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cos ( 6 t ) + cos ( 4 t )

2 cos ( 5 t ) cos t

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sin ( 3 x ) + sin ( 7 x )

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cos ( 7 x ) + cos ( 7 x )

2 cos ( 7 x )

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sin ( 3 x ) sin ( 3 x )

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cos ( 3 x ) + cos ( 9 x )

2 cos ( 6 x ) cos ( 3 x )

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sin h sin ( 3 h )

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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 )

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cos ( 45° ) sin ( 15° )

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sin ( −345° ) sin ( −15° )

1 4 ( 3 2 )

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sin ( 195° ) cos ( 15° )

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sin ( −45° ) sin ( −15° )

1 4 ( 3 1 )

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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° )

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2 sin ( 100° ) sin ( 20° )

cos ( 80° ) cos ( 120° )

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2 sin ( −100° ) sin ( −20° )

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sin ( 213° ) cos ( )

1 2 ( sin ( 221° ) + sin ( 205° ) )

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2 cos ( 56° ) cos ( 47° )

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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° )

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cos ( 58° ) cos ( 12° )

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sin ( 101° ) sin ( 32° )

2 cos ( 66.5° ) sin ( 34.5° )

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cos ( 100° ) + cos ( 200° )

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sin ( −1° ) + sin ( −2° )

2 sin ( −1.5° ) cos ( 0.5° )

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For the following exercises, prove the identity.

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

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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 )

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

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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

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

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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 ) )

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

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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

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sin ( ) sin ( )

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cos ( 44° ) cos ( 22° )

2 sin ( 33° ) sin ( 11° ) , 0.2078

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cos ( 176° ) sin ( )

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sin ( −14° ) sin ( 85° )

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

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Technology

For the following exercises, algebraically determine whether each of the given equation is an 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 )

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

It is an identity.

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

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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.

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

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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 )

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

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sin ( 3 x ) sin x sin x

2 cos ( 2 x )

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

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

sin ( 14 x )

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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 )

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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 ) .

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For the following exercises, prove the identity.

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

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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

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

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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

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

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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

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

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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

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Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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