Angle Of Depression Quiz: Air Traffic Control Angle Of Depression

1) If horizontal distance doubles while θ stays the same, altitude doubles.

True

False

h = d·tanθ is linear in d; h₂ = 2d·tanθ = 6.46 is twice 3.23.

Explanation

h = d·tanθ is linear in d; h₂ = 2d·tanθ = 6.46 is twice 3.23.

2) The angle of depression from the tower equals the angle of elevation from the plane to the tower (assuming level ground).

True

False

They are alternate interior angles with parallel horizontals, hence equal.

Explanation

They are alternate interior angles with parallel horizontals, hence equal.

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4) If the angle of depression doubles while distance stays the same, the altitude also exactly doubles.

True

False

h = d·tanθ; tan is nonlinear, so doubling θ does not double tanθ.

Explanation

h = d·tanθ; tan is nonlinear, so doubling θ does not double tanθ.

5) Angle of depression 24°; horizontal distance 15.2 km. Altitude? (km)

7.07

1/6

1/7

h = 15.2·tan24° = 6.77 km.

Explanation

h = 15.2·tan24° = 6.77 km.

6) If θ is measured in degrees but your calculator is in radians mode, tanθ will be evaluated incorrectly.

True

False

Always match the angle unit to calculator mode.

Explanation

Always match the angle unit to calculator mode.

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8) Select all correct formulas to compute altitude h given horizontal distance d and angle of depression θ.

H = d·tanθ

H = d/ tanθ

H = d·sinθ

H = d·tan(90°−θ)

H = d·cotθ

tanθ = h/d ⇒ h = d·tanθ. Others compute different legs; tan(90°−θ)=cotθ=adjacent/opposite, not altitude.

Explanation

tanθ = h/d ⇒ h = d·tanθ. Others compute different legs; tan(90°−θ)=cotθ=adjacent/opposite, not altitude.

9) Which statements are always true in this context? Select all that apply.

Altitude is the side opposite the angle of depression in the right triangle model

Horizontal separation is the adjacent side relative to θ

Using θ in radians or degrees is fine as long as the tan function uses the same unit

Altitude can be computed with h = d·tanθ

Altitude equals d·cotθ

Opposite = altitude, adjacent = horizontal. Use consistent units. h = d·tanθ; d·cotθ is not altitude.

Explanation

Opposite = altitude, adjacent = horizontal. Use consistent units. h = d·tanθ; d·cotθ is not altitude.

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10) Spot the errors (select all that are incorrect statements).

Tanθ = d/h

Tanθ = h/d

H = d·tanθ

If you mistakenly compute h = d/ tanθ you get the slant height

Angle of depression equals angle of elevation

Correct identity is tanθ = h/d. h = d/tanθ gives adjacent^2/opposite, not slant.

Explanation

Correct identity is tanθ = h/d. h = d/tanθ gives adjacent^2/opposite, not slant.

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11) A tower spots an aircraft with an angle of depression of 18° and a horizontal separation of 12.5 km. What is the aircraft’s altitude? (km)

1/4

1/3

1/5

Altitude = d·tanθ = 12.5·tan18° = 4.06 km.

Explanation

Altitude = d·tanθ = 12.5·tan18° = 4.06 km.

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13) Angle of depression 28°, distance 5.4 km. Altitude? (km)

2.67

3.07

1/2

1/3

h = 5.4·tan28° = 2.87 km.

Explanation

h = 5.4·tan28° = 2.87 km.

14) If the altitude is 2.8 km and horizontal distance is 9.6 km, what is the angle of depression (nearest degree)?

15°

16°

17°

18°

tanθ = h/d = 2.8/9.6 ⇒ θ = arctan(2.8/9.6) ≈ 16.26°, so ≈ 16°.

Explanation

tanθ = h/d = 2.8/9.6 ⇒ θ = arctan(2.8/9.6) ≈ 16.26°, so ≈ 16°.

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16) Which sets of known values are sufficient to compute altitude?

(d, θ)

(d only)

(θ only)

(slant distance, θ) using h = (slant)·sinθ

(slant distance only)

Compute altitude from d and θ via h=d·tanθ or from slant R and θ via h=R·sinθ.

Explanation

Compute altitude from d and θ via h=d·tanθ or from slant R and θ via h=R·sinθ.

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17) If the angle of depression is 0°, the altitude must be 0.

True

False

tan0° = 0 ⇒ h = d·tan0° = 0, meaning the plane is at tower-eye level.

Explanation

tan0° = 0 ⇒ h = d·tan0° = 0, meaning the plane is at tower-eye level.

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19) For the same horizontal distance d, which angle of depression results in the largest altitude?

10°

15°

25°

35°

h = d·tanθ increases with θ on (0°, 90°).

Explanation

h = d·tanθ increases with θ on (0°, 90°).

20) Which equations correctly relate altitude h, distance d, and angle of depression θ? Select all that apply.

Tanθ = h/d

Cosθ = d/ (slant distance)

Sinθ = h/ (slant distance)

Cotθ = d/h

H = d·tanθ

All are standard right-triangle identities: tanθ=h/d; cosθ=adjacent/hypotenuse; sinθ=opposite/hypotenuse; cotθ=d/h; thus h=d·tanθ.

Explanation

All are standard right-triangle identities: tanθ=h/d; cosθ=adjacent/hypotenuse; sinθ=opposite/hypotenuse; cotθ=d/h; thus h=d·tanθ.

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Angle of Depression Quiz: Air Traffic Control Angle of Depression