Bearings and trigonometric applications

IB Mathematics Applications & Interpretation SL· 45 min read

1. Three-Figure Bearings: Definition and Notation★★☆☆☆⏱ 15 min

📘 Definition

Three-figure bearing

Written as 3 digits e.g. 045°, 180°, 315°

An angle measured clockwise from the north direction at the starting point, used to describe direction between two points.

Example:

A direction 45° east of north is 045°, not 45°.

Bearings are always relative to the starting point, so the north line must be drawn at the point you are measuring from, not the destination. This is one of the most common sources of error in exam problems.

📐 Worked Example

Sketch and label a bearing of 125° from point A.

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Step 1: Draw a vertical line through point A, mark the top end with an arrow to indicate north.
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Step 2: Use a protractor to measure 125° starting from north, turning clockwise.
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Step 3: Draw a straight line from A along this angle, and label the bearing 125°.

2. Right-Angled Triangle Bearing Problems★★☆☆☆⏱ 20 min

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Many basic bearing problems form right-angled triangles when you draw horizontal (east-west) and vertical (north-south) lines from the destination. You can use SOHCAHTOA to find how far north or east the destination is from the starting point.

📐 Worked Example

A hiker walks 8 km from camp on a bearing of 025°. How far north is the hiker from camp after this walk?

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Step 1: Draw north line at camp, draw 025° clockwise, mark hiker position H, so CH = 8 km (C = camp).
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Step 2: We need the north distance, which is the side adjacent to the 25° angle in the right triangle formed by north, CH, and the east line through H.
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Step 3: Use cosine for adjacent side:
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$\text{North distance} = CH \times \cos(25^\circ) = 8 \times 0.9063 = 7.25 \text{ km (3 s.f.)}$

3. Non-Right Triangle Bearing Problems★★★☆☆⏱ 30 min

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When you have three points connected by two bearings, the resulting triangle is usually not right-angled. The key step is to calculate the internal angle of the triangle using the fact that all north lines are parallel.

📐 Worked Example

Town B is 12 km from town A on a bearing of 070°. Town C is 18 km from B on a bearing of 150°. Find the distance from A to C.

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Step 1: Draw parallel north lines at A and B. Mark AB = 12 km, BC = 18 km.
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Step 2: Calculate reverse bearing of A from B: 070° + 180° = 250°.
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Step 3: Find internal angle at B: 250° - 150° = 100°.
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Step 4: We have two sides and included angle, so use cosine rule:
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$AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(100^\circ)$
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$AC^2 = 12^2 + 18^2 - 2(12)(18)\cos(100^\circ) = 144 + 324 - 432(-0.1736) = 545.1$
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Step 5: Take the square root: $A C = 545.1 \approx 23.3$ km (3 s.f.)

4. Common Pitfalls

Wrong move:

Drawing the north line at the destination instead of the starting point when measuring bearing.

Why:

Bearings are always measured from the starting point, so this will give the wrong angle.

Correct move:

Always draw a new north line at every point you measure a bearing from.

Wrong move:

Writing a bearing of 45° as 45° instead of 045°.

Why:

IB exam markers require three-figure notation, you will lose a mark for incorrect notation.

Correct move:

Always add a leading zero to get three digits for bearings less than 100°.

Wrong move:

Measuring the bearing counter-clockwise from north.

Why:

By definition, bearings are always measured clockwise from north.

Correct move:

Always start at north and turn clockwise to measure your bearing angle.

Wrong move:

Forgetting that north lines are parallel when calculating internal angles.

Why:

This leads to incorrect internal angles, and wrong results for sides or bearings.

Correct move:

Always calculate the reverse bearing first, then use angle addition to find the internal angle of the triangle.

5. Quick Reference Cheatsheet

Concept	When to Use	Key Rule
Three-figure bearing	All direction questions	3 digits, 000° ≤ θ < 360°, clockwise from north
SOHCAHTOA	Right-angled triangle problems	$sin θ = O / H$ , $cos θ = A / H$ , $tan θ = O / A$
Sine Rule	2 angles + 1 side, 2 sides + non-included angle	$a / sin A = b / sin B = c / sin C$
Cosine Rule	2 sides + included angle, 3 sides	$a^{2} = b^{2} + c^{2} - 2 b c cos A$
Reverse bearing	Finding bearing of start from end	Add/subtract 180° to original bearing

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · Paper 1
6 mark navigation distance problem
2023 · Paper 2
8 mark bearing direction problem

What's Next

Now that you've mastered bearings and their trigonometric applications, you can move on to more advanced topics including 3D trigonometry, which relies on the same sine and cosine rule skills you practiced here. Bearings also form the foundation for many real-world navigation and distance modelling problems common to IB AI SL, which appear frequently in both Paper 1 and Paper 2. This topic builds directly on your knowledge of non-right triangle trigonometry and is a common exam question worth 5-8 marks. Mastering it will prepare you for context-rich problems that require diagram drawing and information extraction.