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For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval [ 0 , 2 π ) . Round to four decimal places.

3 cot 2 x + cot x = 1

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csc 2 x 3 csc x 4 = 0

0.2527 , 2.8889 , 4.7124

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For the following exercises, graph each side of the equation to find the approximate solutions on the interval [ 0 , 2 π ) .

20 cos 2 x + 21 cos x + 1 = 0

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sec 2 x 2 sec x = 15

1.3694 , 1.9106 , 4.3726 , 4.9137

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Practice test

For the following exercises, simplify the given expression.

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

1

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sin ( x ) cos ( 2 x ) sin ( x ) cos ( 2 x )

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c s c ( θ ) cot ( θ ) ( sec 2 θ 1 )

sec ( θ )

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cos 2 ( θ ) sin 2 ( θ ) ( 1 + cot 2 ( θ ) ) ( 1 + tan 2 ( θ ) )

1

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For the following exercises, find the exact value.

cos ( 7 π 12 )

2 6 4

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tan ( sin 1 ( 2 2 ) + tan 1 3 )

2 3

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2 sin ( π 4 ) sin ( π 6 )

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cos ( 4 π 3 + θ )

1 2 cos ( θ ) 3 2 sin ( θ )

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tan ( π 4 + θ )

1 + tan ( θ ) 1 + tan ( θ )

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For the following exercises, simplify each expression. Do not evaluate.

cos 2 ( 32° ) tan 2 ( 32° )

1 cos ( 64 ) 2

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cot ( θ 2 )

± 1 + cos ( θ ) 1 cos ( θ )

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

cos 2 x sin 2 x 1 = 0

0 , π

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cos 2 x = cos x 4 sin 2 x + 2 sin x 3 = 0

sin 1 ( 1 4 ( 13 1 ) ) , π sin 1 ( 1 4 ( 13 1 ) )

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cos ( 2 x ) + sin 2 x = 0

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2 sin 2 x sin x = 0

0 , π 6 , 5 π 6 , π

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Rewrite the expression as a product instead of a sum: cos ( 2 x ) + cos ( 8 x ) .

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

8 cos ( 15 x ) sin ( 3 x )

4 [ sin ( 18 x ) sin ( 12 x ) ]

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

2 ( sin ( 8 θ ) sin ( 4 θ ) )

4 sin ( 2 θ ) cos ( 6 θ )

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Find all solutions of tan ( x ) 3 = 0.

π 3 + k π

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Find the solutions of sec 2 x 2 sec x = 15 on the interval [ 0 , 2 π ) algebraically; then graph both sides of the equation to determine the answer.

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For the following exercises, find all solutions exactly on the interval 0 θ π

2 cos ( θ 2 ) = 1

120

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Find sin ( 2 θ ) , cos ( 2 θ ) , and tan ( 2 θ ) given cot θ = 3 4 and θ is on the interval [ π 2 , π ] .

24 25 , 7 25 , 24 7

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Find sin ( θ 2 ) , cos ( θ 2 ) , and tan ( θ 2 ) given cos θ = 7 25 and θ is in quadrant IV.

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Rewrite the expression sin 4 x with no powers greater than 1.

1 8 ( 3 + cos ( 4 x ) 4 cos ( 2 x ) )

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For the following exercises, prove the identity.

tan 3 x tan x sec 2 x = tan ( x )

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

sin ( 3 x ) cos x sin ( 2 x ) = sin ( x + 2 x ) cos x ( 2 sin x cos x ) = sin x cos ( 2 x ) + sin ( 2 x ) cos x 2 sin x cos 2 x = sin x ( cos 2 x sin 2 x ) + 2 sin x cos x cos x 2 sin x cos 2 x = sin x cos 2 x sin 3 + 0 = cos 2 x sin x sin 3 x = cos 2 x sin x sin 3 x

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

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Plot the points and find a function of the form y = A cos ( B x + C ) + D that fits the given data.

x 0 1 2 3 4 5
y −2 2 −2 2 −2 2

y = 2 cos ( π x + π )

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The displacement h ( t ) in centimeters of a mass suspended by a spring is modeled by the function h ( t ) = 1 4 sin ( 120 π t ) , where t is measured in seconds. Find the amplitude, period, and frequency of this displacement.

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A woman is standing 300 feet away from a 2000-foot building. If she looks to the top of the building, at what angle above horizontal is she looking? A bored worker looks down at her from the 15 th floor (1500 feet above her). At what angle is he looking down at her? Round to the nearest tenth of a degree.

81.5° , 78.7°

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Two frequencies of sound are played on an instrument governed by the equation n ( t ) = 8 cos ( 20 π t ) cos ( 1000 π t ) . What are the period and frequency of the “fast” and “slow” oscillations? What is the amplitude?

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The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July. Construct a function that models this behavior. During what period is there more than 10 inches of snowfall?

6 + 5 cos ( π 6 ( 1 x ) ) . From November 23 to February 6.

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A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring t seconds after being released. At what time will the spring come to rest? In this case, use 1 cm amplitude as rest.

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Water levels near a glacier currently average 9 feet, varying seasonally by 2 inches above and below the average and reaching their highest point in January. Due to global warming, the glacier has begun melting faster than normal. Every year, the water levels rise by a steady 3 inches. Find a function modeling the depth of the water t months from now. If the docks are 2 feet above current water levels, at what point will the water first rise above the docks?

D ( t ) = 2 cos ( π 6 t ) + 108 + 1 4 t , 93.5855 months (or 7.8 years) from now

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Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
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
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
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
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
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
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
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.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply

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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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