Year 11 Higher · Angles, bearings and trigonometry
Understand what a bearing describes
Learn how to work with bearings, similar shapes and right-angled triangle methods, and to construct a line at right angles through a point.
Definition
A bearing describes a direction as an angle measured clockwise from north, written with three figures, so a journey can be recorded exactly using distances and directions. Because every north line points the same way, north lines are parallel and the usual angle rules apply between them, and right-angled triangle methods turn a bearings diagram into calculated lengths and angles. Similar shapes carry the same exactness: equal angles and one fixed ratio between corresponding sides give missing measurements without measuring. Accurate scale drawings depend on precise construction, including drawing a line at right angles through a point with compasses.
Why Do I Have to Learn This?
Angles measure turn, from door hinges to skate ramps. Reading them helps you build, draw and navigate accurately.
Key Rules:
- 1. A bearing is the angle measured clockwise from north at the point you are travelling from, always written with three figures: 070°, not 70°.
Remember:
- •A bearing is three figures, clockwise from north; parallel north lines and right-angled triangle methods do the rest.
Don’t Forget...
- •Measuring a bearing from the wrong point, or anticlockwise from north. The bearing of L from P is measured at P, clockwise from P's north line; measuring at L instead, or turning anticlockwise, gives a different angle entirely.
- •Applying the scale factor to angles as well as lengths in similar shapes. Similarity multiplies every length by the same factor but leaves every angle unchanged; a triangle with different angles would be a different shape, not a similar one.
- •Reaching for Pythagoras when the question involves an angle, or picking the wrong trig ratio. Pythagoras links the three sides only; once an angle is given or wanted you need sin, cos or tan, chosen by which two of opposite, adjacent and hypotenuse appear in the question.
Acute angle
An angle smaller than 90 degrees.
An angle of 45° is acute.
Worked Example:
A ship sails 12 km due east from a port P, then 5 km due north to reach a lighthouse L. Work out the bearing of L from P, to the nearest degree.
Sketch the journey: 12 km due east then 5 km due north forms a right-angled triangle, with the right angle at the turning point.
The angle at P between due east and the direct line PL has opposite side 5 and adjacent side 12, so tan of the angle is 512.
Inverse tan of 512 gives 22.6° to 1 decimal place.
Bearings are measured clockwise from north, and due east is 090°, so the bearing of L from P is 90° - 22.6° = 67.4°.
Answer: 067° to the nearest degree
Why It Works:
North lines at different points are parallel, so the parallel-line angle facts transfer a bearing from one point to another exactly. Trigonometric ratios are constant for a given angle because every right-angled triangle containing that angle is similar, and that same similarity is what lets one scale factor carry every length between similar shapes.
Another Example:
Triangles ABC and PQR are similar, with AB corresponding to PQ. AB = 6 cm, PQ = 9 cm, BC = 8 cm and angle A = 40°. Find the length QR and the size of angle P.
Find the scale factor from the matching pair of sides: 96 = 1.5.
QR corresponds to BC, so QR = 8 x 1.5 = 12 cm.
Angles do not scale in similar shapes, so angle P = angle A = 40°.
Answer: QR = 12 cm and angle P = 40°
Animated Example:
Free Understand what a bearing describes Worksheet
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Before this: Angles (Year 10 Higher)
Let’s Practise the Concept
Step 1 of 3
Put them in order
Tap the numbers from smallest to largest.
In order so far: nothing yet