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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

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
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
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard

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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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