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csc 2 t = 3

sin 1 ( 3 3 ) , π sin 1 ( 3 3 ) , π + sin 1 ( 3 3 ) , 2 π sin 1 ( 3 3 )

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2 sin θ = −1

7 π 6 , 11 π 6

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tan x sin x + sin ( x ) = 0

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9 sin ω 2 = 4 sin 2 ω

sin 1 ( 1 4 ) , π sin 1 ( 1 4 )

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1 2 tan ( ω ) = tan 2 ( ω )

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For the following exercises, use basic identities to simplify the expression.

sec x cos x + cos x 1 sec x

1

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sin 3 x + cos 2 x sin x

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For the following exercises, determine if the given identities are equivalent.

sin 2 x + sec 2 x 1 = ( 1 cos 2 x ) ( 1 + cos 2 x ) cos 2 x

Yes

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tan 3 x csc 2 x cot 2 x cos x sin x = 1

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Sum and Difference Identities

For the following exercises, find the exact value.

tan ( 7 π 12 )

2 3

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sin ( 70° ) cos ( 25° ) cos ( 70° ) sin ( 25° )

2 2

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cos ( 83° ) cos ( 23° ) + sin ( 83° ) sin ( 23° )

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For the following exercises, prove the identity.

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

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

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

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For the following exercise, simplify the expression.

tan ( 1 2 x ) + tan ( 1 8 x ) 1 tan ( 1 8 x ) tan ( 1 2 x )

tan ( 5 8 x )

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For the following exercises, find the exact value.

cos ( sin 1 ( 0 ) cos 1 ( 1 2 ) )

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tan ( sin 1 ( 0 ) + sin 1 ( 1 2 ) )

3 3

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Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.

Find sin ( 2 θ ) , cos ( 2 θ ) , and tan ( 2 θ ) given cos θ = 1 3 and θ is in the interval [ π 2 , π ] .

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Find sin ( 2 θ ) , cos ( 2 θ ) , and tan ( 2 θ ) given sec θ = 5 3 and θ is in the interval [ π 2 , π ] .

24 25 , 7 25 , 24 7

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sec ( 3 π 8 )

2 ( 2 + 2 )

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For the following exercises, use [link] to find the desired quantities.

Image of a right triangle. The base is 24, the height is unknown, and the hypotenuse is 25. The angle opposite the base is labeled alpha, and the remaining acute angle is labeled beta.

sin ( 2 β ) , cos ( 2 β ) , tan ( 2 β ) , sin ( 2 α ) , cos ( 2 α ) , and  tan ( 2 α )

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sin ( β 2 ) , cos ( β 2 ) , tan ( β 2 ) , sin ( α 2 ) , cos ( α 2 ) , and  tan ( α 2 )

2 10 , 7 2 10 , 1 7 , 3 5 , 4 5 , 3 4

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For the following exercises, prove the identity.

2 cos ( 2 x ) sin ( 2 x ) = cot x tan x

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

cot x cos ( 2 x ) = cot x ( 1 2 sin 2 x ) = cot x cos x sin x ( 2 ) sin 2 x = 2 sin x cos x + cot x = sin ( 2 x ) + cot x

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For the following exercises, rewrite the expression with no powers.

cos 2 x sin 4 ( 2 x )

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tan 2 x sin 3 x

10 sin x 5 sin ( 3 x ) + sin ( 5 x ) 8 ( cos ( 2 x ) + 1 )

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Sum-to-Product and Product-to-Sum Formulas

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

cos ( π 3 ) sin ( π 4 )

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2 sin ( 2 π 3 ) sin ( 5 π 6 )

3 2

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2 cos ( π 5 ) cos ( π 3 )

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For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

sin ( π 12 ) sin ( 7 π 12 )

2 2

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cos ( 5 π 12 ) + cos ( 7 π 12 )

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For the following exercises, change the functions from a product to a sum or a sum to a product.

sin ( 9 x ) cos ( 3 x )

1 2 ( sin ( 6 x ) + sin ( 12 x ) )

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cos ( 7 x ) cos ( 12 x )

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sin ( 11 x ) + sin ( 2 x )

2 sin ( 13 2 x ) cos ( 9 2 x )

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cos ( 6 x ) + cos ( 5 x )

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Solving Trigonometric Equations

For the following exercises, find all exact solutions on the interval [ 0 , 2 π ) .

tan x + 1 = 0

3 π 4 , 7 π 4

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2 sin ( 2 x ) + 2 = 0

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For the following exercises, find all exact solutions on the interval [ 0 , 2 π ) .

2 sin 2 x sin x = 0

0 , π 6 , 5 π 6 , π

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cos 2 x cos x 1 = 0

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2 sin 2 x + 5 sin x + 3 = 0

3 π 2

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cos x 5 sin ( 2 x ) = 0

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1 sec 2 x + 2 + sin 2 x + 4 cos 2 x = 0

No solution

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Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
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classification of plants, gymnosperm features.
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What is Atoms
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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.
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how is urine formed in human
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malaria is caused by an insect called mosquito.
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Malaria is cause by female anopheles mosquito
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Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
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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
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stimulates the follicle to release the mature ovum into the oviduct
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endocrine secrete hormone and regulate body process
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while pituitary gland is an example of endocrine system and it's found in the Brain
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Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
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biology is the study of life.
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Biology is the study of how living organisms live and survive in a specific environment
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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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