Coordinates and trigonometry · Ages 14–15

Angles of elevation and depression worksheet

Part of: Trigonometry

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Coordinates and trigonometry · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.From the ground 44 m from a tower's foot, the angle of elevation of its top is 41°. Correct to 1 decimal place, how tall is the tower, in m?
Answer:
2.From the top of a cliff 116 m high, a boat at sea is at an angle of depression of 9°. Correct to 1 decimal place, how far is the boat from the foot of the cliff, in m?
Answer:
3.From the top of a cliff 92 m high, a boat at sea is at an angle of depression of 13°. Correct to 1 decimal place, how far is the boat from the foot of the cliff, in m?
Answer:
4.Ming's eyes are 1.5 m above the ground, and he stands 131 m from a tree 57 m tall, looking at its top. To the nearest 0.1°, what is the angle of elevation?
Answer:
5.A building is 50 m tall. From the ground 98 m away, what is the angle of elevation of its top, to the nearest 0.1°?
Answer:
6.From the top of a lighthouse 42 m high, two boats in the same direction are at angles of depression of 49° and 19°. Correct to 1 decimal place: the nearer boat is m from the lighthouse's foot, and the boats are m apart.
Fill in each blank
Answer:
7.From point A on the ground the top of a tower is at an elevation of 21°; walking 64 m nearer, to B, the elevation is 52°. Correct to 1 decimal place, how tall is the tower, in m?
Answer:
8.From the top of a lighthouse 148 m high, two boats in the same direction are at angles of depression of 32° and 20°. Correct to 1 decimal place: the nearer boat is m from the lighthouse's foot, and the boats are m apart.
Fill in each blank
Answer:

9.From point A, point B is at an angle of depression of 26°. What is the angle of elevation of A from B?
- A. 26°
- B. 154°
- C. 52°
- D. 64°
Choose the right answer
Answer:
10.From point A, point B is at an angle of depression of 10°. What is the angle of elevation of A from B?
- A. 80°
- B. 170°
- C. 10°
- D. 20°
Choose the right answer
Answer:
11.From the ground 40 m from a tower 44 m tall, the angle of elevation of its top is more than 38°.
True or false?
Answer:
12.From the ground 37 m from a tower 23 m tall, the angle of elevation of its top is more than 27°.
True or false?
Answer:

13.A drone 46 m above the ground sees a car below at an angle of depression of 45°. Correct to 1 decimal place, how far is the drone from the car in a straight line, in m?
Working:
Answer:
14.A survey team sees a peak at an elevation of 22° from A, then walks 37 m nearer to B, where the elevation is 47°. Correct to 1 decimal place: the peak is m high, and B is m from the point under it.
Fill in each blank
Working:
Answer:
15.Ming stands 63 m from a building, his eyes 1.7 m above the ground, and sees its top at an elevation of 53°. Correct to 1 decimal place, how tall is the building, in m?
Working:
Answer:
16.A lighthouse is 87 m tall, and its keeper sees a boat from the top at an angle of depression of 17°. Correct to 1 decimal place, how far is the boat from the lighthouse's foot, in m?
Working:
Answer:
17.A drone 59 m above the ground sees a car below at an angle of depression of 65°. Correct to 1 decimal place, how far is the drone from the car in a straight line, in m?
Working:
Answer:
18.Ming stands 75 m from a building, his eyes 1.6 m above the ground, and sees its top at an elevation of 50°. Correct to 1 decimal place, how tall is the building, in m?
Working:
Answer:

Show the answers and working

Angles of elevation and depression worksheet: Answers and working

1. 38.2 m
tan 41° = height ÷ 44, so the height = 44 × tan 41° ≈ 38.2 m.
2. 732.4 m
The depression equals the elevation at the boat (alternate angles): tan 9° = 116 ÷ distance, distance = 116 ÷ tan 9° ≈ 732.4 m.
3. 398.5 m
The depression equals the elevation at the boat (alternate angles): tan 13° = 92 ÷ distance, distance = 92 ÷ tan 13° ≈ 398.5 m.
4. 23°
The elevation is measured from his eye's horizontal: the top is 57 − 1.5 = 55.5 m above his eye, tan θ = 55.5131, θ ≈ 23°.
5. 27°
tan θ = 5098, so θ = tan⁻¹(5098) ≈ 27°.
6. 36.5, 85.5
The bigger depression is the nearer: 42 ÷ tan 49° ≈ 36.5; the other is 42 ÷ tan 19°; their difference, rounded after, ≈ 85.5 m.
7. 35.1 m
Let the height be h: A is h ÷ tan 21° from the foot and B h ÷ tan 52°, 64 apart: h(1tan 21° − 1tan 52°) = 64, so h ≈ 35.1 m.
8. 236.8, 169.8
The bigger depression is the nearer: 148 ÷ tan 32° ≈ 236.8; the other is 148 ÷ tan 20°; their difference, rounded after, ≈ 169.8 m.

9. A. 26°
The horizontals at A and B are parallel and the line of sight crosses them: the depression and the elevation are alternate angles, both 26°.
10. C. 10°
The horizontals at A and B are parallel and the line of sight crosses them: the depression and the elevation are alternate angles, both 10°.
11. True
The elevation = tan⁻¹(4440) ≈ 47.7°; compare it with 38°.
12. True
The elevation = tan⁻¹(2337) ≈ 31.9°; compare it with 27°.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 65.1 m
The straight line is the hypotenuse and the height 46 opposite: sin 45° = 46distance, so distance = 46 ÷ sin 45° ≈ 65.1 m.
14. 24, 22.4
Let the height be h: h ÷ tan 22° − h ÷ tan 47° = 37, h ≈ 24; B is h ÷ tan 47° ≈ 22.4 from the foot (with h unrounded).
15. 85.3 m
The part above his eye is 63 × tan 53°; add the eye's 1.7: about 85.3 m.
16. 284.6 m
The depression equals the elevation at the boat: tan 17° = 87distance, so distance = 87 ÷ tan 17° ≈ 284.6 m.
17. 65.1 m
The straight line is the hypotenuse and the height 59 opposite: sin 65° = 59distance, so distance = 59 ÷ sin 65° ≈ 65.1 m.
18. 91 m
The part above his eye is 75 × tan 50°; add the eye's 1.6: about 91 m.

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