Lesson 5: Bearings and Trigonometric Applications

👨🏽‍🏫 Introduction

Imagine navigating a ship or a plane without GPS. How would you know which direction to go? 🧭 Bearings help us describe direction precisely, especially in navigation and surveying.

In this lesson, you’ll learn how to interpret and calculate bearings, and use trigonometry to solve problems involving distances and angles between objects. Perfect for real-world applications in travel, shipping, engineering, and even rescue missions! 🚢✈️

• 🧭 Understand 3-figure bearings and how to read them
• 📐 Use sine and cosine rules in non-right-angled triangles
• 🛠 Solve real-world problems involving direction and distance

🧠 Key Concepts

📌 1. Bearings (3-Figure Notation)

• A bearing is measured from the north line clockwise to the direction of travel.
• Always written as a 3-digit angle (e.g. North-East is 045°, East is 090°, South is 180°, South-West is 225°)

📌 2. Trigonometric Tools Used with Bearings

• Use the sine rule when given two angles and one side (AAS or ASA):
  
• Use the cosine rule when given two sides and included angle (SAS):
  

🔍 Worked Examples

1. 1. Example 1 – Using Bearings:
      A ship sails from port A to port B on a bearing of 060° for 100 km. Sketch this on a diagram starting with north, then mark 60° clockwise. This shows the direction from A to B.
      🟢 Answer: Bearing from A to B is correctly shown as 060°.

1. 1. Example 2 – Using Cosine Rule to Find Distance:
      Two towns, P and Q, are 50 km apart. From a third town R, P is on a bearing of 045° and Q on a bearing of 120°, with PR = 40 km and QR = 60 km. Find the distance PQ.
      Angle between the bearings =
      Apply cosine rule:
      
      
      
      🟢 Answer: PQ ≈ 62.9 km

1. Example 3 – Finding a Bearing from Coordinates:
   A person walks 300 m due north, then 400 m due east. What is the bearing from the starting point to the final point?
   Use tan(θ):
   
   Bearing from start = 090° – 53° = 037°
   🟢 Final Bearing: 037°

✍🏽 Practice Exercises

🅰️ Section A – Bearings Basics

1. What is the bearing of East? South-West?
2. On a map, point A is directly north of point B. What is the bearing of A from B?

🅱️ Section B – Using Cosine Rule

1. Two boats leave the same port. One sails on a bearing of 030° for 80 km, the other on a bearing of 120° for 100 km. Find the distance between the boats.

🅾️ Section C – Apply Tangents to Bearings

1. A person walks 500 m due east, then 300 m due north. Find the bearing from the starting point.

✅ Answers and Explanations

• A1: East = 090°, South-West = 225°
• A2: From B to A (North) = 000°
• B1: Angle between = 120° – 30° = 90°
  Apply cosine rule:
  
  
• C1:
  Bearing = 090° – 31° = 059°

📌 Recap

• 🧭 Bearings describe direction using 3-digit angles measured clockwise from north
• 📐 Cosine and sine rules help solve triangle-based direction problems
• 📊 Tangents help with bearings from coordinate movements

💭 Reflection Prompt

Think about your path to school or work. Can you estimate the bearing from your home to your destination? What if you traveled diagonally—could you use trigonometry to find how far you actually traveled?