Deck 7: Applications of Trigonometric Functions

Question 1

Solve the problem.

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

Answer 1

C

Question 2

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.

Answer 2

B

Question 3

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
Find sin B when b = 4 and c = 7.

Answer 3

A

Question 4

Solve the problem.

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

Question 5

Solve the right triangle using the information given. Round answers to two decimal places, if necessary.

Question 6

Solve the right triangle using the information given. Round answers to two decimal places, if necessary.

Question 7

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.

Question 8

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.

Question 9

Solve the problem.

Question 10

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.

Question 11

Solve the problem.
A surveyor is measuring the distance across a small lake. He has set up his transit on one side of the lake 90 feet from a piling that is directly across from a pier on the other side of the lake. From his transit, the angle between
The piling and the pier is 55°. What is the distance between the piling and the pier to the nearest foot?

A) 74 ft
B) 63 ft
C) 129 ft
D) 52 ft

Question 12

Solve the right triangle using the information given. Round answers to two decimal places, if necessary.

Question 13

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
Find cos A when a = 3 and b = 7.

Question 14

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.

Question 15

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
Find sin A when a = 8 and b = 7.

Question 16

Solve the right triangle using the information given. Round answers to two decimal places, if necessary.

Question 17

Solve the problem.

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

Question 18

Solve the problem.
A radio transmission tower is 210 feet tall. How long should a guy wire be if it is to be attached 13 feet from the top and is to make an angle of 20° with the ground? Give your answer to the nearest tenth of a foot.

A) 614.0 ft
B) 209.6 ft
C) 223.5 ft
D) 576.0 ft

Question 19

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.

Question 20

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.

Question 21

Solve the problem.
Yosemite Falls in California consists of three sections: Upper Yosemite Fall (by itself one of the ten highest
waterfalls in the world), the Middle Cascade, and Lower Yosemite Fall. From a footbridge across the creek 2500
feet from the falls, the angles of elevation to the top and bottom of Upper Yosemite Fall are 45.74° and 24.42°,
respectively. How high is the total series of three falls? How high is Upper Yosemite Fall? Round your answers
to the nearest foot.

Question 22

Solve the triangle.

Question 23

Solve the problem.
Two hikers on opposite sides of a canyon each stand precisely 525 meters above the canyon floor. They each sight a landmark on the canyon floor on a line directly between them. The angles of depression from each hiker
To the landmark meter are 37° and 21°. How far apart are the hikers? Round your answer to the nearest whole
Meter.

A) 2064 m
B) 2065 m
C) 1064 m
D) 2063 m

Question 24

Solve the triangle.

Question 25

Solve the problem.
A twenty-five foot ladder just reaches the top of a house and forms an angle of 41.5° with the wall of the house. How tall is the house? Round your answer to the nearest 0.1 foot.

A) 18.7 ft
B) 18.8 ft
C) 18.6 ft
D) 19 ft

Question 26

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no
triangle at all. Solve any triangle(s) that results.

Question 27

Solve the triangle.

Question 28

Solve the problem.
In 1838, the German mathematician and astronomer Friedrich Wilhelm Bessel was the first person to calculate
the distance to a star other than the Sun. He accomplished this by first determining the parallax of the star, 61
Cygni, at 0.314 arc seconds (Parallax is the change in position of the star measured against background stars as
Earth orbits the Sun. See illustration.) If the distance from Earth to the Sun is about 150,000,000 km and determine the distance d from Earth to 61 Cygni using Bessel's figures.

Question 29

Solve the problem.
Given a triangle with a = 9, b = 11, , what is (are) the possible length(s) of c? Round your answer to two decimal places.

A) 16.42 or 2.44
B) 16.42 or 3.41
C) 6.61
D) 14.21

Question 30

Solve the problem.
A building 200 feet tall casts a 40 foot long shadow. If a person looks down from the top of the building, what is the measure of the angle between the end of the shadow and the vertical side of the building (to the nearest
Degree)? (Assume the person's eyes are level with the top of the building.)

A) 78°
B) 11°
C) 12°
D) 79°

Question 31

Solve the problem.
A forest ranger at Lookout A sights a fire directly north of her position. Another ranger at Lookout B, exactly 2 kilometers directly west of A, sights the same fire at a bearing of N41.2°E. How far is the fire from Lookout A?
Round your answer to the nearest 0.01 km.

A) 2.32 km
B) 2.25 km
C) 2.18 km
D) 2.28 km

Question 32

Solve the triangle.

Question 33

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no
triangle at all. Solve any triangle(s) that results.

Question 34

Solve the problem.
A straight trail with a uniform inclination of 18° leads from a lodge at an elevation of 800 feet to a mountain lake at an elevation of 7,100 feet. What is the length of the trail (to the nearest foot)?

A) 7,465 ft
B) 22,976 ft
C) 20,387 ft
D) 6,624 ft

Question 35

Solve the problem.
A photographer points a camera at a window in a nearby building forming an angle of 42° with the camera platform. If the camera is 52 m from the building, how high above the platform is the window, to the nearest
Hundredth of a meter?

A) 0.9 m
B) 1.11 m
C) 46.82 m
D) 57.75 m

Question 36

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no
triangle at all. Solve any triangle(s) that results.

Question 37

Solve the problem.
John (whose line of sight is 6 ft above horizontal) is trying to estimate the height of a tall oak tree. He first measures the angle of elevation from where he is looking as 35°. He walks 30 feet closer to the tree and finds
That the angle of elevation has increased by 12°. Estimate the height of the tree rounded to the nearest whole
Number.

A) 67 ft
B) 86 ft
C) 61 ft
D) 90 ft

Question 38

Solve the problem.
A sailboat leaves port on a bearing of S72°W. After sailing for two hours at 12 knots, the boat turns 90° toward the south. After sailing for three hours at 9 knots on this course, what is the bearing to the ship from port?
Round your answer to the nearest 0.1°.

A) S24.6°W
B) N23.6°E
C) S23.6°W
D) N24.6°E

Question 39

Solve the problem.
From the edge of a 1000-foot cliff, the angles of depression to two cars in the valley below are 21° and 28°. How far apart are the cars? Round your answers to the nearest 0.1 ft.

A) 724.5 ft
B) 713.4 ft
C) 724.4 ft
D) 714.4 ft

Question 40

Solve the problem.
A tree casts a shadow of 26 meters when the angle of elevation of the sun is 24°. Find the height of the tree to the nearest meter.

A) 10 m
B) 11 m
C) 12 m
D) 13 m

Question 41

Solve the problem.

Question 42

Solve the problem.

Question 43

Solve the problem.
It is 4.7 km from Lighthouse A to Port B. The bearing of the port from the lighthouse is N73°E. A ship has sailed due west from the port and its bearing from the lighthouse is N31°E. How far has the ship sailed from the port?
Round your answer to the nearest 0.1 km.

A) 3.5 km
B) 3.7 km
C) 3.1 km
D) 2.7 km

Question 44

Solve the problem.
Two surveyors 180 meters apart on the same side of a river measure their respective angles to a point between
them on the other side of the river and obtain 54° and 68°. How far from the point (line-of-sight distance) is
each surveyor? Round your answer to the nearest 0.1 meter.

Question 45

Solve the problem.

Question 46

Solve the problem.
A rocket tracking station has two telescopes A and B placed 1.3 miles apart. The telescopes lock onto a rocket and transmit their angles of elevation to a computer after a rocket launch. What is the distance to the rocket
From telescope B at the moment when both tracking stations are directly east of the rocket telescope A reports an
Angle of elevation of 27° and telescope B reports an angle of elevation of 52°?

A) 2.42 mi
B) 1.4 mi
C) 0.75 mi
D) 2.26 mi

Question 47

Solve the triangle. Find the angles α and β first.

Question 48

Solve the problem.
A pier 1250 meters long extends at an angle from the shoreline. A surveyor walks to a point 1500 meters down
the shoreline from the pier and measures the angle formed by the ends of the pier. If is found to be 53°. What
acute angle (correct to the nearest 0.1°) does the pier form with the shoreline? Is there more than one possibility?
If so, how can we know which is the correct one?

Question 49

Solve the triangle. Find the angles α and β first.

Question 50

Solve the problem.

Question 51

Solve the problem.
A flagpole is perpendicular to the horizontal but is on a slope that rises 10° from the horizontal. The pole casts a 43-foot shadow down the slope and angle of elevation of the sun measured from the slope is 36°. How tall is the
Pole? Round your answer to the nearest 0.1 foot.

A) 33.5 ft
B) 36.4 ft
C) 35.4 ft
D) 36.2 ft

Question 52

Solve the problem.
A ship sailing parallel to shore sights a lighthouse at an angle of 12° from its direction of travel. After traveling 3 miles farther, the angle is 20°. At that time, how far is the ship from the lighthouse?

A) 1.82 mi
B) 4.48 mi
C) 7.37 mi
D) 3 mi

Question 53

Solve the problem.
A ship at sea, the Admiral, spots two other ships, the Barstow and the Cauldrew and measures the angle between them at be 45°. They radio the Barstow and by comparing known landmarks, the distance between the
The Admiral and the Barstow is found to be 323 meters. The Barstow reports an angle of 59° between the
Admiral and the Cauldrew. To the nearest meter, what is the distance between the Barstow and the Cauldrew?

A) 81 m
B) 235 m
C) 49 m
D) 266 m

Question 54

Solve the problem.
A guy wire to the top of a tower makes an angle of 64° with the level ground. At a point 26 feet farther from the base of the tower and in line with the base of the wire, the angle of elevation to the top of the tower is 27°. What
Is the length of the guy wire?

A) 38.83 ft
B) 13.13 ft
C) 51.47 ft
D) 19.61 ft

Question 55

Solve the problem.

Question 56

Solve the problem.
A surveyor standing 56 meters from the base of a building measures the angle to the top of the building and finds it to be 39°. The surveyor then measures the angle to the top of the radio tower on the building and finds
That it is 49°. How tall is the radio tower?

A) 19.07 m
B) 9.87 m
C) 6.78 m
D) 7.02 m

Question 57

Solve the triangle. Find the angles α and β first.

Question 58

Solve the problem.

Question 59

Solve the problem.
An airplane is sighted at the same time by two ground observers who are 5 miles apart and both directly west of the airplane. They report the angles of elevation as 14° and 24°. How high is the airplane?

A) 2.03 mi
B) 2.83 mi
C) 1.21 mi
D) 4.76 mi

Question 60

Solve the triangle. Find the angles α and β first.

Question 61

Solve the problem.
A famous golfer tees off on a straight 390 yard par 4 and slices his drive to the right. The drive goes 280 yards from the tee. Using a 7-iron on his second shot, he hits the ball 180 yards and it lands inches from the hole. How
Many degrees (to the nearest degree) to the right of the line from the tee to the hole did he slice his drive?

A) 25°
B) 53°
C) 41°
D) 114°

Question 62

Find the area of the triangle. If necessary, round the answer to two decimal places.
Find the area of the shaded portion ( see illustration) of a circle of radius 25 cm, formed by a central angle of
115°. Round your answer to the nearest square cm. [Hint: Subtract the area of the triangle from the area of the sector of the circle to obtain the area of the shaded
portion.]

Question 63

Find the area of the triangle. If necessary, round the answer to two decimal places.
A new homeowner has a triangular-shaped back yard. Two of the three sides measure 65 ft and 80 ft and form an included angle of 125°. The owner wants to approximate the area of the yard, so that he can determine the
Amount of fertilizer and grass seed to be purchased. Find the area of the yard rounded to the nearest square
Foot.

A) 5200 sq. ft
B) 2130 sq. ft
C) 4260 sq. ft
D) 2129 sq. ft

Question 64

Solve the triangle. Find the angles α and β first.

Question 65

Find the area of the triangle. If necessary, round the answer to two decimal places.

A) 6.89
B) 26.43
C) 24.43
D) 8.89

Question 66

Find the area of the triangle. If necessary, round the answer to two decimal places.

A) 88.80
B) 35.46
C) 70.92
D) 141.84

Question 67

Find the area of the triangle. If necessary, round the answer to two decimal places.

A) 22.82
B) 11.41
C) 12.24
D) 5.71

Question 68

Solve the problem.
A famous golfer tees off on a long, straight 471 yard par 4 and slices his drive 18° to the right of the line from tee to the hole. If the drive went 290 yards, how many yards will the golfer's second shot have to be to reach the
Hole?

A) 752.2 yd
B) 214.8 yd
C) 660.2 yd
D) 419.6 yd

Question 69

Find the area of the triangle. If necessary, round the answer to two decimal places.

A) 26.80
B) 27.01
C) 3.29
D) 53.60

Question 70

Find the area of the triangle. If necessary, round the answer to two decimal places.

A) 3.52
B) 3.50
C) 3.51
D) 3.53

Question 71

Solve the problem.
Island A is 150 miles from island B. A ship captain travels 250 miles from island A and then finds that he is off course and 160 miles from island B. What angle, in degrees, must he turn through to head straight for island B?
Round the answer to two decimal places. (Hint: Be careful to properly identify which angle is the turning
Angle.)

A) 145.08°
B) 110.17°
C) 55.08°
D) 34.92°

Question 72

Find the area of the triangle. If necessary, round the answer to two decimal places.

A) 3.73
B) 3.03
C) 1.75
D) 12.12

Question 73

Solve the problem.
The distance from home plate to dead center field in a certain baseball stadium is 402 feet. A baseball diamond is a square with a distance from home plate to first base of 90 feet. How far is it from first base to dead center
Field?

A) 344.3 ft
B) 470 ft
C) 379.6 ft
D) 327.2 ft

Question 74

Solve the problem.
A plane takes off from an airport on the bearing S29°W. It continues for 20 minutes then changes to bearing S52°W and flies for 2 hours 20 minutes on this course then lands at a second airport. If the plane' s speed is 420
Mph, how far from the first airport is the second airport? Round your answer correct to the nearest mile.

A) 1011 mi
B) 1111 mi
C) 1110 mi
D) 1010 mi

Question 75

Solve the problem.
A ladder leans against a building that has a wall slanting away from the ladder at an angle of 96° with the ground. If the bottom of the ladder is 23 feet from the base of the wall and it reaches a point 52 feet up the wall,
How tall is the ladder to the nearest foot?

A) 61 ft
B) 60 ft
C) 59 ft
D) 58 ft

Question 76

Solve the problem.

A) 65°
B) 50°
C) 60°
D) 62°

Question 77

Find the area of the triangle. If necessary, round the answer to two decimal places.
Penrose tiles are formed from a rhombus WXYZ with sides of length 1 and interior angles 72° and 108°. (Refer to
the illustration.) A point O is chosen on the diagonal 1 unit from Y. Line segments OX and OZ are drawn to the other
vertices. The two resulting tiles are called a kite (figure OXYZ) and a dart (figure OXWZ). Find the area of the kite tile
and the dart tile, correct to the nearest 0.01.

Question 78

Solve the problem.
In flying the 81 miles from Champaign to Peoria, a student pilot sets a heading that is 9° off course and maintains an average speed of 96 miles per hour. After 15 minutes, the instructor notices the course error and
Tells the student to correct his heading. Through what angle will the plane move to correct the heading and how
Many miles away is Peoria when the plane turns?

A) 12.7°; 72.23 mi
B) 12.7°; 57.42 mi
C) 167.3°; 72.23 mi
D) 167.3°; 57.42 mi

Question 79

Solve the triangle. Find the angles α and β first.

Question 80

Solve the problem.
Two points A and B are on opposite sides of a building. A surveyor selects a third point C to place a transit. Point C is 52 feet from point A and 68 feet from point B. The angle ACB is 53°. How far apart are points A and
B?

A) 97.2 ft
B) 107.6 ft
C) 72.1 ft
D) 55.4 ft