Deck 2: Acute Angles and Right Triangles

Question 1

Find the reference angle for the given angle.
39°

A)51°
B)39°
C)129°
D)141°

Answer 1

B

Question 2

Evaluate the function requested. Write your answer as a fraction in lowest terms.

Answer 2

B

Question 3

Solve the problem.
Find the exact value of x in the figure.

Answer 3

D

Question 4

Find all values of , if is in the interval [0, 360°) and has the given function value.
tan ϴ = 1

A)45° and 225°
B)135° and 225°
C)45° and 315°
D)225° and 315°

Question 5

Find the reference angle for the given angle.
129°

A)51°
B)49°
C)39°
D)61°

Question 6

Solve the problem.
From a boat on the lake, the angle of elevation to the top of a cliff is 13°11'. If the base of the cliff is 1683 feet from the boat, how high is the cliff (to the nearest foot)?

A)394 ft
B)407 ft
C)397 ft
D)404 ft

Question 7

Convert the radian measure to degrees. Round to the nearest hundredth if necessary.
Radio direction finders are set up at points A and B, 8.68 mi apart on an east-west line. From A it is found that the bearing of a signal from a transmitter is N 54.3°E, while from B it is N 35.7°W. Find
The distance of the transmitter from B, to the nearest hundredth of a mile.

A)4.57 mi
B)7.05 mi
C)5.07 mi
D)7.55 mi

Question 8

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length using
the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle. Rationalize
the denominator if applicable.

Question 9

Solve the problem.

Question 10

Find the reference angle for the given angle.
-403°

A)47°
B)137°
C)43°
D)133°

Question 11

Solve the problem.
A conservation officer needs to know the width of a river in order to set instruments correctly for a study of pollutants in the river. From point A, the conservation officer walks 100 feet downstream
And sights point B on the opposite bank to determine that ϴ = 30° (see figure). How wide is the river
(round to the nearest foot)?

Question 12

Find the sign of the following.

Question 13

The number represents an approximate measurement. State the range represented by the measurement.
21 ft

A)20.5 ft to 21.5 ft
B)20.75 ft to 21.25 ft
C)20.9 ft to 21.1 ft
D)20 ft to 22 ft

Question 14

Solve the problem.
From the top of a vertical tower, 319 feet above the the surface of the earth, the angle of depression to a doghouse is 23° 3´. How far is it from the doghouse to the foot of the tower? Round your
Answer to the hundredths place when necessary.

A)749.7 ft
B)852.3 ft
C)734.11 ft
D)762.1 ft

Question 15

Without using a calculator, give the exact trigonometric function value with rational denominator.
tan 60°

Question 16

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary.
sin 304°53´

A)-0.8103183
B)-0.8203183
C)-0.8213183
D)-0.8303183

Question 17

Solve the problem.
A fire is sighted due west of lookout A. The bearing of the fire from lookout B, 14.1 miles due south of A, is N 32°57'W. How far is the fire from B (to the nearest tenth of a mile)?

A)18.8 mi
B)19.8 mi
C)16.8 mi
D)17.8 mi

Question 18

Find the exact value of the expression.
cs c 330°

Question 19

Without using a calculator, give the exact trigonometric function value with rational denominator.
cos 30°

Question 20

Solve the problem.
When sitting atop a tree and looking down at his pal Joey, the angle of depression of Mack's line of sight is 51°11'. If Joey is known to be standing 32 feet from the base of the tree, how tall is the tree
(to the nearest foot)?

A)40 ft
B)46 ft
C)44 ft
D)42 ft

Question 21

Determine whether the statement is true or false.
cos 195° = 2 cos 225° cos 30° + 2 sin 225° sin 30°

Question 22

Solve the right triangle. If two sides are given, give angles in degrees and minutes.

Question 23

Decide whether the statement is true or false.
tan 23° < tan 4°

Question 24

Solve the problem.
An airplane travels at 165 km/h for 3 hr in a direction of 174° from a local airport. At the end of this time, how far east of the airport is the plane (to the nearest kilometer)?

A)4736 km
B)492 km
C)4710 km
D)52 km

Question 25

Solve the problem.
A ship travels 82 km on a bearing of 24°, and then travels on a bearing of 114° for 162 km. Find the distance from the starting point to the end of the trip, to the nearest kilometer.

A)244 km
B)33 km
C)182 km
D)75 km

Question 26

Solve the problem.

Question 27

Without using a calculator, give the exact trigonometric function value with rational denominator.
sin 60°

Question 28

Find all values of , if is in the interval [0, 360°) and has the given function value.

Question 29

Solve the problem.
Bob is driving along a straight and level road straight toward a mountain. At some point on his trip he measures the angle of elevation to the top of the mountain and finds it to be 21° 32´. He then
Drives 1 mile (1 mile = 5280 ft)more and measures the angle of elevation to be 35° 7´. Find the
Height of the mountain to the nearest foot.

A)4847 ft
B)47,467 ft
C)474,672 ft
D)4747 ft

Question 30

Find the exact value of the expression.
cos 210°

Question 31

Without using a calculator, give the exact trigonometric function value with rational denominator.
sec 30°

Question 32

Find a value of in [0°, 90°] that satisfies the statement. Leave answer in decimal degrees rounded to seven decimal
places, if necessary.
sec ϴ = 2.7259498

A)21.5211475°
B)68.4788525°
C)54.5922708°
D)20.1452902°

Question 33

Write the function in terms of its cofunction. Assume that any angle in which an unknown appears is an acute angle.

Question 34

Evaluate.

Question 35

Find a solution for the equation. Assume that all angles are acute angles.

Question 36

Find the exact value of the expression.
sec 3105°

Question 37

Find the sign of the following.
sec (-ϴ), given that ϴ is in the interval (180°, 270°).

A)positive
B)negative

Question 38

Solve the problem.
Find a formula for the area of the figure in terms of s.

Question 39

Evaluate.

Question 40

Find all values of , if is in the interval [0, 360°) and has the given function value.

Question 41

Determine whether the statement is true or false.
sin 60° = 2 sin 30°

Question 42

Solve the problem.

Question 43

Decide whether the statement is true or false.
sin 86° > sin 24°

Question 44

Solve the problem.
The angle of elevation from a point on the ground to the top of a tower is 39° 16´. The angle of elevation from a point 114 feet farther back from the tower is 24° 38´. Find the height of the tower.
Round to the nearest foot.

A)130 ft
B)115 ft
C)119 ft
D)1190 ft

Question 45

Find all values of , if is in the interval [0, 360°) and has the given function value.

Question 46

Determine whether the statement is true or false.

Question 47

Solve the problem.
From a boat on the river below a dam, the angle of elevation to the top of the dam is 14°57'. If the dam is 1286 feet above the level of the river, how far is the boat from the base of the dam (to the
Nearest foot)?

A)4816 ft
B)4796 ft
C)4806 ft
D)4786 ft

Question 48

Write the function in terms of its cofunction. Assume that any angle in which an unknown appears is an acute angle.
cot 31.1°

A)tan 31.1°
B)tan 148.9°
C)cot 58.9°
D)tan 58.9°

Question 49

Solve the problem.
Find h as indicated in the figure. Round to the nearest meter.

Question 50

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length using
the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle. Rationalize
the denominator if applicable.

Question 51

Solve the right triangle.

Question 52

Find the exact value of the expression.

Question 53

Use a calculator to decide whether the statement is true or false.
sin (2 ·30°)= 2 · sin 30°

Question 54

Find a solution for the equation. Assume that all angles are acute angles.

Question 55

Without using a calculator, give the exact trigonometric function value with rational denominator.
csc 45°

Question 56

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary.
2 sin 44° 14' cos 44° 14' - sin 88° 28'

A)1
B)0
C)2
D)-1

Question 57

Solve the problem.
Find a formula for the area of the figure in terms of s.

Question 58

The number represents an approximate measurement. State the range represented by the measurement.
17.3 m

A)17.29 m to 17.31 m
B)17.275 m to 17.325 m
C)17.25 m to 17.35 m
D)16.3 m to 18.3 m

Question 59

Solve the right triangle. If two sides are given, give angles in degrees and minutes.

Question 60

Solve the problem.
Find h as indicated in the figure. Round to the nearest foot.

Question 61

An observer for a radar station is located at the origin of a coordinate system. For the point given, find the bearing of an
airplane located at that point. Express the bearing using both methods.
(0, )

Question 62

Find the sign of the following.

Question 63

Find the sign of the following.
cos (ϴ + 180°), given that ϴ is in the interval (90°, 180°).

A)positive
B)negative

Question 64

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary.
cos 47° cos 133° - sin 47° sin 133°

A)2
B)0
C)1
D)-1

Question 65

Determine whether the statement is true or false.
sin 390° = sin 120° cos 270° + cos 120° sin 270°

Question 66

Solve the right triangle.
A = 19° 17´, c = 287 ft, C = 90° Round side lengths to two decimal places, if necessary.

A)B = 70° 42´, a = 97.98 ft, b = 272.1 ft
B)B = 70° 43´, a = 94.21 ft , b = 270.01 ft
C)B = 71° 42´, a = 94.78 ft, b = 266.9 ft
D)B = 70° 43´, a = 94.78 ft, b = 270.9 ft

Question 67

Solve the right triangle.
a = 3.6 m, B = 30.3°, C = 90° Round values to one decimal place.

A)A = 59.7°, b = 4.4 m, c = 5.7 m
B)A = 59.7°, b = 2.1 m, c = 4.2 m
C)A = 59.7°, b = 1 m, c = 3.7 m
D)A = 59.7°, b = 4.4 m, c = 4.2 m

Question 68

Solve the problem.
Any offset between a stationary radar gun and a moving target creates a "cosine effect" that reduces the radar mileage reading by the cosine of the angle between the gun and the vehicle. That is, the
Radar speed reading is the product of the actual reading and the cosine of the angle. Find the radar
Reading to the nearest hundredth for the auto shown in the figure.

Question 69

Find the sign of the following.

Question 70

Solve the right triangle. If two sides are given, give angles in degrees and minutes.

Question 71

Solve the problem.
The index of refraction for air, Ia, is 1.0003. The index of refraction for water, Iw, is 1.3. If

Question 72

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length using
the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle. Rationalize
the denominator if applicable.

Question 73

Find a solution for the equation. Assume that all angles are acute angles.
tan(3ϴ + 16°)= cot(ϴ + 4°)

A)19°
B)16.5°
C)17.5°
D)17°

Question 74

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary.
cos 39° 5' cos 50° 55' - sin 39° 5' sin 50° 55'

A)2
B)-1
C)0
D)1

Question 75

Solve the right triangle. If two sides are given, give angles in degrees and minutes.

Question 76

Find a value of in [0°, 90°] that satisfies the statement. Leave answer in decimal degrees rounded to seven decimal
places, if necessary.
csc ϴ = 1.7437345

A)60.1665137°
B)34.9933809°
C)29.8334863°
D)55.0066191°

Question 77

Determine whether the statement is true or false.
cos 60° = cos 180° - cos 120°

Question 78

Solve the problem.
Any offset between a stationary radar gun and a moving target creates a "cosine effect" that reduces the radar mileage reading by the cosine of the angle between the gun and the vehicle. That is, the
Radar speed reading is the product of the actual reading and the cosine of the angle. Find the radar
Reading to the nearest hundredth for the auto shown in the figure.

Question 79

Solve the problem.
From a balloon 1037 feet high, the angle of depression to the ranger headquarters is 58°41'. How far is the headquarters from a point on the ground directly below the balloon (to the nearest foot)?

A)626 ft
B)621 ft
C)636 ft
D)631 ft

Question 80

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length using
the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle. Rationalize
the denominator if applicable.