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

From the top of the building, the angle of depression equals the angle of elevation from the car.

Use the tangent ratio:

tan 25° = opposite / adjacent = 60 / x

Solve for x:

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

Hence, the horizontal distance is approximately 128 m.

Explanation

From the top of the building, the angle of depression equals the angle of elevation from the car.

Use the tangent ratio:

tan 25° = opposite / adjacent = 60 / x

Solve for x:

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

Hence, the horizontal distance is approximately 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

Use the tangent ratio:

tan 37° = height / 45

Solve for height:

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

Hence, the tower’s height is approximately 34 m.

Explanation

Use the tangent ratio:

tan 37° = height / 45

Solve for height:

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

Hence, the tower’s height is approximately 34 m.

3) 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

The drone is 120 m high, and the angle of depression is 15°, so use tangent:

tan 15° = 120 / x

Solve for x:

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

Closest option is 450 m, so the horizontal distance is approximately 450 m.

Explanation

The drone is 120 m high, and the angle of depression is 15°, so use tangent:

tan 15° = 120 / x

Solve for x:

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

Closest option is 450 m, so the horizontal distance is approximately 450 m.

4) 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

Use sine to find the vertical height:

height = hypotenuse × sin(angle)

Here:

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

Hence, the ladder reaches approximately 19.0 ft up the wall.

Explanation

Use sine to find the vertical height:

height = hypotenuse × sin(angle)

Here:

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

Hence, the ladder reaches approximately 19.0 ft up the wall.

5) 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

Use the tangent ratio with the height as opposite:

tan 30° = 50 / x

Solve for x:

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

Hence, the boat is approximately 87 m from the base of the cliff.

Explanation

Use the tangent ratio with the height as opposite:

tan 30° = 50 / x

Solve for x:

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

Hence, the boat is approximately 87 m from the base of the cliff.

6) 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 be the slant distance (line of sight).

The vertical side is 25 m, and angle at the top is 40°.

Use sine:

sin 40° = 25 / s

Solve for s:

s = 25 / sin 40° ≈ 25 / 0.6428 ≈ 38.9

Hence, the line-of-sight distance is approximately 39 m.

Explanation

Let s be the slant distance (line of sight).

The vertical side is 25 m, and angle at the top is 40°.

Use sine:

sin 40° = 25 / s

Solve for s:

s = 25 / sin 40° ≈ 25 / 0.6428 ≈ 38.9

Hence, the line-of-sight distance is approximately 39 m.

7) A 30 m tower forms a right triangle with a line of sight making an angle θ below the horizontal. If tan θ = 30 /x and if θ = 28° and the tower is 30 m tall, compute x to the nearest meter.

15 m

56 m

63 m

57 m

For θ = 28°:

tan 28° = 30 / x

Solve for x:

x = 30 / tan 28° ≈ 30 / 0.5317 ≈ 56.4

Hence, the horizontal distance is approximately 56 m.

Explanation

For θ = 28°:

tan 28° = 30 / x

Solve for x:

x = 30 / tan 28° ≈ 30 / 0.5317 ≈ 56.4

Hence, the horizontal distance is approximately 56 m.

8) 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

The altitude is 1500 m, and the angle of depression is 6°.

Use tangent:

tan 6° = 1500 / x

Solve for x:

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

Closest option is 14,325 m, so the horizontal distance is approximately 14,325 m.

Explanation

The altitude is 1500 m, and the angle of depression is 6°.

Use tangent:

tan 6° = 1500 / x

Solve for x:

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

Closest option is 14,325 m, so the horizontal distance is approximately 14,325 m.

9) 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°

Use the tangent ratio with rise and run:

tan θ = 0.9 / 7.2 = 0.125

Find the angle:

θ = arctan(0.125) ≈ 7.1°

Rounded to the nearest degree:

θ ≈ 7°.

Explanation

Use the tangent ratio with rise and run:

tan θ = 0.9 / 7.2 = 0.125

Find the angle:

θ = arctan(0.125) ≈ 7.1°

Rounded to the nearest degree:

θ ≈ 7°.

10) 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

Let H be the observer’s height above the base of the flagpole and d the horizontal distance.

From the base angle: tan 14° = H / d

From the top angle: tan 12° = (H − 10) / d

Subtract the two:

H/d − (H − 10)/d = tan 14° − tan 12°

10/d = tan 14° − tan 12°

So:

d = 10 / (tan 14° − tan 12°) ≈ 272.5

Now find H:

H = d × tan 14° ≈ 272.5 × 0.2493 ≈ 67.9

Hence, the observer is approximately 67.9 m above the base.

Explanation

Let H be the observer’s height above the base of the flagpole and d the horizontal distance.

From the base angle: tan 14° = H / d

From the top angle: tan 12° = (H − 10) / d

Subtract the two:

H/d − (H − 10)/d = tan 14° − tan 12°

10/d = tan 14° − tan 12°

So:

d = 10 / (tan 14° − tan 12°) ≈ 272.5

Now find H:

H = d × tan 14° ≈ 272.5 × 0.2493 ≈ 67.9

Hence, the observer is approximately 67.9 m above the base.

11) A security camera is mounted 8 m high. The angle of depression to a door 20 m away is closest to:

18°

22°

21°

24°

The vertical drop is 8 m and horizontal distance is 20 m.

Use tangent:

tan θ = opposite / adjacent = 8 / 20 = 0.4

Now find θ:

θ = arctan(0.4) ≈ 21.8°

Closest option is 22°, so the angle of depression is approximately 22°.

Explanation

The vertical drop is 8 m and horizontal distance is 20 m.

Use tangent:

tan θ = opposite / adjacent = 8 / 20 = 0.4

Now find θ:

θ = arctan(0.4) ≈ 21.8°

Closest option is 22°, so the angle of depression is approximately 22°.

12) A roadway descends at a 5° angle over a 400 m horizontal distance. What is the change in elevation?

35 m

31 m

40 m

29 m

Use tangent with the horizontal run:

drop = 400 × tan 5°

Compute:

drop ≈ 400 × 0.0875 ≈ 35.0

Hence, the change in elevation is approximately 35 m.

Explanation

Use tangent with the horizontal run:

drop = 400 × tan 5°

Compute:

drop ≈ 400 × 0.0875 ≈ 35.0

Hence, the change in elevation is approximately 35 m.

13) 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

Use the tangent ratio:

tan 37° = height / 45

Solve for height:

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

Hence, the tower’s height is approximately 34 m.

Explanation

Use the tangent ratio:

tan 37° = height / 45

Solve for height:

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

Hence, the tower’s height is approximately 34 m.

14) 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

The string is the hypotenuse and the height is the opposite:

height = 75 × sin 48°

Compute:

height ≈ 75 × 0.7431 ≈ 55.7

Hence, the kite’s height is approximately 56 m above the ground.

Explanation

The string is the hypotenuse and the height is the opposite:

height = 75 × sin 48°

Compute:

height ≈ 75 × 0.7431 ≈ 55.7

Hence, the kite’s height is approximately 56 m above the ground.

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

35°

44°

55°

56°

Use tangent with height as opposite and shadow as adjacent:

tan θ = 35 / 24 ≈ 1.4583

Now find θ:

θ = arctan(35 / 24) ≈ arctan(1.4583) ≈ 56°

Hence, the sun’s angle of elevation is approximately 56°.

Explanation

Use tangent with height as opposite and shadow as adjacent:

tan θ = 35 / 24 ≈ 1.4583

Now find θ:

θ = arctan(35 / 24) ≈ arctan(1.4583) ≈ 56°

Hence, the sun’s angle of elevation is approximately 56°.

16) 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

Use tangent:

tan 18° = 45 / x

Solve for x:

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

Hence, the horizontal distance is approximately 139 m.

Explanation

Use tangent:

tan 18° = 45 / x

Solve for x:

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

Hence, the horizontal distance is approximately 139 m.

17) 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

The vertical side is 30 m and the horizontal distance is x.

Opposite = 30, adjacent = x.

So:

tan θ = opposite / adjacent = 30 / x

Hence, the correct relationship is tan(θ) = 30 / x.

Explanation

The vertical side is 30 m and the horizontal distance is x.

Opposite = 30, adjacent = x.

So:

tan θ = opposite / adjacent = 30 / x

Hence, the correct relationship is tan(θ) = 30 / x.

18) 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

The observer is 12 m above ground, and the angle of elevation to the top is 25°.

First find the vertical rise from the platform to the top of the tree:

rise = 30 × tan 25° ≈ 30 × 0.4663 ≈ 14.0

Total tree height = platform height + rise

= 12 + 14.0 = 26.0

Hence, the tree is approximately 26 m tall.

Explanation

The observer is 12 m above ground, and the angle of elevation to the top is 25°.

First find the vertical rise from the platform to the top of the tree:

rise = 30 × tan 25° ≈ 30 × 0.4663 ≈ 14.0

Total tree height = platform height + rise

= 12 + 14.0 = 26.0

Hence, the tree is approximately 26 m tall.

19) 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

Let s be the line-of-sight distance.

The vertical side is 18 m, angle of depression is 32°.

Use sine:

sin 32° = 18 / s

Solve for s:

s = 18 / sin 32° ≈ 18 / 0.5299 ≈ 34.0

Hence, the line-of-sight distance is approximately 34 m.

Explanation

Let s be the line-of-sight distance.

The vertical side is 18 m, angle of depression is 32°.

Use sine:

sin 32° = 18 / s

Solve for s:

s = 18 / sin 32° ≈ 18 / 0.5299 ≈ 34.0

Hence, the line-of-sight distance is approximately 34 m.

20) 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

Use the tangent ratio with horizontal distance 1.4 km:

height gain = 1.4 × tan 22°

Compute:

height gain ≈ 1.4 × 0.4040 ≈ 0.5656 km

Rounded:

height gain ≈ 0.56 km.

Explanation

Use the tangent ratio with horizontal distance 1.4 km:

height gain = 1.4 × tan 22°

Compute:

height gain ≈ 1.4 × 0.4040 ≈ 0.5656 km

Rounded:

height gain ≈ 0.56 km.

Angle of Depression: Setting Up Right-Triangle Relationships