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