# Bearings and trigonometric applications

> IB Mathematics Applications & Interpretation SL · Geometry and Trigonometry
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u3-bearings-and-trigonometric-applications/

This subtopic covers three-figure bearing notation, drawing and interpreting bearings, and applying sine and cosine rules to solve real-world navigation problems common in IB AI SL exams.

**Prerequisites:** [Sine and cosine rules for non-right triangles](https://www.owlsprep.com/study/ib-math-ai-sl-u3-sine-and-cosine-rules/); [Right-angled trigonometry](https://www.owlsprep.com/study/ib-math-ai-sl-u3-right-angled-trigonometry/)

## Learning objectives

- Correctly interpret and draw three-figure bearings
- Apply sine and cosine rules to solve bearing problems
- Solve real-world navigation problems involving bearings
- Calculate unknown distances and directions from given bearing data

## Three-Figure Bearings: Definition and Notation

**Three-figure bearing** — An angle measured clockwise from the north direction at the starting point, used to describe direction between two points.

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

*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.

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

> **tip**
>
> Always write bearings with three digits. Add a leading zero for angles less than 100° (e.g. 030° not 30°) to earn full marks for notation.

## Right-Angled Triangle Bearing Problems

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.

> **mnemonic**
>
> SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent

**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?

1. Step 1: Draw north line at camp, draw 025° clockwise, mark hiker position H, so CH = 8 km (C = camp).
2. 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.
3. Step 3: Use cosine for adjacent side:
4. $$\text{North distance} = CH \times \cos(25^\circ) = 8 \times 0.9063 = 7.25 \text{ km (3 s.f.)}$$

*Calculator:* allowed

## Non-Right Triangle Bearing Problems

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.

> **tip**
>
> To find the reverse bearing (bearing of point A from point B, when you know bearing of B from A), add 180° if the original bearing is less than 180°, or subtract 180° if it is more than 180°.

**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.

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

**Exam command terms**

- **Find the bearing** — Give your answer as a three-digit angle between 000° and 359°, rounded to the nearest degree unless stated otherwise

- **Calculate the distance** — Give your answer to 3 significant figures, following IB convention

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Drawing the north line at the destination instead of the starting point when measuring bearing.
  - Why it fails: Bearings are always measured from the starting point, so this will give the wrong angle.
  - Correct: Always draw a new north line at every point you measure a bearing from.
- **Wrong:** Writing a bearing of 45° as 45° instead of 045°.
  - Why it fails: IB exam markers require three-figure notation, you will lose a mark for incorrect notation.
  - Correct: Always add a leading zero to get three digits for bearings less than 100°.
- **Wrong:** Measuring the bearing counter-clockwise from north.
  - Why it fails: By definition, bearings are always measured clockwise from north.
  - Correct: Always start at north and turn clockwise to measure your bearing angle.
- **Wrong:** Forgetting that north lines are parallel when calculating internal angles.
  - Why it fails: This leads to incorrect internal angles, and wrong results for sides or bearings.
  - Correct: Always calculate the reverse bearing first, then use angle addition to find the internal angle of the triangle.

## 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 | $\sin\theta = O/H$, $\cos\theta = A/H$, $\tan\theta = O/A$ |
| 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 | $a^2 = b^2 + c^2 - 2bc\cos A$ |
| Reverse bearing | Finding bearing of start from end | Add/subtract 180° to original bearing |

## 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.

- [Sinusoidal functions: amplitude, period, translation](https://www.owlsprep.com/study/ib-math-ai-sl-u3-sinusoidal-functions-amplitude-period-translation/)
- [Voronoi diagrams and closest site problems](https://www.owlsprep.com/study/ib-math-ai-sl-u3-voronoi-diagrams-and-closest-site/)
- [Triangle area formula for non-right triangles](https://www.owlsprep.com/study/ib-math-ai-sl-u3-triangle-area-formula-for-non/)

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