Mathematics · Lesson 21

Bearings, angles of elevation & depression

JAMBWAECNECO

What the syllabus asks

JAMB: Trigonometry: angles of elevation and depression, bearings, application to heights and distances

WAEC: Trigonometry: bearings (three-figure and compass), angles of elevation and depression, solving problems in two dimensions using trigonometric ratios, sine and cosine rules

NECO: Trigonometry: bearings (three-figure and compass), angles of elevation and depression, solving problems in two dimensions using trigonometric ratios, sine and cosine rules

What you will learn

Write and convert three-figure and compass bearings.
Find back bearings.
Solve distance and bearing problems using right-angled triangles.
Solve bearing problems using the sine and cosine rules.
Solve angle of elevation and depression problems.
Handle problems with two observation points.

Key points

Three-figure bearings: clockwise from North, three digits (045°, 210°).
Compass: N θ E = θ; S θ E = 180° − θ; S θ W = 180° + θ; N θ W = 360° − θ.
Back bearing = bearing ± 180°.
A journey d on bearing θ: d cos θ North, d sin θ East.
Interior angle at a turn: use back bearings and parallel North lines.
Right angle → SOH CAH TOA/Pythagoras; otherwise sine and cosine rules.
Elevation (up) and depression (down) are measured from the horizontal; they are equal as alternate angles.
Two-point problems: share the height h, write two equations, solve; add observer’s eye height if given.

Start here: What a bearing is

Pilots, ship captains, surveyors and soldiers need a precise way to describe direction. "Turn left a bit" is not good enough at sea. A bearing gives direction as an angle measured from a fixed reference: North.

Three-figure bearing: the angle measured clockwise from North, written with three digits (from 000° to 360°).

So East is 090°, South is 180°, West is 270°, and North-East is 045°. An angle of 5° clockwise from North is written 005°, not 5°.

The phrase "the bearing of B from A" means: stand at A, face North, and turn clockwise until you face B. The point after "from" is where the North line is drawn.

Worked example 21.1

Write as three-figure bearings: (a) due West; (b) South-East; (c) 20° East of South.

(a) West is three right angles clockwise from North: 270°.
(b) South-East is halfway between East (090°) and South (180°): 135°.
(c) Start at South (180°) and turn 20° towards East, which is anticlockwise: 180° − 20° = 160°.

270°; 135°; 160°

⚠️ Common mistake: measuring anticlockwise, or from East (as in ordinary angle work). Bearings are always from North, clockwise.

Try it yourself. Write the three-figure bearing of North-West.

Show solution

315°.

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Practice questions

1. Convert N 65° W to a three-figure bearing.

Show worked answer

295°.

2. The bearing of P from Q is 048°. Find the bearing of Q from P.

Show worked answer

228°.

3. A boat sails 15 km on 150°. How far South and East is it?

Show worked answer

South 15 cos 30° = 12.99 km; East 15 sin 30° = 7.5 km.

4. From 25 m, the elevation of a building’s top is 50°. Find its height.

Show worked answer

25 tan 50° = 29.79 m.

5. From a cliff 100 m high, the depression of a ship is 15°. How far is the ship?

Show worked answer

100/tan 15° = 373.2 m.

Key words

Bearing: The direction of one point from another, measured as an angle from North.
Three-figure bearing: A bearing measured clockwise from North, written with three digits.
Compass bearing: A direction given as N/S, an angle, then E/W, e.g. S 25° E.
Back bearing: The bearing of the starting point from the destination.
Angle of elevation: The angle above the horizontal from an observer to an object higher up.
Angle of depression: The angle below the horizontal from an observer to an object lower down.
Horizontal: A level line, parallel to the ground.
Line of sight: The straight line from the observer’s eye to the object.
Sine rule: a/sin A = b/sin B = c/sin C.
Cosine rule: a² = b² + c² − 2bc cos A.
Clinometer: An instrument for measuring angles of elevation.

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