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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir 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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