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Algebraic

For the following exercises, find all solutions exactly on the interval 0 θ < 2 π .

2 sin θ = 2

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2 sin θ = 3

π 3 , 2 π 3

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2 cos θ = 2

3 π 4 , 5 π 4

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tan x = 1

π 4 , 5 π 4

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4 sin 2 x 2 = 0

π 4 , 3 π 4 , 5 π 4 , 7 π 4

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For the following exercises, solve exactly on [ 0 , 2 π ) .

2 cos θ = 2

π 4 , 7 π 4

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2 sin θ = −1

7 π 6 , 11 π 6

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2 sin θ = 3

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2 sin ( 3 θ ) = 1

π 18 , 5 π 18 , 13 π 18 , 17 π 18 , 25 π 18 , 29 π 18

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2 cos ( 3 θ ) = 2

3 π 12 , 5 π 12 , 11 π 12 , 13 π 12 , 19 π 12 , 21 π 12

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2 sin ( π θ ) = 1

1 6 , 5 6 , 13 6 , 17 6 , 25 6 , 29 6 , 37 6

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2 cos ( π 5 θ ) = 3

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

sec ( x ) sin ( x ) 2 sin ( x ) = 0

0 , π 3 , π , 5 π 3

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tan ( x ) 2 sin ( x ) tan ( x ) = 0

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2 cos 2 t + cos ( t ) = 1

π 3 , π , 5 π 3

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2 tan 2 ( t ) = 3 sec ( t )

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2 sin ( x ) cos ( x ) sin ( x ) + 2 cos ( x ) 1 = 0

π 3 , 3 π 2 , 5 π 3

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tan 2 ( x ) = 1 + 2 tan ( x )

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8 sin 2 ( x ) + 6 sin ( x ) + 1 = 0

π sin 1 ( 1 4 ) , 7 π 6 , 11 π 6 , 2 π + sin 1 ( 1 4 )

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For the following exercises, solve with the methods shown in this section exactly on the interval [ 0 , 2 π ) .

sin ( 3 x ) cos ( 6 x ) cos ( 3 x ) sin ( 6 x ) = −0.9

1 3 ( sin 1 ( 9 10 ) ) , π 3 1 3 ( sin 1 ( 9 10 ) ) , 2 π 3 + 1 3 ( sin 1 ( 9 10 ) ) , π 1 3 ( sin 1 ( 9 10 ) ) , 4 π 3 + 1 3 ( sin 1 ( 9 10 ) ) , 5 π 3 1 3 ( sin 1 ( 9 10 ) )

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sin ( 6 x ) cos ( 11 x ) cos ( 6 x ) sin ( 11 x ) = −0.1

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

0

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6 sin ( 2 t ) + 9 sin t = 0

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9 cos ( 2 θ ) = 9 cos 2 θ 4

π 6 , 5 π 6 , 7 π 6 , 11 π 6

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cos ( 2 t ) = sin t

3 π 2 , π 6 , 5 π 6

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cos ( 6 x ) cos ( 3 x ) = 0

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For the following exercises, solve exactly on the interval [ 0 , 2 π ) . Use the quadratic formula if the equations do not factor.

tan 2 x 3 tan x = 0

0 , π 3 , π , 4 π 3

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sin 2 x + sin x 2 = 0

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sin 2 x 2 sin x 4 = 0

There are no solutions.

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5 cos 2 x + 3 cos x 1 = 0

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3 cos 2 x 2 cos x 2 = 0

cos 1 ( 1 3 ( 1 7 ) ) , 2 π cos 1 ( 1 3 ( 1 7 ) )

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5 sin 2 x + 2 sin x 1 = 0

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tan 2 x + 5 tan x 1 = 0

tan 1 ( 1 2 ( 29 5 ) ) , π + tan 1 ( 1 2 ( 29 5 ) ) , π + tan 1 ( 1 2 ( 29 5 ) ) , 2 π + tan 1 ( 1 2 ( 29 5 ) )

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tan 2 x tan x 2 = 0

There are no solutions.

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For the following exercises, find exact solutions on the interval [ 0 , 2 π ) . Look for opportunities to use trigonometric identities.

sin 2 x cos 2 x sin x = 0

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sin 2 x + cos 2 x = 0

There are no solutions.

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sin ( 2 x ) sin x = 0

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cos ( 2 x ) cos x = 0

0 , 2 π 3 , 4 π 3

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2 tan x 2 sec 2 x sin 2 x = cos 2 x

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

π 4 , 3 π 4 , 5 π 4 , 7 π 4

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10 sin x cos x = 6 cos x

sin 1 ( 3 5 ) , π 2 , π sin 1 ( 3 5 ) , 3 π 2

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−3 sin t = 15 cos t sin t

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4 cos 2 x 4 = 15 cos x

cos 1 ( 1 4 ) , 2 π cos 1 ( 1 4 )

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8 sin 2 x + 6 sin x + 1 = 0

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8 cos 2 θ = 3 2 cos θ

π 3 , cos 1 ( 3 4 ) , 2 π cos 1 ( 3 4 ) , 5 π 3

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6 cos 2 x + 7 sin x 8 = 0

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12 sin 2 t + cos t 6 = 0

cos 1 ( 3 4 ) , cos 1 ( 2 3 ) , 2 π cos 1 ( 2 3 ) , 2 π cos 1 ( 3 4 )

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tan x = 3 sin x

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cos 3 t = cos t

0 , π 2 , π , 3 π 2

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Graphical

For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.

6 sin 2 x 5 sin x + 1 = 0

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8 cos 2 x 2 cos x 1 = 0

π 3 , cos −1 ( 1 4 ) , 2 π cos −1 ( 1 4 ) , 5 π 3

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100 tan 2 x + 20 tan x 3 = 0

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2 cos 2 x cos x + 15 = 0

There are no solutions.

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20 sin 2 x 27 sin x + 7 = 0

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2 tan 2 x + 7 tan x + 6 = 0

π + tan −1 ( −2 ) , π + tan −1 ( 3 2 ) , 2 π + tan −1 ( −2 ) , 2 π + tan −1 ( 3 2 )

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130 tan 2 x + 69 tan x 130 = 0

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Technology

For the following exercises, use a calculator to find all solutions to four decimal places.

sin x = 0.27

2 π k + 0.2734 , 2 π k + 2.8682

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tan x = −0.34

π k 0.3277

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For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval [ 0 , 2 π ) . Round to four decimal places.

tan 2 x + 3 tan x 3 = 0

0.6694 , 1.8287 , 3.8110 , 4.9703

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6 tan 2 x + 13 tan x = −6

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tan 2 x sec x = 1

1.0472 , 3.1416 , 5.2360

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sin 2 x 2 cos 2 x = 0

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2 tan 2 x + 9 tan x 6 = 0

0.5326 , 1.7648 , 3.6742 , 4.9064

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4 sin 2 x + sin ( 2 x ) sec x 3 = 0

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Extensions

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

csc 2 x 3 csc x 4 = 0

sin 1 ( 1 4 ) , π sin 1 ( 1 4 ) , 3 π 2

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sin 2 x cos 2 x 1 = 0

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sin 2 x ( 1 sin 2 x ) + cos 2 x ( 1 sin 2 x ) = 0

π 2 , 3 π 2

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3 sec 2 x + 2 + sin 2 x tan 2 x + cos 2 x = 0

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

There are no solutions.

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tan 2 x 1 sec 3 x cos x = 0

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sin ( 2 x ) sec 2 x = 0

0 , π 2 , π , 3 π 2

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sin ( 2 x ) 2 csc 2 x = 0

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2 cos 2 x sin 2 x cos x 5 = 0

There are no solutions.

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

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Real-world applications

An airplane has only enough gas to fly to a city 200 miles northeast of its current location. If the pilot knows that the city is 25 miles north, how many degrees north of east should the airplane fly?

7.2

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If a loading ramp is placed next to a truck, at a height of 4 feet, and the ramp is 15 feet long, what angle does the ramp make with the ground?

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If a loading ramp is placed next to a truck, at a height of 2 feet, and the ramp is 20 feet long, what angle does the ramp make with the ground?

5.7

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A woman is watching a launched rocket currently 11 miles in altitude. If she is standing 4 miles from the launch pad, at what angle is she looking up from horizontal?

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An astronaut is in a launched rocket currently 15 miles in altitude. If a man is standing 2 miles from the launch pad, at what angle is she looking down at him from horizontal? (Hint: this is called the angle of depression.)

82.4

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A woman is standing 8 meters away from a 10-meter tall building. At what angle is she looking to the top of the building?

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A man is standing 10 meters away from a 6-meter tall building. Someone at the top of the building is looking down at him. At what angle is the person looking at him?

31.0

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A 20-foot tall building has a shadow that is 55 feet long. What is the angle of elevation of the sun?

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A 90-foot tall building has a shadow that is 2 feet long. What is the angle of elevation of the sun?

88.7

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A spotlight on the ground 3 meters from a 2-meter tall man casts a 6 meter shadow on a wall 6 meters from the man. At what angle is the light?

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A spotlight on the ground 3 feet from a 5-foot tall woman casts a 15-foot tall shadow on a wall 6 feet from the woman. At what angle is the light?

59.0

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For the following exercises, find a solution to the following word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree.

A person does a handstand with his feet touching a wall and his hands 1.5 feet away from the wall. If the person is 6 feet tall, what angle do his feet make with the wall?

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A person does a handstand with her feet touching a wall and her hands 3 feet away from the wall. If the person is 5 feet tall, what angle do her feet make with the wall?

36.9

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A 23-foot ladder is positioned next to a house. If the ladder slips at 7 feet from the house when there is not enough traction, what angle should the ladder make with the ground to avoid slipping?

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Questions & Answers

what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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