Key Idea: This topic turns a word problem about heights, distances and directions into a triangle you can solve. The exam pattern is always the same: read → draw the diagram → pick the rule (right-angled trig, or the sine/cosine rule).
📐 Elevation, depression & bearings
| Term | What it is | Watch for |
|---|---|---|
| Angle of elevation | Measured up from the horizontal to a point above you | From the horizontal, not the vertical |
| Angle of depression | Measured down from the horizontal to a point below you | = the elevation back up (alternate angles) |
| Three-figure bearing | Angle measured clockwise from North, 3 digits | 5° is 005°; sweep clockwise, never anticlockwise |
N = 000°, E = 090°, S = 180°, W = 270°. A direction like N40°E = 040°; S30°W = 180° + 30° = 210°; South-East = 135°. Back bearing = ±180° (A from B given B from A): add 180° if it's under 180°, subtract if it's 180° or more. Keep it between 000° and 360°.
🔺 Which rule? (Paper 1 by-hand, Paper 2 with GDC)
| The triangle is… | Use | When |
|---|---|---|
| Right-angled | SOH-CAH-TOA (usually tan) | One angle + one side; height vs. ground distance |
| Non-right, 2 sides + included angle | Cosine rule for the third side | Two journey legs on different bearings |
| Non-right, 3 sides (angle wanted) | Cosine rule rearranged for the angle | All three distances known |
| Non-right, side + its opposite angle | Sine rule | An angle and the side facing it are paired |
✏️ IB-style worked examples
IB-style question — elevation, find a height
From a point 65 m from the base of a tower, the angle of elevation to the top is 38°. Find the height of the tower.
Step by step:
Height is opposite, 65 m is adjacent → use tan.
Solve for the height.
About 50.8 m. (Add eye height if the angle is taken from a person's eye.)
IB-style question — depression, find a distance
From the top of a 55 m cliff, the angle of depression to a buoy is 24°. Find the buoy's distance from the base of the cliff.
Step by step:
Depression (24°) equals the elevation from the buoy, so it sits inside the triangle.
Rearrange for the distance.
About 124 m.
IB-style question — two observers, cosine rule for an angle
An object is seen from two points. The slant distances to it are 120 m and 150 m, and the two points are 90 m apart. Find the angle at the object.
Step by step:
Three sides, angle wanted → cosine rule rearranged.
Evaluate, then inverse-cosine.
About 36.9°.
IB-style question — bearing of one place, then its back bearing
A direction is given as 'North-West'. (a) Write it as a three-figure bearing. (b) If the bearing of B from A is 050°, find the bearing of A from B.
Step by step:
(a) NW is 45° west of North (000°) — that is 45° back from a full turn of 360°.
(b) 050° is under 180°, so add 180° for the back bearing.
(a) 315°. (b) 230°.
IB-style question — journey on two bearings, cosine rule
A ship sails 9 km on a bearing of 050°, then 6 km on a bearing of 110°. Find its direct distance from the start.
Step by step:
Interior angle at the turn, from the North lines: 180° − (110° − 50°).
Two legs + included angle → cosine rule.
Square-root for the distance.
About 13.1 km.
Important: Bearings are measured clockwise from North, always with three digits. Sketch the North arrow first and sweep clockwise — never anticlockwise, never from South, never from the East line. A bearing drawn the wrong way sends every later angle and side astray.
Tap each card to reveal the answer.
Elevation vs. depression — measured from what? Both from the horizontal — up = elevation, down = depression (never the vertical).
Angle of depression to a boat is 18° — what's the elevation from the boat? 18° — depression equals the elevation back (alternate angles).
Write the direction 'due West' as a bearing 270° — three-quarters of a turn clockwise from North.
Bearing of Y from X is 200° — bearing of X from Y? 020° — it's 180° or more, so subtract 180°.
Two legs of 7 km and 4 km with a 60° turn between — which rule? Cosine rule — two sides and the included (interior) angle.
Write the bearing for an angle of 7° clockwise from North 007° — bearings always use three digits.
Exam Tips
- Always: read → draw the diagram → choose the rule. The sketch is where the marks start.
- Bearings go clockwise from North and use three digits — 5° is 005°.
- Depression equals the elevation back up; mark that equal angle inside your triangle.
- Right-angled → SOH-CAH-TOA (often tan). Non-right → sine or cosine rule.
- For journeys, find the interior angle at the turn from the North lines — it's rarely just the difference of the two bearings.