Bearings & Navigation — SL 3.4 Bearings and navigation...

SECTION 1

Historical Context & Motivation

Long before GPS satellites orbited the Earth, sailors, explorers, and pilots needed reliable methods to describe direction. The question was deceptively simple: if you are standing at one point and need to travel to another, how do you communicate the exact direction to follow? Early civilizations answered this with landmarks and stars, but as voyages grew longer and oceans offered no landmarks at all, a precise numerical system became essential. Bearings emerged as that system — a way to express any direction as a single number measured clockwise from north.

~1100

The Magnetic Compass Arrives in Europe

Chinese-invented magnetic compasses reach European navigators, giving sailors a fixed reference direction (magnetic north) for the first time on open water.

1500s

The 32-Point Compass Rose

Mariners develop a 32-point compass rose dividing the circle into fine increments such as "north-northeast." While useful, this system lacked the precision that growing empires demanded for long-distance voyages.

1700s

Degree-Based Bearings Adopted

The compass points are replaced by degree measurements (0° to 360°), giving navigators a continuous, precise scale. Three-figure bearings (e.g., 045°, 270°) become the standard in military and maritime contexts.

1900s–Today

Modern Navigation & GPS

Aviation and satellite technology inherit the bearing system. Pilots file flight plans using bearings, search-and-rescue teams communicate positions with bearings, and GPS devices still display bearing information alongside coordinates.

The core question this topic addresses is: given distances and directions described as bearings, how can we use trigonometry to calculate unknown distances, angles, or positions? This is exactly what IB SL 3.4 explores — turning real navigation scenarios into solvable triangle problems.

SECTION 2

Core Principles & Definitions

Before solving navigation problems, you need to master a few foundational ideas. A bearing is an angle measured clockwise from north, always written as a three-figure number. So due east is written 090°, not 90°. Due south is 180°, and due west is 270°. This convention eliminates any ambiguity about which direction is meant.

Always From North

Every bearing starts at the north line (0° / 360°) at the point you are measuring from. Draw a vertical north line at your starting position before doing anything else.

Always Clockwise

Rotate clockwise from north to the direction of travel. Anti-clockwise angles must be converted: an anti-clockwise angle of 40° from north equals a bearing of 360° − 40° = 320°.

Always Three Figures

Bearings are always given with three digits. North itself is 000° (or sometimes 360°). Single-digit bearings get two leading zeros; two-digit bearings get one. For example, 7° becomes 007°.

Bearing OF vs. Bearing FROM

"The bearing of B from A" means: stand at A, face north, and measure the clockwise angle to the line AB. It describes the direction you would travel to go from A to B.

Back Bearings

The reverse direction is called the back bearing. If the bearing of B from A is θ, then the bearing of A from B equals θ + 180° (if θ < 180°) or θ − 180° (if θ ≥ 180°).

✦ KEY TAKEAWAY

Think of bearings like reading a clock face laid on top of a map. The 12 o'clock position is north (000°), the 3 o'clock position is east (090°), the 6 o'clock position is south (180°), and the 9 o'clock position is west (270°). You always read the angle going clockwise, just as clock hands move.

SECTION 3

Visual Explanation — The Bearing Compass

A compass diagram showing the bearing of point B from point A. The north line is drawn vertically from A, and the angle is measured clockwise to the line AB, giving a bearing of 048°. Key compass directions and their three-figure equivalents are shown for reference.

In the diagram above, notice how the north arrow is drawn at point A — the point we are measuring from. The bearing 048° tells us that if you faced north at A and rotated clockwise by 48°, you would be looking directly at B. This diagram is the template for every bearing problem you will encounter. The first step in any problem is always to draw the north line at the relevant point and then mark the clockwise angle.

SECTION 4

Mathematical Framework

Navigation problems in SL 3.4 almost always reduce to solving triangles. Once you sketch the journey, label distances, and mark the bearings, you'll find that the angles inside the triangle can be deduced from the bearings using properties of parallel lines (all north lines are parallel). The two main tools are the sine rule and the cosine rule, along with simple right-angle trigonometry when the triangle is right-angled.

SINE RULE

a / sin A = b / sin B = c / sin C

Used when you know two angles and one side (AAS), or two sides and a non-included angle (SSA). Here a is the side opposite angle A, and so on.

COSINE RULE (FINDING A SIDE)

c² = a² + b² − 2ab cos C

Used when you know two sides and the included angle (SAS). Here C is the angle between sides a and b.

COSINE RULE (FINDING AN ANGLE)

cos C = (a² + b² − c²) / (2ab)

Rearranged form used when you know all three sides (SSS) and need to find an angle.

BACK BEARING

Back bearing = θ + 180° (if θ < 180°) or θ − 180° (if θ ≥ 180°)

The back bearing is the direction from B back to A. It is always exactly 180° different from the forward bearing.

💡 Converting Bearings to Triangle Angles

Since all north lines are parallel, you can use co-interior angles (which sum to 180°) and alternate angles (which are equal) to convert bearing information into the interior angles of your triangle. Draw the north line at every vertex and label every angle you can find — this is the key step that connects bearings to trigonometry.

SECTION 5

Detailed Breakdown — From Bearings to Triangle Angles

The trickiest part of bearing problems is not the trigonometry itself — it's translating the bearings into angles inside a triangle. Let's walk through this critical skill with a detailed diagram. Consider a ship that sails from port A on a bearing of 065° to point B, then changes course to a bearing of 140° to reach point C. To find the angle at B inside triangle ABC, we need to use the relationship between the two bearing lines and the parallel north lines.

A ship sails from A on bearing 065° to B, then from B on bearing 140° to C. The interior angle at B is found using the back bearing of A from B (245°) minus the forward bearing from B to C (140°), giving ∠B = 105°.

Here is the step-by-step method to find interior triangle angles from bearings. First, at vertex B, draw a north line. The back bearing of A from B is 065° + 180° = 245°. This means the line BA points in the 245° direction from B's north. The forward bearing from B to C is 140°. The angle between these two lines — measured going clockwise from the BA direction to the BC direction — is 245° − 140° = 105°. This is the interior angle ∠ABC of the triangle.

Step 1: Draw north lines at every vertex of the triangle.
Step 2: Calculate back bearings where needed (add or subtract 180°).
Step 3: Use the difference between the back bearing and the forward bearing at each vertex to find the interior angle.
Step 4: Verify that all interior angles sum to 180°.

SECTION 6

Worked Example — Search and Rescue

A coast guard station at point P spots a distress signal. A rescue boat leaves P on a bearing of 072° and travels 15 km to reach a checkpoint Q. From Q, the boat changes course to a bearing of 150° and travels 22 km to reach the stranded vessel at R. Find the distance PR and the bearing of R from P.

Finding the Direct Distance PR and Bearing of R from P

Step 1 — Sketch and LabelDraw triangle PQR. Mark the north line at P and at Q. Label PQ = 15 km and QR = 22 km. The bearing from P to Q is 072° and the bearing from Q to R is 150°.

Step 2 — Find the Interior Angle at QAt vertex Q, draw the north line and identify the two directions: the direction back to P and the direction forward to R. The back bearing of P from Q is 072° + 180° = 252°. The forward bearing from Q to R is 150°. Since 150° is less than 252°, both directions are measured clockwise from north at Q, with the QP direction at 252° and the QR direction at 150°. The interior angle ∠PQR is the clockwise angle from the QR direction to the QP direction, which is simply 252° − 150° = 102°.

∠PQR = 102°

Step 3 — Apply the Cosine Rule to Find PRUsing c² = a² + b² − 2ab cos C where a = QR = 22 km, b = PQ = 15 km, and C = ∠PQR = 102°: PR² = 22² + 15² − 2(22)(15) cos 102° PR² = 484 + 225 − 660 × (−0.2079) PR² = 709 + 137.2 = 846.2 PR = √846.2 ≈ 29.1 km

PR ≈ 29.1 km

Step 4 — Use the Sine Rule to Find ∠QPRsin(∠QPR) / QR = sin(∠PQR) / PR sin(∠QPR) / 22 = sin 102° / 29.1 sin(∠QPR) = 22 × sin 102° / 29.1 sin(∠QPR) = 22 × 0.9781 / 29.1 = 0.7393 ∠QPR = sin⁻¹(0.7393) ≈ 47.7°

∠QPR ≈ 47.7°

Step 5 — Find the Bearing of R from PThe angle ∠QPR ≈ 47.7° is the angle at P inside the triangle, between sides PQ and PR. To find the bearing of R from P, we need to determine whether R lies clockwise or counter-clockwise of the direction PQ when viewed from P. Refer to the sketch: from Q, the route turned to a bearing of 150°, which is clockwise of the back bearing 252° (i.e., 150° < 252°), meaning R is to the right (clockwise) of the line PQ when viewed from P. Therefore the bearing of R from P is found by adding ∠QPR to the bearing of PQ: Bearing of R from P = bearing of Q from P + ∠QPR = 072° + 47.7° = 119.7° ≈ 120° To confirm the direction: on the sketch, R is to the south-east of P, and a bearing of 120° (between east at 090° and south-east at 135°) is consistent with this position. If instead R had been to the counter-clockwise (left) side of PQ from P, we would subtract: bearing = 072° − ∠QPR. Always check your sketch to determine which case applies.

Bearing of R from P ≈ 120°

SECTION 7

Common Strengths, Limitations & Pitfalls

Key tips and common errors in bearing problems

Strength / Tip	Common Pitfall	How to Avoid It
Bearings provide a universal, unambiguous direction system	Forgetting to write three figures (e.g., writing 72° instead of 072°)	Always pad with leading zeros. If the IB examiner sees '72°' labelled as a bearing, marks may be deducted.
North lines are always parallel, enabling angle rules	Measuring the bearing anti-clockwise or from south	Write 'FROM NORTH, CLOCKWISE' on your diagram as a reminder until it becomes automatic.
Problems reduce to standard triangle trig (sine/cosine rules)	Using the wrong rule: sine rule when you have SAS, or cosine rule when you have AAS	Check what you know: two sides + included angle → cosine rule; two angles + one side → sine rule.
Back bearings are a quick mental check	Getting the back bearing wrong (e.g., always adding 180° even when θ ≥ 180°)	If θ < 180°, add 180°. If θ ≥ 180°, subtract 180°. The result must be between 000° and 360°.
Diagrams make the problem visual and manageable	Trying to solve without a diagram	ALWAYS draw a clear, labelled sketch before writing any equations. This is the single most important habit.

✦ KEY TAKEAWAY

Think of a bearing problem like assembling flat-pack furniture: you must follow the instructions in order. Step 1: draw a sketch. Step 2: mark all north lines. Step 3: convert bearings to interior angles. Step 4: choose the right trig rule. Skip any step and things won't fit together — but follow the order and the answer falls into place.

SECTION 8

Connection to Advanced Topics

SL 3.4 bearings form the foundation for more advanced navigation concepts that appear in HL topics and in university-level mathematics and physics. Understanding how directions translate to triangle problems prepares you for 3D trigonometry, vector navigation, and coordinate geometry on the sphere.

From SL 3.4 to advanced navigation

SL 3.4 Concept	Advanced Extension
Two-dimensional bearings between points	Three-dimensional bearings including altitude (angles of elevation / depression combined with bearings)
Sine and cosine rules in flat triangles	Spherical trigonometry for great-circle navigation on Earth's curved surface
Back bearings (adding/subtracting 180°)	Vector reversal and unit direction vectors in mechanics
Static triangle problems (fixed positions)	Relative velocity problems — where two moving objects create a changing triangle over time

In the IB HL course, you may encounter Voronoi diagrams and optimization on maps, which also rely on understanding how directions and distances define regions. Outside of IB, fields like aerospace engineering, maritime law (determining territorial waters), and even video game development (pathfinding algorithms) all build on the same bearing and triangle concepts you are learning now.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

A hiker walks due west. What is the three-figure bearing of their direction of travel? Explain why bearings are measured clockwise from north rather than anti-clockwise.

PROBLEM 2 — BASIC CALCULATION

The bearing of town B from town A is 125°. What is the bearing of A from B?

PROBLEM 3 — INTERMEDIATE

A yacht sails from harbour H on a bearing of 040° for 30 km to point A. It then sails on a bearing of 160° for 45 km to point B. Find the distance HB, giving your answer to one decimal place.

PROBLEM 4 — APPLIED

A rescue helicopter at base P must fly to a climber stranded at point R. The helicopter first detects the climber by radio signal from station Q, which is 20 km from P on a bearing of 310°. Station Q reports the climber is 35 km away on a bearing of 050° from Q. Find the direct distance PR and the bearing the helicopter should fly from P to reach R.

PROBLEM 5 — CRITICAL THINKING

Three lighthouses form a triangle. Lighthouse A is due north of lighthouse C. The bearing of B from A is 110° and the bearing of B from C is 040°. If the distance from A to B is 18 km, find the distance from C to B and the distance from A to C. Verify your answers by checking that the interior angles of the triangle sum to 180°.

SUMMARY

Lesson Summary

A bearing is an angle measured clockwise from north, always written as a three-figure number (e.g., 045°, 180°, 310°). To find a back bearing, add 180° if the bearing is less than 180°, or subtract 180° if it is 180° or more. Navigation problems are solved by drawing a clear sketch, marking north lines at every vertex, converting bearings into interior triangle angles using parallel line rules, and then applying the sine rule or cosine rule as appropriate.

The key to success in SL 3.4 bearing problems is a systematic approach: always sketch first, label all known distances and bearings, convert bearings to interior angles, choose the correct trigonometric rule based on the information available (SAS → cosine rule, AAS → sine rule), and verify your answer makes sense in the context of the map. These skills connect directly to real-world applications in aviation, maritime navigation, surveying, and search-and-rescue operations.