Practice Maths

L16 — Angles of Elevation and Depression

Key Terms

Angle of elevation
The angle measured upward from the horizontal to a line of sight to an object above.
Angle of depression
The angle measured downward from the horizontal to a line of sight to an object below.
Horizontal
The flat reference line from which elevation and depression angles are always measured.
Vertical height
The opposite side in the right triangle formed by the observer, horizontal, and line of sight.
Horizontal distance
The adjacent side in the right triangle; the flat distance between observer and object.
Alternate angles
The angle of elevation from A to B equals the angle of depression from B to A (alternate interior angles on parallel lines).

Definitions

  • The angle of elevation is the angle measured upward from the horizontal to a line of sight.
  • The angle of depression is the angle measured downward from the horizontal to a line of sight.
  • Both angles are always measured from the horizontal, never from the vertical.
  • The angle of elevation from A to B equals the angle of depression from B to A (alternate interior angles).
elev. Angle of Elevation top dep. Angle of Depression
Both angles are measured from the horizontal line of sight.

Key Formula

Draw a right triangle: the opposite side is the vertical height difference; the adjacent side is the horizontal distance. The angle at the observer is the elevation or depression angle.

tan θ = height ÷ horizontal distance

Hot Tip: Always measure from the horizontal, never from the vertical. If a problem states an angle “from the vertical,” subtract from 90° first. Also remember: the angle of elevation and depression between the same two points are always equal.

Setting Up the Right Triangle

For any elevation or depression problem:

  1. Identify the observer and the object.
  2. Draw a horizontal line from the observer.
  3. The right triangle has: opposite = vertical difference, adjacent = horizontal distance, angle = elevation/depression angle.
  4. Choose the appropriate trig ratio and solve.

Worked Example 1 — Angle of Elevation

From a point 30 m from the base of a tower, the angle of elevation of the top is 62°. Find the height of the tower.

tan 62° = height / 30 ⇒ height = 30 × tan 62° = 30 × 1.8807 ≈ 56.4 m

Worked Example 2 — Angle of Depression

From the top of an 80 m cliff, the angle of depression of a boat is 28°. Find the horizontal distance to the boat.

tan 28° = 80 / distance ⇒ distance = 80 / tan 28° = 80 / 0.5317 ≈ 150.5 m

Worked Example 3 — Eye Height Adjustment

From eye level 1.7 m above the ground, the angle of elevation of a building’s roof is 48°. The horizontal distance is 25 m. Find the total height of the building.

Height above eye: 25 × tan 48° ≈ 25 × 1.1106 ≈ 27.77 m

Total building height = 27.77 + 1.7 ≈ 29.47 m

  1. Find the height (angle of elevation). Fluency

    Use the formula: height = horizontal × tan(θ). Give answers to 2 d.p.

    • (a) Horizontal distance = 40 m, angle of elevation = 30°.
    • (b) Horizontal distance = 25 m, angle of elevation = 55°.
    • (c) Horizontal distance = 80 m, angle of elevation = 12°.
    • (d) Horizontal distance = 15 m, angle of elevation = 45°.

  2. Find the horizontal distance. Fluency

    • (a) Height = 50 m, angle of elevation = 40°.
    • (b) Height = 20 m, angle of elevation = 65°.
    • (c) Height = 100 m, angle of elevation = 30°.
    • (d) Height = 35 m, angle of elevation = 50°.

  3. Find the angle of elevation. Fluency

    • (a) Height = 30 m, horizontal distance = 40 m.
    • (b) Height = 10 m, horizontal distance = 10 m.
    • (c) Height = 15 m, slant distance (line of sight) = 20 m.
    • (d) Height = 8 m, horizontal distance = 12 m.

  4. Angle of depression problems. Fluency

    • (a) From the top of a 60 m cliff, a boat is seen at a horizontal distance of 100 m. Find the angle of depression.
    • (b) A plane at height 2000 m observes a tower 1500 m away horizontally. Find the angle of depression to the tower’s top (assume the plane and tower top are at the same height relative to ground — the angle is from horizontal down to the tower).
    • (c) From a window 8 m above the ground, the angle of depression to a car is 20°. How far away is the car (horizontal distance)?
    • (d) From a cliff 45 m high, the angle of depression to a boat is 35°. Find the horizontal distance to the boat.

  5. Read from the diagram: building height. Understanding

    The diagram shows a person whose eyes are 1.6 m above the ground. They look up at the top of a building at an angle of elevation of 55°. The horizontal distance from the person to the building is 20 m.

    1.6 m h 20 m 55°
    • (a) Write a trigonometric equation for the height above eye level that the building extends.
    • (b) Calculate the height above eye level.
    • (c) Find the total height h of the building.
    • (d) Why must the person’s eye height (1.6 m) be added separately?

  6. Two angles from the same point. Understanding

    From a point P on level ground, the angles of elevation of two buildings are 30° and 50°. Both buildings are on the same side of P, at horizontal distances of 60 m and 60 m respectively (directly behind each other).

    • (a) Find the height of the shorter building (angle = 30°, distance = 60 m).
    • (b) Find the height of the taller building (angle = 50°, distance = 60 m).
    • (c) What is the difference in height between the two buildings?
    • (d) If a person stood on top of the shorter building and looked at the top of the taller building, would the angle be an elevation or depression? Explain.

  7. Elevation = Depression. Understanding

    A lighthouse is 50 m tall. From its top, the angle of depression to a boat is 22°.

    • (a) Find the horizontal distance from the lighthouse to the boat.
    • (b) From the boat, what is the angle of elevation of the top of the lighthouse?
    • (c) Explain why the two angles in (a) and (b) are equal.
    • (d) As the boat approaches, the angle of depression increases or decreases? Why?

  8. Ramp and slope angles. Understanding

    A wheelchair ramp rises 0.6 m over a horizontal run of 4 m.

    • (a) Find the angle the ramp makes with the horizontal.
    • (b) Find the length of the ramp surface (slant length).
    • (c) Australian Standards require ramps to have a gradient no steeper than 1:8 (rise:run). Does this ramp comply?
    • (d) What maximum angle does the 1:8 gradient correspond to?

  9. Lighthouse and ship. Problem Solving

    A lighthouse keeper at the top of a 45 m lighthouse observes two ships directly in front. The angles of depression are 18° to Ship A and 32° to Ship B.

    • (a) Find the horizontal distance from the lighthouse to Ship A.
    • (b) Find the horizontal distance from the lighthouse to Ship B.
    • (c) Hence find the distance between the two ships.
    • (d) Which ship is closer to the lighthouse?

  10. Inaccessible height. Problem Solving

    To find the height of a cliff, a surveyor measures angles of elevation from two points A and B on level ground, both on the same line directly away from the cliff. From A, the angle of elevation is 52°. From B, 80 m further from A, the angle of elevation is 28°.

    • (a) Let the horizontal distance from A to the cliff base be d metres. Write an expression for the cliff height h using the angle from A.
    • (b) Write another expression for h using the angle from B (the horizontal distance from B is d + 80).
    • (c) Set the two expressions equal and solve for d.
    • (d) Hence find the height of the cliff.
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