   sin ( 4 t − 2 t 2 ) 2   sin ( 4 t + 2 t 2 )   cos ( 4 t − 2 t 2 ) = − 2   sin ( 3 t ) sin   t 2   sin ( 3 t ) cos   t = − 2 sin ( 3 t ) sin   t 2 sin ( 3 t ) cos   t = − sin   t cos   t = − tan   t
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

what is biology
Hajah Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
what is biology
Victoria Reply
HOW CAN MAN ORGAN FUNCTION
Alfred Reply
the diagram of the digestive system
Assiatu Reply
allimentary cannel
Ogenrwot
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 2

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask