<< Chapter < Page Chapter >> Page >

Explain a situation where we would convert an equation from a product to a sum, and give an example.

Got questions? Get instant answers now!

Algebraic

For the following exercises, rewrite the product as a sum or difference.

16 sin ( 16 x ) sin ( 11 x )

8 ( cos ( 5 x ) cos ( 27 x ) )

Got questions? Get instant answers now!

20 cos ( 36 t ) cos ( 6 t )

Got questions? Get instant answers now!

2 sin ( 5 x ) cos ( 3 x )

sin ( 2 x ) + sin ( 8 x )

Got questions? Get instant answers now!

10 cos ( 5 x ) sin ( 10 x )

Got questions? Get instant answers now!

sin ( x ) sin ( 5 x )

1 2 ( cos ( 6 x ) cos ( 4 x ) )

Got questions? Get instant answers now!

For the following exercises, rewrite the sum or difference as a product.

cos ( 6 t ) + cos ( 4 t )

2 cos ( 5 t ) cos t

Got questions? Get instant answers now!

sin ( 3 x ) + sin ( 7 x )

Got questions? Get instant answers now!

cos ( 7 x ) + cos ( 7 x )

2 cos ( 7 x )

Got questions? Get instant answers now!

sin ( 3 x ) sin ( 3 x )

Got questions? Get instant answers now!

cos ( 3 x ) + cos ( 9 x )

2 cos ( 6 x ) cos ( 3 x )

Got questions? Get instant answers now!

sin h sin ( 3 h )

Got questions? Get instant answers now!

For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

cos ( 45° ) cos ( 15° )

1 4 ( 1 + 3 )

Got questions? Get instant answers now!

cos ( 45° ) sin ( 15° )

Got questions? Get instant answers now!

sin ( −345° ) sin ( −15° )

1 4 ( 3 2 )

Got questions? Get instant answers now!

sin ( 195° ) cos ( 15° )

Got questions? Get instant answers now!

sin ( −45° ) sin ( −15° )

1 4 ( 3 1 )

Got questions? Get instant answers now!

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

cos ( 23° ) sin ( 17° )

Got questions? Get instant answers now!

2 sin ( 100° ) sin ( 20° )

cos ( 80° ) cos ( 120° )

Got questions? Get instant answers now!

2 sin ( −100° ) sin ( −20° )

Got questions? Get instant answers now!

sin ( 213° ) cos ( )

1 2 ( sin ( 221° ) + sin ( 205° ) )

Got questions? Get instant answers now!

2 cos ( 56° ) cos ( 47° )

Got questions? Get instant answers now!

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

sin ( 76° ) + sin ( 14° )

2 cos ( 31° )

Got questions? Get instant answers now!

cos ( 58° ) cos ( 12° )

Got questions? Get instant answers now!

sin ( 101° ) sin ( 32° )

2 cos ( 66.5 ° ) sin ( 34.5 ° )

Got questions? Get instant answers now!

cos ( 100° ) + cos ( 200° )

Got questions? Get instant answers now!

sin ( −1° ) + sin ( −2° )

2 sin ( −1.5° ) cos ( 0.5° )

Got questions? Get instant answers now!

For the following exercises, prove the identity.

cos ( a + b ) cos ( a b ) = 1 tan a tan b 1 + tan a tan b

Got questions? Get instant answers now!

4 sin ( 3 x ) cos ( 4 x ) = 2 sin ( 7 x ) 2 sin x

2 sin ( 7 x ) 2 sin x = 2 sin ( 4 x + 3 x ) 2 sin ( 4 x 3 x ) = 2 ( sin ( 4 x ) cos ( 3 x ) + sin ( 3 x ) cos ( 4 x ) ) 2 ( sin ( 4 x ) cos ( 3 x ) sin ( 3 x ) cos ( 4 x ) ) = 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) = 4 sin ( 3 x ) cos ( 4 x )

Got questions? Get instant answers now!

6 cos ( 8 x ) sin ( 2 x ) sin ( 6 x ) = −3 sin ( 10 x ) csc ( 6 x ) + 3

Got questions? Get instant answers now!

sin x + sin ( 3 x ) = 4 sin x cos 2 x

sin x + sin ( 3 x ) = 2 sin ( 4 x 2 ) cos ( 2 x 2 ) =
2 sin ( 2 x ) cos x = 2 ( 2 sin x cos x ) cos x =
4 sin x cos 2 x

Got questions? Get instant answers now!

2 ( cos 3 x cos x sin 2 x ) = cos ( 3 x ) + cos x

Got questions? Get instant answers now!

2 tan x cos ( 3 x ) = sec x ( sin ( 4 x ) sin ( 2 x ) )

2 tan x cos ( 3 x ) = 2 sin x cos ( 3 x ) cos x = 2 ( .5 ( sin ( 4 x ) sin ( 2 x ) ) ) cos x
= 1 cos x ( sin ( 4 x ) sin ( 2 x ) ) = sec x ( sin ( 4 x ) sin ( 2 x ) )

Got questions? Get instant answers now!

cos ( a + b ) + cos ( a b ) = 2 cos a cos b

Got questions? Get instant answers now!

Numeric

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.

cos ( 58 ) + cos ( 12 )

2 cos ( 35 ) cos ( 23 ) ,  1 .5081

Got questions? Get instant answers now!

sin ( 2 ) sin ( 3 )

Got questions? Get instant answers now!

cos ( 44 ) cos ( 22 )

2 sin ( 33 ) sin ( 11 ) ,   0.2078

Got questions? Get instant answers now!

cos ( 176 ) sin ( 9 )

Got questions? Get instant answers now!

sin ( 14 ) sin ( 85 )

1 2 ( cos ( 99 ) cos ( 71 ) ) ,   0.2410

Got questions? Get instant answers now!

Technology

For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.

2 sin ( 2 x ) sin ( 3 x ) = cos x cos ( 5 x )

Got questions? Get instant answers now!

cos ( 10 θ ) + cos ( 6 θ ) cos ( 6 θ ) cos ( 10 θ ) = cot ( 2 θ ) cot ( 8 θ )

It is and identity.

Got questions? Get instant answers now!

sin ( 3 x ) sin ( 5 x ) cos ( 3 x ) + cos ( 5 x ) = tan x

Got questions? Get instant answers now!

2 cos ( 2 x ) cos x + sin ( 2 x ) sin x = 2 sin x

It is not an identity, but 2 cos 3 x is.

Got questions? Get instant answers now!

sin ( 2 x ) + sin ( 4 x ) sin ( 2 x ) sin ( 4 x ) = tan ( 3 x ) cot x

Got questions? Get instant answers now!

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

sin ( 9 t ) sin ( 3 t ) cos ( 9 t ) + cos ( 3 t )

tan ( 3 t )

Got questions? Get instant answers now!

2 sin ( 8 x ) cos ( 6 x ) sin ( 2 x )

Got questions? Get instant answers now!

sin ( 3 x ) sin x sin x

2 cos ( 2 x )

Got questions? Get instant answers now!

cos ( 5 x ) + cos ( 3 x ) sin ( 5 x ) + sin ( 3 x )

Got questions? Get instant answers now!

sin x cos ( 15 x ) cos x sin ( 15 x )

sin ( 14 x )

Got questions? Get instant answers now!

Extensions

For the following exercises, prove the following sum-to-product formulas.

sin x sin y = 2 sin ( x y 2 ) cos ( x + y 2 )

Got questions? Get instant answers now!

cos x + cos y = 2 cos ( x + y 2 ) cos ( x y 2 )

Start with cos x + cos y . Make a substitution and let x = α + β and let y = α β , so cos x + cos y becomes
cos ( α + β ) + cos ( α β ) = cos α cos β sin α sin β + cos α cos β + sin α sin β = 2 cos α cos β

Since x = α + β and y = α β , we can solve for α and β in terms of x and y and substitute in for 2 cos α cos β and get 2 cos ( x + y 2 ) cos ( x y 2 ) .

Got questions? Get instant answers now!

For the following exercises, prove the identity.

sin ( 6 x ) + sin ( 4 x ) sin ( 6 x ) sin ( 4 x ) = tan ( 5 x ) cot x

Got questions? Get instant answers now!

cos ( 3 x ) + cos x cos ( 3 x ) cos x = cot ( 2 x ) cot x

cos ( 3 x ) + cos x cos ( 3 x ) cos x = 2 cos ( 2 x ) cos x 2 sin ( 2 x ) sin x = cot ( 2 x ) cot x

Got questions? Get instant answers now!

cos ( 6 y ) + cos ( 8 y ) sin ( 6 y ) sin ( 4 y ) = cot y cos ( 7 y ) sec ( 5 y )

Got questions? Get instant answers now!

cos ( 2 y ) cos ( 4 y ) sin ( 2 y ) + sin ( 4 y ) = tan y

cos ( 2 y ) cos ( 4 y ) sin ( 2 y ) + sin ( 4 y ) = 2 sin ( 3 y ) sin ( y ) 2 sin ( 3 y ) cos y = 2 sin ( 3 y ) sin ( y ) 2 sin ( 3 y ) cos y = tan y

Got questions? Get instant answers now!

sin ( 10 x ) sin ( 2 x ) cos ( 10 x ) + cos ( 2 x ) = tan ( 4 x )

Got questions? Get instant answers now!

cos x cos ( 3 x ) = 4 sin 2 x cos x

cos x cos ( 3 x ) = 2 sin ( 2 x ) sin ( x ) = 2 ( 2 sin x cos x ) sin x = 4 sin 2 x cos x

Got questions? Get instant answers now!

( cos ( 2 x ) cos ( 4 x ) ) 2 + ( sin ( 4 x ) + sin ( 2 x ) ) 2 = 4 sin 2 ( 3 x )

Got questions? Get instant answers now!

tan ( π 4 t ) = 1 tan t 1 + tan t

tan ( π 4 t ) = tan ( π 4 ) tan t 1 + tan ( π 4 ) tan ( t ) = 1 tan t 1 + tan t

Got questions? Get instant answers now!

Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask