Deck 1: Trigonometry

Question 1

The designers of a water park are creating a new slide and have sketched some preliminary drawings.The length of the ladder is a = 26 feet, and its angle of elevation is 60º (see figure).
​ ​
Find the angle of depression from the top of the slide to the end of the slide at the ground in terms of the horizontal distance d the rider travels.
​

A)
B)
C)
D)
E)

Answer 1

Question 2

For the simple harmonic motion described by the trigonometric function, find the least positive value of t for which d = 0. ​ ​

A)
B)
C)
D)
E)

Answer 2

Question 3

A ladder 17 feet long leans against the side of a house.Find the height from the top of the ladder to the ground if the angle of elevation of the ladder is 80º.Approximate the answer to one decimal place. ​

A)19.7 ft
B)20.7 ft
C)18.7 ft
D)16.7 ft
E)17.7 ft

Answer 3

16.7 ft

Question 4

Solve the right triangle shown in the figure for all unknown sides and angles.Round your answers to two decimal places. ​ ​
​ ​

A)
B) ​
C)
D)
E)

Question 5

For the simple harmonic motion described by the trigonometric function, find the least positive value of t for which d = 0. ​
​ ​

A)
B)
C) ​
D) ​
E)

Question 6

A cellular telephone tower that is 150 feet tall is placed on top of a mountain that is 1200 feet above sea level.What is the angle of depression from the top of the tower to a cell phone user who is 6 horizontal miles away and 400 feet above sea level? Round your answer to two decimal places. ​

A)3.72º
B)1.72º
C)5.72º
D)4.72º
E)2.72º

Question 7

You are holding one of the tethers attached to the top of a giant character balloon in a parade.Before the start of the parade the balloon is upright and the bottom is floating approximately 20 feet above ground level.You are standing approximately a = 130 feet ahead of the balloon (see figure). ​ ​
Find the length l of the tether you are holding in terms of h, the height h of the balloon from top to bottom.
​

A)
B) ​
C)
D) ​
E)

Question 8

An engineer erects a 111-foot cellular telephone tower.Find the angle of elevation to the top of the tower at a point on level ground 62 feet from its base.Round your answer to one decimal place. ​

A)60.8º
B)64.8º​
C)63.8º
D)61.8º
E)62.8º

Question 9

The sun is 25º above the horizon.Find the length of a shadow cast by a building that is feet tall (see figure).Approximate the answer to two decimal places. ​ ​

A)281.79 ft
B)278.79 ft
C)279.79 ft
D)280.79 ft
E)282.79 ft

Question 10

A police department has set up a speed enforcement zone on a straight length of highway.A patrol car is parked parallel to the zone, a = 210 feet from one end and b = 130 feet from the other end (see figure). ​ ​
Find the length l of the zone.Round your answer to two decimal places.
​

A)286.98 ft
B)276.98 ft
C)266.98 ft
D)246.98 ft
E)256.98 ft

Question 11

Find the altitude of the isosceles triangle shown in the figure.Round your answer to two decimal places. ​ ​

A)2.00
B)11.00
C)5.50
D)22.00​
E)7.50

Question 12

The sun is 20º above the horizon.Find the length of a shadow cast by a park statue that is 30 feet tall.Approximate the answer to two decimal places. ​

A)82.42 ft
B)84.42 ft
C)85.42 ft
D)83.42 ft
E)86.42 ft

Question 13

The height of an outdoor basketball backboard is feet, and the backboard casts a shadow feet long. Find the angle of elevation of the sun.Round your answer to one decimal place.
​

A)36.0º
B)37.0º
C)35.0º
D)39.0º
E)38.0º

Question 14

A passenger in an airplane at an altitude of a = 20 kilometers sees two towns directly to the east of the plane.The angles of depression to the towns are 28º and 55º (see figure).How far apart are the towns? How far apart are the ships? Round your answer to one decimal place. ​ ​

A)23.6 km
B)24.6 km
C)25.6 km
D)26.6 km
E)27.6 km

Question 15

You are holding one of the tethers attached to the top of a giant character balloon in a parade.Before the start of the parade the balloon is upright and the bottom is floating approximately 20 feet above ground level.You are standing approximately a = 130 feet ahead of the balloon (see figure). ​ ​
Find an expression for the angle of elevation from you to the top of the balloon.
​

A)
B)
C)
D)
E)

Question 16

The length of a shadow of a tree is 120 feet when the angle of elevation of the sun is 33º.Approximate the height of the tree.Approximate the answer to one decimal place. ​

A)77.9 ft
B)81.9 ft
C)79.9 ft
D)78.9 ft
E)80.9 ft

Question 17

The designers of a water park are creating a new slide and have sketched some preliminary drawings.The length of the ladder is a = 24 feet, and its angle of elevation is 60º (see figure).
​ ​
Find the height of the slide.Round your answer to two decimal places.
​

A)About 21.78 ft
B)About 22.78 ft
C)About 23.78 ft
D)About 20.78 ft
E)About 24.78 ft

Question 18

A Global Positioning System satellite orbits a = 13,000 miles above Earth's surface (see figure).Find the angle of depression from the satellite to the horizon.Assume the radius of Earth is 4000 miles.Round your answer to two decimal places. ​ ​

A)77.39º
B)78.39º
C)79.39º
D)76.39º
E)80.39º

Question 19

An observer in a lighthouse a = 340 feet above sea level observes two ships directly offshore. The angles of depression to the ships are 4º and 6.5º (see figure).How far apart are the ships? Round your answer to one decimal place.
​ ​

A)1878.1 ft
B)1879.1 ft
C)1880.1 ft
D)1881.1 ft
E)1882.1 ft

Question 20

You are holding one of the tethers attached to the top of a giant character balloon in a parade.Before the start of the parade the balloon is upright and the bottom is floating approximately 20 feet above ground level.You are standing approximately a = 100 feet ahead of the balloon (see figure).Round your answer to two decimal places. ​ ​
Find the height h of the balloon if the angle of elevation to the top of the balloon is 35º.
​

A)55.02 ft
B)56.02 ft
C)53.02 ft
D)57.02 ft
E)54.02 ft

Question 21

Find the distance y across the flat sides of a hexagonal nut (see figure). cm Round your answer to two decimal places.
​ ​

A) cm
B) cm
C) cm
D) cm
E) cm

Question 22

For the simple harmonic motion described by the trigonometric function, find the maximum displacement. ​ ​

A) ​
B)
C) ​
D)
E)

Question 23

For the simple harmonic motion described by the trigonometric function, find the maximum displacement. ​ ​

A) ​
B)
C)
D)
E)

Question 24

A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches.Its motion (in ideal conditions) is modeled by ​ where y is measured in feet and t is the time in seconds.
What is the period of the oscillations?
​

A)
B)
C)
D)
E)

Question 25

For the simple harmonic motion described by the trigonometric function, find the frequency per second. ​ ​

A)
B)
C)
D)
E)

Question 26

For the simple harmonic motion described by the trigonometric function, find the frequency per second. ​ ​

A)
B)
C)
D)
E)

Question 27

For the simple harmonic motion described by the trigonometric function, find the frequency per second. ​ ​

A)16 ​
B)π ​
C)12
D)2π
E)6

Question 28

Determine the angle between the diagonal of a cube and the diagonal of its base, as shown in the figure, where .Round your answer to one decimal place.
​ ​

A)38.3^o
B)36.3^o
C)39.3^o
D)35.3^o
E)37.3^o

Question 29

Find the angle between two nonvertical lines and .The angle satisfies the equation ​ ​
Where and are slopes of and , respectively.
(Assume that .)
​
​ ​
Round your answer to one decimal place.
​

A)54.1^o
B)​53.1^o
C)55.1^o
D)52.1^o
E)56.1o

Question 30

Find a model for simple harmonic motion satisfying the specified conditions. ​
Displacement (t = 0): 0
Amplitude: 5 centimeters
Period: 2 seconds
​

A)
B)​
C) ​
D) ​
E) ​

Question 31

Find a model for simple harmonic motion satisfying the specified conditions. ​
Displacement (t = 0): 0
Amplitude: 5 meters
Period: 10 seconds
​

A) ​
B) ​
C)​ ​
D) ​
E) ​

Question 32

For the simple harmonic motion described by the trigonometric function, find the maximumvalue of d when . ​ Round your answer to nearest whole number.
​

A)1
B)10
C)6
D)5
E)0

Question 33

For the simple harmonic motion described by the trigonometric function, find the maximum displacement. ​ ​

A) ​
B)
C)
D)
E)

Question 34

A point on the end of a tuning fork moves in simple harmonic motion described by ​ ​
Find ω given that the tuning fork for middle C has a frequency 270 of vibrations per second.
​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 35

Find the angle α between two nonvertical lines and .The angle α satisfies the equation ​ ​
Where and are slopes of and , respectively.
(Assume that .)
​ : 5x - 4y = 5 : x + y = 1
​
Round your answer to one decimal place.
​

A)83.7º
B)84.7º
C)85.7º
D)86.7º
E)87.7º

Question 36

Find the length of the sides of a regular pentagon inscribed in a circle of radius 26 inches.Round your answer to one decimal place. ​

A)31.6 in.
B)33.6 in.
C)30.6 in.
D)32.6 in.
E)34.6 in.

Question 37

For the simple harmonic motion described by the trigonometric function, find the least positive value of t for which ​ ​

A)
B)
C)
D)
E)

Question 38

Find the lengths of all the unknown members of the truss. Round your answer to one decimal place.
​ ​ ​

A)
B)
C)
D)
E)

Question 39

Find the length of the sides of a regular hexagon inscribed in a circle of radius 29 inches. ​

A)30 in.
B)31 in.
C)32 in.
D)33 in.
E)29 in.

Question 40

Determine the angle between the diagonal of a cube and its edge, as shown in the figure, where .Round your answer to one decimal place. ​ ​

A)56.7^o
B)57.7^o
C)55.7^o
D)54.7^o
E)58.7^o ​

Question 41

The angle of elevation of the sun is 26^o.Find the length, l, of a shadow cast by a tree that is 38 feet tall.Round your answer to two decimal places.

A) feet
B) feet
C) feet
D) feet
E) feet

Question 42

Evaluate the expression.Round your result to two decimal places. ​ ​

A)-1.53
B)-0.53
C)2.47
D)0.47
E)1.47

Question 43

If the sides of a rectangular solid are as shown, and , determine the angle, θ, between the diagonal of the base of the solid and the diagonal of the solid.Round answer to two decimal places.

A)21.91^o
B)24.09^o
C)17.21^o
D)26.28^o
E)19.86^o

Question 44

A plane is 52 miles west and 46 miles north of an airport.The pilot wants to fly directly to the airport.What bearing should the pilot take? Answer should be given in degrees and minutes.

A)
B)
C)
D)
E)

Question 45

Evaluate the expression.Round your result to two decimal places. ​ ​

A)4.21
B)2.21
C)3.21
D)0.21
E)1.21

Question 46

N 24 E means 30 degrees north of east.
​

Question 47

Evaluate the expression.Round your result to two decimal places. ​ ​

A)2.12
B)0.12
C)1.12
D)-0.88
E)3.12

Question 48

Evaluate the expression.Round your result to two decimal places. ​ ​

A)1.87
B)-2.13
C)-1.13
D)0.87
E)-0.13

Question 49

After leaving the runway, a plane's angle of ascent is 17^o and its speed is 278 feet per second.How many minutes will it take for the airplane to climb to a height of 11,500 feet? Round answer to two decimal places.

A)0.69 minutes
B)2.36 minutes
C)0.72 minutes
D)1.25 minutes
E)1.81 minutes

Question 50

The Leaning Tower of Pisa is not vertical, but if you know the angle of elevation θ to the top of the tower when you stand 20 feet away from it, you can find its height h using the formula
​ ​

Question 51

If and , determine the value of B.Round to two decimal places.

A)68.55^o ​
B)66.87^o ​
C)28.13^o ​
D)21.45^o ​
E)23.13^o ​

Question 52

A granular substance such as sand naturally settles into a cone-shaped pile when poured from a small aperture.Its height depends on the humidity and adhesion between granules.The angle of elevation of a pile, θ, is called the angle of repose.If the height of a pile of sand is 11 feet and its diameter is approximately 34 feet, determine the angle of repose.Round answer to nearest degree.

A)29^o
B)30^o
C)31^o
D)32^o
E)33^o

Question 53

A land developer wants to find the distance across a small lake in the middle of his proposed development.The bearing from A to B is .The developer leaves point A and travels 74 yards perpendicular to to point C.The bearing from C to point B is .Determine the distance, AB, across the small lake.Round distance to nearest yard.

A)169 yards
B)114 yards
C)139 yards
D)154 yards
E)121 yards

Question 54

Evaluate the expression.Round your result to two decimal places. ​ ​

A)2.71
B)-0.29
C)1.71
D)0.71
E)-1.29

Question 55

After leaving the runway, a plane's angle of ascent is 17^o and its speed is 280 feet per second.How many minutes will it take for the airplane to climb to a height of 11,000 feet? Round your answer to two decimal places. ​

A)1.72 minutes
B)1.19 minutes
C)2.24 minutes
D)0.65 minutes
E)0.68 minutes

Question 56

Find the altitude of the isosceles triangle shown below if and .Round answer to two decimal places.

A)33.18 meters
B)6.23 meters
C)9.98 meters
D)16.59 meters
E)9.42 meters

Question 57

Evaluate the expression.Round your result to two decimal places. ​ ​

A)3.54
B)0.54
C)1.54
D)2.54
E)-0.46

Question 58

If and , determine the value of A.Round to two decimal places.

A)48.19^o ​
B)56.31^o ​
C)33.69^o ​
D)41.81^o ​
E)53.19^o ​

Question 59

Evaluate the expression.Round your result to two decimal places. ​ ​

A)-0.66
B)2.34
C)1.34
D)3.34
E)0.34

Question 60

Find the altitude of the isosceles triangle shown below if and .Round your answer to two decimal places.

A)5.12 centimeters
B)6.25 centimeters
C)3.13 centimeters
D)2.46 centimeters
E)1.38 centimeters

Question 61

Find an algebraic expression that is equivalent to the expression.. ​ ​

A)
B)
C)
D)
E)

Question 62

Use an inverse trigonometric function to write θ as a function of x. ​ ​
A = x + 5
B = 10
​

A)
B)
C)
D)
E)

Question 63

Evaluate the expression.Round your result to two decimal places. ​ ​

A)-3.53
B)0.47
C)-0.53
D)-1.53
E)-2.53

Question 64

A television camera at ground level is filming the lift-off of a space shuttle at a point meters from the launch pad (see figure).Let θ be the angle of elevation to the shuttle and let s be the height of the shuttle.Write θ as a function of s. ​ ​

A)
B)
C)
D)
E)

Question 65

Find an algebraic expression that is equivalent to the expression. ​ ​

A)
B)
C)
D)
E)

Question 66

Use an inverse trigonometric function to write θ as a function of x. ​ ​
A = x
B = 6
​

A)
B)
C)
D)
E)

Question 67

A boat is pulled in by means of a winch located on a dock feet above the deck of the boat (see figure).Let θ be the angle of elevation from the boat to the winch and let s be the length of the rope from the winch to the boat.Write θ as a function of s. ​ ​

A)
B)
C)
D)
E)

Question 68

Evaluate the expression.Round your result to two decimal places. ​ ​

A)1.73
B)-0.27
C)-1.27
D)2.73
E)0.73

Question 69

Use an inverse trigonometric function to write θ as a function of x. ​ ​
A = 5x
B = x + 9
​

A)
B)
C)
D)
E)

Question 70

An airplane flies at an altitude of miles toward a point directly over an observer.Consider θ and x as shown in the figure.Write θ as a function of x. ​ ​

A)
B)
C)
D)
E)

Question 71

Use the properties of inverse trigonometric functions to evaluate the expression. ​ ​

A)
B)
C)0.4
D)
E)

Question 72

Use an inverse trigonometric function to write θ as a function of x. ​ ​
A = x
B = 10
​

A)
B)
C)
D)
E)

Question 73

Evaluate the expression.Round your result to two decimal places. ​ ​

A)0.30
B)1.30
C)4.30
D)3.30
E)2.30

Question 74

Use an inverse trigonometric function to write θ as a function of x.​ ​ ​

A)
B)
C)
D)
E)

Question 75

Evaluate the expression.Round your result to two decimal places. ​ ​

A)-0.53
B)-1.53
C)0.47
D)-3.53
E)-2.53

Question 76

Evaluate the expression.Round your result to two decimal places. ​ ​

A)2.20
B)0.20
C)3.20
D)-0.80
E)1.20

Question 77

Use an inverse trigonometric function to write θ as a function of x. ​ ​
A = x + 6
B = 2
​

A)
B)
C)
D)
E)

Question 78

Find an algebraic expression that is equivalent to the expression.. ​ ​

A)
B)
C)
D)
E)

Question 79

Use the properties of inverse trigonometric functions to evaluate the expression. ​ ​

A)1
B)-1
C)
D)0
E)

Question 80

Use the properties of inverse trigonometric functions to evaluate the expression. ​ ​

A)
B)
C)
D)
E)40