---
title: "Bearing Problems with the Sine and Cosine Rules"
description: "Turn a journey described by bearings into a triangle, then solve it with the sine or cosine rule instead of drawing it to scale."
canonical: https://lightmysky.com/learn/mathematics/bearing-problems-with-the-sine-and-cosine-rules-mt_YQFdU4aiFP
source: https://lightmysky.com/learn/mathematics/bearing-problems-with-the-sine-and-cosine-rules-mt_YQFdU4aiFP.md
retrieved: 2026-09-12
---

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# Bearing Problems with the Sine and Cosine Rules

Turn a journey described by bearings into a triangle, then solve it with the sine or cosine rule instead of drawing it to scale.

Subject: Mathematics · Area: Geometry · Ages 15 to 16
Page: https://lightmysky.com/learn/mathematics/bearing-problems-with-the-sine-and-cosine-rules-mt_YQFdU4aiFP

## Ready when they can

- Draw the triangle a two-leg journey makes and mark the angle each bearing contributes
- Choose the sine or cosine rule from what the sketch gives, then compute the return distance
- Convert a computed angle back into a three-figure bearing for the answer

## Lesson: Bearings journeys without scale drawings

Bearings run clockwise from north with three figures, so 40 degrees becomes 040. A two-leg journey makes a triangle: draw each leg, then mark the angle each bearing contributes at the turning point. The turn comes from subtracting the bearings, so a 060 leg followed by 150 differs by 90.

Let the sketch tell you which rule fits. Two sides with their included angle, the classic SAS layout, calls for the cosine rule. An angle paired with its opposite side calls for the sine rule. For 60 km and 80 km meeting at 90 degrees, the return side is 100 km.

**Example.** A yacht sails 60 km, turns through 90 degrees, then sails 80 km. The cosine rule on the SAS layout gives the return distance: root of 3600 plus 6400, which is 100 km. Keep full calculator accuracy until the final rounding.

**Tip.** A computed angle is not automatically the bearing. Draw the north line at the return point and translate clockwise, so 40 degrees becomes 040. Sense-check at the end: a north east return should read between 000 and 090.

**Recap.** Sketch the journey triangle, pick the rule the sketch offers, solve the return, then translate angles back into three-figure bearings.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Bearings and Scale Drawings](https://lightmysky.com/learn/mathematics/bearings-and-scale-drawings-mt_9q32jC0Fk_)
- [The Sine Rule](https://lightmysky.com/learn/mathematics/the-sine-rule-mt_cTt_-9KZho)
- [Choosing a Triangle Method](https://lightmysky.com/learn/mathematics/choosing-a-triangle-method-mt_nubZzZjVvn)
- [The Cosine Rule](https://lightmysky.com/learn/mathematics/the-cosine-rule-mt_yy73gK5CEP)

## Opens up

- [Forces as Vectors: Resolving and Resultants](https://lightmysky.com/learn/mathematics/forces-as-vectors-resolving-and-resultants-mt_-qDRgd2g8v)