Geometry NameTrigonometric Application DateDay ______ BlockWrite a trigonometric ratio for each problem, then find the missing value. 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. 348 𝑓𝑡 B. 2524 𝑓𝑡 C. 2476 𝑓𝑡 D. 17,963 𝑓𝑡 E. 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, 𝜃= 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?

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?

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. 21 ft b. 23 ft c. 45 ft d. 90 ft e. 107 ft

A) 1000 miles B) 1500 miles C) 2000 miles D) 2598 milesE) 3000 miles

a. 72 ft b. 75 ft c. 200 ft d. 270 ft e. 1044 ft

Draw a picture (when necessary) and write an equation for each problem. Round your answers to the nearest tenth. 5. The angle of elevation to the top of the Empire State Building in New York is from a point on the ground 1 mile from the base of the building. Find the height of the Empire State Building in feet.

6. Use the diagram below to find the distance across the suspension bridge.

7.

9.

8.

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. 0.45 km/hr b. 0.9 km/hr c. 3.1 km/hr d. 3.4 km/hr e. 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. 1 km/hr b. 2 km/hr c. 3 km/hr d. 4 km/hr e. 5 km/hr

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. 685 miles b. 1485 miles c. 1810 miles d. 2000 miles e. 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. 690 miles b. 750 miles c. 800 miles d. 1090 miles e. 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. 750 miles b. 1040 miles c. 1500 miles d. 2820 miles e. 3020 miles

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. 1385 miles b. 1500 miles c. 1520 miles d. 1560 miles e. 1600 miles

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. 348 𝑓𝑡 B. 2524 𝑓𝑡 C. 2476 𝑓𝑡 D. 𝟏𝟕, 𝟗𝟔𝟑 𝒇𝒕 E. 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 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. 72 ft B. 75 ft C. 200 ft D. 270 ft E. 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. 21 ft B. 23 ft C. 45 ft D. 90 ft E. 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?

A. 0.45 km/hr B. 0.9 km/hr C. 3.1 km/hr D. 𝟑. 𝟒 km/hr E. 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. 1 km/hr B. 2 km/hr C. 3 km/hr D. 4 km/hr E. 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?

A. 685 miles B. 1485 miles C. 1810 miles D. 2000 miles E. 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. 690 miles B. 750 miles C. 800 miles D. 1090 miles E. 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?

A. 750 miles B. 1040 miles C. 1500 miles D. 2820 miles E. 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. 1385 miles B. 1500 miles C. 1520 miles D. 1560 miles E. 1600 miles Trap Answer Rationale: A is the solution to 3000 sin 27.5 (distance between Miami and Boston)

Trigonometry Word Problems 10