Trigonometry Word Problems
1
Name:_____________________
1.)
A hiker sets out to climb a mountain with a 2,5000 𝑓𝑡 vertical ascent as shown in the diagram below. If
the angle of incline, 𝛼 = 8° what is to total distance he has to travel to reach the summit, to the nearest whole foot?
A.
B.
C.
D.
E.
348 𝑓𝑡
2524 𝑓𝑡
2476 𝑓𝑡
17,963 𝑓𝑡
20,000 𝑓𝑡
2.)
An airplane departs Miami every morning and flies to Boston and then from Boston to Los Angeles as
shown in the diagram below. If the plane takes off from Los Angeles at an angle, 𝛼 = 30° and the distance from
Boston to Los Angeles is approximately 3,000 miles, what is the distance, to the nearest mile, between Miami and
Los Angeles?
A.
B.
C.
D.
E.
1000 miles
1500 miles
2000 miles
2598 miles
3000 miles
3.)
A man who is 5 feet tall looks up at an angle of 15°and sees a flag as shown in the diagram below. If the
man is standing 280 𝑓𝑡 from the base of the flag pole (𝑏 = 280 𝑓𝑡) then what is the value of 𝑎 in the diagram?
A.
B.
C.
D.
E.
72 ft
75 ft
200 ft
270 ft
1044 ft
Trigonometry Word Problems
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4.)
A tree casts a shadow near dusk as shown in the following diagram. If the tree is 50 feet long and the sun
sits 25° above the horizon (𝛼 = 25°), how long is the shadow cast by the tree, to the nearest foot?
A.
B.
C.
D.
E.
21 ft
23 ft
45 ft
90 ft
107 ft
5.)
A sailboat is traveling due west as shown in the diagram below. A man on the front of the boat looks up at
11 A.M in the afternoon and looks up at an angle, 𝜃 = 20°, and sees a hot air balloon attached to the ocean floor by
a string. 2 hours later the man realizes they are directly under the balloon, which hasn’t moved, and it is exactly
2,500 meters above the water line. Approximately how fast is the boat traveling, in kilometers per hour?
A.
B.
C.
D.
E.
0.45 km/hr
0.9 km/hr
3.1 km/hr
3.4 km/hr
6.8 km/hr
6.)
A hot air balloon sits in a fixed position as a boat travels due westward towards the balloon, as in the
diagram below. A sailor notices that the balloon appears higher in the sky as the boat travels west. If the balloon is
2 kilometers above the ocean, and 𝜃! = 15° at 11 AM and 𝜃! = 30° at 3 PM how fast is the boat traveling, to the
nearest kilometer per hour?
A.
B.
C.
D.
E.
1 km/hr
2 km/hr
3 km/hr
4 km/hr
5 km/hr
Trigonometry Word Problems
7.)
An airplane has a route map as shown below. If the plane travels 1640 miles and arrives in Albuquerque
from Detroit and then departs at a 25° angle as shown on the map below (𝜃 = 25°) towards Atlanta, what is the
approximate distance between Albuquerque and Atlanta?
A.
B.
C.
D.
E.
685 miles
1485 miles
1810 miles
2000 miles
3880 miles
8.)
Three US Cities are shown below on a map that is approximately to scale. If the angle centered at
Washington DC is 50° and the distance from Chicago to Monroe is 840 miles, which of the following gives the
closest approximation to the distance between Washington DC and Monroe?
A.
B.
C.
D.
E.
690 miles
750 miles
800 miles
1090 miles
1300 miles
9.)
A triangle is drawn between three major US cities on a map drawn approximately to scale below. If the
angle centered at New York is 15°and the distance between Portland and New York is 2,920 miles, which of the
following is the closest approximation to the distance between New York and San Francisco?
A.
B.
C.
D.
E.
750 miles
1040 miles
1500 miles
2820 miles
3020 miles
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Trigonometry Word Problems
10.)
Four major US cities are shown on a map that is approximately drawn to scale below. If the distance
between Boston and Los Angeles is approximately 3000 miles, what is the best approximation to the distance
between New Orleans and Boston?
A.
B.
C.
D.
E.
1385 miles
1500 miles
1520 miles
1560 miles
1600 miles
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Trigonometry Word Problems
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Teacher Instructions: These questions were created for ACT and SAT preparation, but
they are also useful practice for any multiple choice, standardized test such as the CST or
STAR testing. Each question has a trap answer, and that information is encoded above
each question.
I would suggest that you require higher achieving students to identify each trap answer
and give their reasons for identifying it as such.
Above each question, information is encoded as this: Question Number/Easy Medium or
Hard/SubTopic/Topic/Correct Answer/Trap Answer
Elementary Trigonometry
1/EASY/Elementary Trigonometry /Trig Word Problems/D (Correct Answer)/B (Trap Answer)
A hiker sets out to climb a mountain with a 2,5000 𝑓𝑡 vertical ascent as shown in the diagram below. If the angle of
incline, 𝛼 = 8° what is to total distance he has to travel to reach the summit, to the nearest whole foot?
A.
B.
C.
D.
E.
348 𝑓𝑡
2524 𝑓𝑡
2476 𝑓𝑡
𝟏𝟕, 𝟗𝟔𝟑 𝒇𝒕
20,000 𝑓𝑡
Trap Answer Rationale: B is if they use Cosine on accident, only I put a rounding error in A and not in C b/c I’ve
noticed they do that sometimes on the ACT as a signal that you should go back and check your work.
2/E/Elementary Trigonometry /Trig Word Problems/D/B
An airplane departs Miami every morning and flies to Boston and then from Boston to Los Angeles as shown in the
diagram below. If the plane takes off from Los Angeles at an angle, 𝛼 = 30° and the distance from Boston to Los
Angeles is approximately 3,000 miles, what is the distance, to the nearest mile, between Miami and Los Angeles?
A. 1000 miles
B. 1500 miles
Trigonometry Word Problems
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C. 2000 miles
D. 𝟐𝟓𝟗𝟖 miles
E. 3000 miles
Trap Answer Rationale: B is if they use cos instead of sin
3/E/Elementary Trigonometry /Trig Word Problems/B/E
A man who is 5 feet tall looks up at an angle of 15°and sees a flag as shown in the diagram below. If the man is
standing 280 𝑓𝑡 from the base of the flag pole (𝑏 = 280 𝑓𝑡) then what is the value of 𝑎 in the diagram?
A.
B.
C.
D.
E.
72 ft
75 ft
200 ft
270 ft
1044 ft
Trap Answer Rationale: E doesn’t make sense intuitively, but if they do 280/tan 15 that is what they get. The other
answers are 280 sin 15 and 280 cos 15
4/E/ Elementary Trigonometry /Trig Word Problems /E/B
A tree casts a shadow near dusk as shown in the following diagram. If the tree is 50 feet long and the sun sits 25°
above the horizon (𝛼 = 25°), how long is the shadow cast by the tree, to the nearest foot?
A.
B.
C.
D.
E.
21 ft
23 ft
45 ft
90 ft
107 ft
Trap Answer Rationale: B is the answer to 50 * tan 25
5/H/ Elementary Trigonometry /Trig Word Problems /D/E
A sailboat is traveling due west as shown in the diagram below. A man on the front of the boat looks up at 11 A.M
in the afternoon and looks up at an angle, 𝜃 = 20°, and sees a hot air balloon attached to the ocean floor by a string.
2 hours later the man realizes they are directly under the balloon, which hasn’t moved, and it is exactly 2,500 meters
above the water line. Approximately how fast is the boat traveling, in kilometers per hour?
Trigonometry Word Problems
A.
B.
C.
D.
E.
7
0.45 km/hr
0.9 km/hr
3.1 km/hr
𝟑. 𝟒 km/hr
6.8 km/hr
Trap Answer Rationale: E is a trap b/c it is the total distance traveled in 2 hours, so the km/hr needs to be divided by
2.
6/H/ Elementary Trigonometry /Trig Word Problems /A/D
A hot air balloon sits in a fixed position as a boat travels due westward towards the balloon, as in the diagram below.
A sailor notices that the balloon appears higher in the sky as the boat travels west. If the balloon is 2 kilometers
above the ocean, and 𝜃! = 15° at 11 AM and 𝜃! = 30° at 3 PM how fast is the boat traveling, to the nearest
kilometer per hour?
A.
B.
C.
D.
E.
1 km/hr
2 km/hr
3 km/hr
4 km/hr
5 km/hr
Trap Answer Rationale: The boat travels 4 kilometers in 4 hours, so D is the total distance.
7/E/ Elementary Trigonometry /Trig Word Problems /B/A
An airplane has a route map as shown below. If the plane travels 1640 miles and arrives in Albuquerque from
Detroit and then departs at a 25° angle as shown on the map below (𝜃 = 25°) towards Atlanta, what is the
approximate distance between Albuquerque and Atlanta?
Trigonometry Word Problems
A.
B.
C.
D.
E.
8
685 miles
1485 miles
1810 miles
2000 miles
3880 miles
Trap Answer Rationale: A is the distance between Atlanta and Detroit
8/M/ Elementary Trigonometry /Trig Word Problems /D/A
Three US Cities are shown below on a map that is approximately to scale. If the angle centered at Washington DC
is 50° and the distance from Chicago to Monroe is 840 miles, which of the following gives the closest
approximation to the distance between Washington DC and Monroe?
A.
B.
C.
D.
E.
690 miles
750 miles
800 miles
1090 miles
1300 miles
Trap Answer Rationale: A gives the other length (DC – Chicago)
9/M/ Elementary Trigonometry /Trig Word Problems /E/D
A triangle is drawn between three major US cities on a map drawn approximately to scale below. If the angle
centered at New York is 15°and the distance between Portland and New York is 2,920 miles, which of the following
is the closest approximation to the distance between New York and San Francisco?
Trigonometry Word Problems
A.
B.
C.
D.
E.
750 miles
1040 miles
1500 miles
2820 miles
3020 miles
Trap Answer Rationale: If they do 2920*cos 15 instead of 2920/cos 15 they get D
10/M/ Elementary Trigonometry /Trig Word Problems /D/A
Four major US cities are shown on a map that is approximately drawn to scale below. If the distance between
Boston and Los Angeles is approximately 3000 miles, what is the best approximation to the distance between New
Orleans and Boston?
A.
B.
C.
D.
E.
1385 miles
1500 miles
1520 miles
1560 miles
1600 miles
Trap Answer Rationale: A is the solution to 3000 sin 27.5 (distance between Miami and Boston)
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Trigonometry Word Problems
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