The key idea: Both angles are measured from the horizontal line through the observer's eye. Look up → angle of elevation. Look down → angle of depression.
| Type | Direction observer looks | Where the angle sits |
|---|---|---|
| Elevation | Upward, toward an object above | Between horizontal and the line of sight, above horizontal |
| Depression | Downward, toward an object below | Between horizontal and the line of sight, below horizontal |
[Diagram: ] - Available in full study mode
Alternate angles are equal: When you draw parallel horizontal lines (one through the observer, one through the object), the angle of elevation from the bottom equals the angle of depression from the top. This is used frequently in IB problems.
Worked example — height of a building
From a point 40 m from the base of a building, the angle of elevation to the top is 52°. Find the height of the building.
Step by step
- Draw a right triangle. The horizontal = 40 m (adjacent), the height = unknown (opposite), angle = 52°.
- Use tan.
- Solve.
Final answer
Height ≈ 51.2 m.
Learn what examiners really want
See exactly what to write to score full marks. Our AI shows you model answers and the key phrases examiners look for.
Worked example — distance to a boat
An observer stands on a cliff 80 m above sea level. The angle of depre ion to a boat is 34°. Find the horizontal distance from the base of the cliff to the boat.
Step by step
- Angle of depre ion = 34°. The alternate angle at the boat (from horizontal) is also 34°.
- The vertical = 80 m (opposite to 34°), horizontal = d (adjacent).
- Rearrange.
Final answer
Horizontal distance ≈ 118.6 m.
Set up the triangle correctly: The angle of depre ion is always measured from the horizontal, NOT from the vertical. Draw the diagram first.
Worked example — height from two positions
From point A, the angle of elevation to the top of a tower is 30°. From point B, 50 m closer, the angle of elevation is 50°. Find the height of the tower.
Step by step
- Let h = height of tower, d = horizontal distance from B to tower.
- From B: tan 50° = h/d → h = d tan 50°.
- From A: tan 30° = h/(d+50) → h = (d+50)tan30°.
- Set equal: d tan50° = (d+50)tan30°. Expand and solve for d.
- Compute d.
- Find h.
Final answer
Height of tower ≈ 56.0 m.