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 |
Interactive: switch between elevation, depression, and both-angles view
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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.
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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 = opposite / adjacent.
- Solve.
Final answer
Height ≈ 51.2 m.
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Worked example — distance to a boat
An observer stands on a cliff 80 m above sea level.
The angle of depression to a boat is 34°.
Find the horizontal distance from the base of the cliff to the boat.
Step by step
- Angle of depression = 34°. By alternate angles (parallel horizontals), the angle of elevation from the boat to the observer is also 34°.
- From the boat: 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.
- From A: tan 30° = h/(d + 50).
- Set equal and solve for d.
- Compute d.
- Find h.
Final answer
Height of tower ≈ 56.0 m.
What a bearing means: A bearing is a direction measured clockwise from North, always written with three figures — 040°, not 40°.
To solve a bearings journey: sketch it, draw a North line at each turn, find the interior angle of the triangle, then use the cosine rule (for a distance) or the sine rule (for an angle, e.g. the bearing back).
IB-style question — bearings navigation
A yacht sails 7 km on a bearing of 040°, then 10 km on a bearing of 100°.
(a) Find the direct distance of the yacht from its starting point.
(b) Find the bearing it must steer to sail straight back to the start.
Step by step
- (a) Sketch the journey and mark the North line at the turn. The yacht arrives heading 040°, so it looks back along 040° + 180° = 220°, and leaves on 100°. The interior angle of the triangle is the difference.
- Apply the cosine rule to the two legs (7 and 10) with 120° between them.
- Square root.
- (b) Find the angle at C (between the return leg and the second leg) with the sine rule.
- At C the yacht faces 100°, so it looks back toward B along 100° + 180° = 280°. The start A lies 24.2° before that (see the diagram), giving the bearing of A from C.
Final answer
(a) ≈ 14.8 km. (b) ≈ 256°.
The journey from the question — 7 km on 040°, then 10 km on 100°. North lines are dashed at each turn; the purple dashed leg is the direct distance back to the start (part a).
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IB-style question — two angles of elevation
A drone hovers above level ground. From a point A the angle of elevation to the drone is 35°. From a point B, 50 m nearer (on the same line), it is 48°.
Find the height of the drone above the ground.
Step by step
- Use the slanted triangle ABD (D = drone). The angle at B inside the triangle is the supplement of 48°.
- Sine rule with AB = 50 to find the slant distance BD.
- Height = BD sin 48° (right triangle below the drone).
Final answer
The drone is about 94.7 m above the ground.
The slanted triangle ABD: solve with the sine rule for BD, then BD sin 48° gives the drone's height.
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