Angle of Depression Setup � Identify Known/Unknowns & Choose the Right Trig Ratio (HSG-SRT.C.8)

1) From the top of a 60 m building, the angle of depression to a car on level ground is 25°. What is the horizontal distance to the car?

28 m

128 m

140 m

64 m

Angle of depression = angle of elevation.

tan 25° = 60 / x

x = 60 / tan 25° ≈ 60 / 0.4663 ≈ 128.7

Hence, x ≈ 128 m.

Explanation

Angle of depression = angle of elevation.

tan 25° = 60 / x

x = 60 / tan 25° ≈ 60 / 0.4663 ≈ 128.7

Hence, x ≈ 128 m.

2) A surveyor stands 45 m from a tower’s base. The angle of elevation to the top is 37°. Estimate the tower’s height.

34 m

27 m

68 m

55 m

tan 37° = height / 45

height = 45 × tan 37° ≈ 45 × 0.7536 ≈ 33.9

Hence, height ≈ 34 m.

Explanation

tan 37° = height / 45

height = 45 × tan 37° ≈ 45 × 0.7536 ≈ 33.9

Hence, height ≈ 34 m.

3) A surveyor stands 45 m from a tower’s base. The angle of elevation to the top is 37°. Estimate the tower’s height.

34 m

27 m

68 m

55 m

Same setup as Q2.

height = 45 × tan 37° ≈ 33.9

Hence, height ≈ 34 m.

Explanation

Same setup as Q2.

height = 45 × tan 37° ≈ 33.9

Hence, height ≈ 34 m.

4) A drone is 120 m above the ground. The angle of depression to its launch point is 15°. Find the horizontal distance to the launch point.

31 m

450 m

464 m

120 m

tan 15° = 120 / x

x = 120 / tan 15° ≈ 120 / 0.2679 ≈ 448.5

Closest option: 450 m.

Hence, x ≈ 450 m.

Explanation

tan 15° = 120 / x

x = 120 / tan 15° ≈ 120 / 0.2679 ≈ 448.5

Closest option: 450 m.

Hence, x ≈ 450 m.

5) A 20 ft ladder leans against a wall making a 72° angle with the ground. How high up the wall does it reach?

19.0 ft

18.0 ft

6.2 ft

21.0 ft

height = 20 × sin 72° ≈ 20 × 0.9511 ≈ 19.0

Hence, height ≈ 19.0 ft.

Explanation

height = 20 × sin 72° ≈ 20 × 0.9511 ≈ 19.0

Hence, height ≈ 19.0 ft.

6) You look down from a 50 m cliff at a boat. The angle of depression is 30°. What is the boat’s horizontal distance from the base of the cliff?

29 m

43 m

87 m

100 m

tan 30° = 50 / x

x = 50 / tan 30° = 50 / (√3/3) ≈ 50 / 0.5774 ≈ 86.6

Hence, x ≈ 87 m.

Explanation

tan 30° = 50 / x

x = 50 / tan 30° = 50 / (√3/3) ≈ 50 / 0.5774 ≈ 86.6

Hence, x ≈ 87 m.

7) A kite string makes a 48° angle of elevation with the ground and is 75 m long. Assuming the string is taut, what is the kite’s height above ground?

50 m

56 m

60 m

49 m

height = 75 × sin 48° ≈ 75 × 0.7431 ≈ 55.7

Hence, height ≈ 56 m.

Explanation

height = 75 × sin 48° ≈ 75 × 0.7431 ≈ 55.7

Hence, height ≈ 56 m.

8) A 35 m building casts a 24 m shadow. What is the sun’s angle of elevation?

35°

44°

55°

56°

tan θ = 35 / 24 ≈ 1.4583

θ = arctan(1.4583) ≈ 56°

Hence, θ ≈ 56°.

Explanation

tan θ = 35 / 24 ≈ 1.4583

θ = arctan(1.4583) ≈ 56°

Hence, θ ≈ 56°.

9) From a ship, the angle of elevation to the top of a 45 m lighthouse is 18°. How far is the ship from the lighthouse base (horizontal distance)?

139 m

145 m

142 m

123 m

tan 18° = 45 / x

x = 45 / tan 18° ≈ 45 / 0.3249 ≈ 138.5

Hence, x ≈ 139 m.

Explanation

tan 18° = 45 / x

x = 45 / tan 18° ≈ 45 / 0.3249 ≈ 138.5

Hence, x ≈ 139 m.

10) A rescue worker at the top of a 25 m cliff sees a raft below with an angle of depression of 40°. What is the slant (line-of-sight) distance to the raft?

16 m

39 m

25 m

23 m

Let s = line-of-sight (hypotenuse).

sin 40° = 25 / s ⇒ s = 25 / sin 40° ≈ 25 / 0.6428 ≈ 38.9

Hence, s ≈ 39 m.

Explanation

Let s = line-of-sight (hypotenuse).

sin 40° = 25 / s ⇒ s = 25 / sin 40° ≈ 25 / 0.6428 ≈ 38.9

Hence, s ≈ 39 m.

11) (Use this for Q11–12) A 30 m tower forms a right triangle with a line of sight making an angle θ below the horizontal. Which trigonometric relationship correctly solves for the horizontal distance x from the tower to the ground point in terms of θ?

Sin(θ) = x / 30

Cos(θ) = 30 / x

Tan(θ) = 30 / x

Tan(θ) = x / 30

Opposite = 30, adjacent = x.

tan θ = 30 / x.

Hence, tan(θ) = 30 / x.

Explanation

Opposite = 30, adjacent = x.

tan θ = 30 / x.

Hence, tan(θ) = 30 / x.

12) If θ = 28° and the tower is 30 m tall, compute x to the nearest meter.

15 m

56 m

63 m

57 m

x = 30 / tan 28° ≈ 30 / 0.5317 ≈ 56.4

Hence, x ≈ 56 m.

Explanation

x = 30 / tan 28° ≈ 30 / 0.5317 ≈ 56.4

Hence, x ≈ 56 m.

13) An observer stands on a 12 m platform and sees the top of a tree at an angle of elevation 25°. The horizontal distance to the tree is 30 m. What is the tree’s height?

23 m

26 m

24 m

25 m

Rise = 30 × tan 25° ≈ 14.0

Tree height = 12 + 14.0 = 26.0

Hence, ≈ 26 m.

Explanation

Rise = 30 × tan 25° ≈ 14.0

Tree height = 12 + 14.0 = 26.0

Hence, ≈ 26 m.

14) A plane descends toward an airport. From an altitude of 1500 m, the pilot has an angle of depression of 6° to the runway. What is the horizontal distance to the threshold?

14,325 m

1,505 m

28,648 m

15,000 m

x = 1500 / tan 6° ≈ 1500 / 0.1051 ≈ 14,274

Closest option: 14,325 m.

Hence, ≈ 14,325 m.

Explanation

x = 1500 / tan 6° ≈ 1500 / 0.1051 ≈ 14,274

Closest option: 14,325 m.

Hence, ≈ 14,325 m.

15) A wheelchair ramp rises 0.9 m over a run of 7.2 m. What is the ramp’s angle of elevation to the nearest degree?

4°

7°

6°

8°

tan θ = 0.9 / 7.2 = 0.125

θ = arctan(0.125) ≈ 7.1° ⇒ nearest 7°.

Hence, θ ≈ 7°.

Explanation

tan θ = 0.9 / 7.2 = 0.125

θ = arctan(0.125) ≈ 7.1° ⇒ nearest 7°.

Hence, θ ≈ 7°.

16) From a balcony 18 m above ground, the angle of depression to a parked car is 32°. What is the line-of-sight distance to the car?

34 m

26 m

21 m

33 m

s = 18 / sin 32° ≈ 18 / 0.5299 ≈ 34.0

Hence, s ≈ 34 m.

Explanation

s = 18 / sin 32° ≈ 18 / 0.5299 ≈ 34.0

Hence, s ≈ 34 m.

17) A hiker looks up at a peak. The angle of elevation is 22°, and the horizontal distance to the base is 1.4 km. How much higher is the peak than the hiker?

0.53 km

0.58 km

0.52 km

0.56 km

Height gain = 1.4 × tan 22° ≈ 0.5656 km

Hence, ≈ 0.56 km.

Explanation

Height gain = 1.4 × tan 22° ≈ 0.5656 km

Hence, ≈ 0.56 km.

18) A 10 m flagpole stands on level ground. From a point uphill, the angle of depression to the top of the pole is 12° and to the base is 14°. Estimate the observer’s height above the base.

67,9 m

238.5 m

23.9 m

47.5 m

tan 14° = H / d, tan 12° = (H − 10) / d

10 = d(tan14° − tan12°) ⇒ d ≈ 272.5

H = d × tan14° ≈ 67.9 m

Therefore, ≈ 67.9 m.

Explanation

tan 14° = H / d, tan 12° = (H − 10) / d

10 = d(tan14° − tan12°) ⇒ d ≈ 272.5

H = d × tan14° ≈ 67.9 m

Therefore, ≈ 67.9 m.