How to Solve: Bearings with Trigonometry

How to Solve: Bearings with Trigonometry

Introduction

"You’re a pilot navigating a storm, a ship captain avoiding rocks, or an engineer designing a bridge—bearings with trigonometry are how you find your way. Master this, and you’ll ace every navigation question on your exam."

What You Need To Know First

1. Right-angled triangle trigonometry (SOHCAHTOA)
2. The four quadrants and angles greater than 90° (ASTC rule)
3. How to draw and interpret bearings (three-figure notation, e.g., 045°)

Key Vocabulary

Formulas To Know

Step-by-Step Method

Step 1: Draw the Diagram

• Sketch a North line (vertical) at the starting point.
• Draw the bearing as a line clockwise from North.
• Label all given distances and angles.

Step 2: Identify the Triangle

• If the problem involves two bearings and a distance, you’ll likely have a non-right-angled triangle.
• If one angle is 90°, use SOHCAHTOA.
• If no right angle, use Sine Rule or Cosine Rule.

Step 3: Find Missing Angles

• Use bearing rules:
• If two bearings meet, the angle between them is the difference (e.g., 045° and 120° → 120° – 45° = 75°).
• If bearings cross, use alternate angles or supplementary angles.
• For back bearings, add/subtract 180°.

Step 4: Apply Trigonometry

• Right-angled triangle? → Use SOHCAHTOA.
• Non-right-angled triangle? → Use Sine Rule (for angles/sides) or Cosine Rule (for a side when you have two sides and the included angle).

Step 5: Calculate the Unknown

• Solve for the missing side or angle.
• Round to 1 decimal place (unless specified otherwise).

Step 6: Check Units and Direction

• Ensure the answer is in km, m, or ° as required.
• If asked for a bearing, give it as a three-figure number (e.g., 050°, not 50°).

Worked Examples

Example 1 – Basic (Right-Angled Triangle)

Question: A ship sails 10 km on a bearing of 030°. How far East is it from its starting point?

Solution: 1. Draw the diagram:
- North line at start.
- Bearing 030° (30° clockwise from North).
- 10 km line at 30°.
- Forms a right-angled triangle with East (x) and North (y) components.

1. Identify the triangle:
2. Right angle between North and East.

3. Angle at start = 30°.

4. Use SOHCAHTOA:

5. We need the East component (x) → Adjacent to 30°.
6. cos 30° = Adjacent / Hypotenuse → cos 30° = x / 10
7. x = 10 × cos 30° = 10 × 0.8660 = 8.7 km East

What we did and why: - We used cosine because the East distance is adjacent to the 30° angle. - The hypotenuse was the 10 km sailing distance.

Example 2 – Medium (Non-Right-Angled Triangle)

Question: Two ships leave a port at the same time. - Ship A sails 15 km on a bearing of 050°. - Ship B sails 20 km on a bearing of 110°. How far apart are the two ships?

Solution: 1. Draw the diagram:
- North line at port.
- Ship A: 15 km at 050°.
- Ship B: 20 km at 110°.
- Angle between paths = 110° – 50° = 60°.

1. Identify the triangle:
2. No right angle → Cosine Rule needed.

3. Sides: a = 15 km, b = 20 km, angle C = 60°.

4. Apply Cosine Rule:

5. c² = a² + b² – 2ab cos C
6. c² = 15² + 20² – 2(15)(20)cos 60°
7. c² = 225 + 400 – 600 × 0.5
8. c² = 625 – 300 = 325
9. c = √325 = 18.0 km apart

What we did and why: - We used the Cosine Rule because we had two sides and the included angle. - The angle between the ships was 60° (110° – 50°).

Example 3 – Exam Style (Disguised Problem)

Question: A hiker walks 8 km on a bearing of 140°, then turns and walks 6 km on a bearing of 230°. How far is the hiker from the starting point?

Solution: 1. Draw the diagram:
- North line at start.
- First leg: 8 km at 140° (Quadrant II).
- Second leg: 6 km at 230° (Quadrant III).
- Angle between paths = 230° – 140° = 90° (right angle!).

1. Identify the triangle:
2. Right-angled triangle formed.

3. Legs: 8 km and 6 km.

4. Use Pythagoras’ Theorem:

5. Distance = √(8² + 6²) = √(64 + 36) = √100 = 10 km

What we did and why: - The 90° angle was hidden in the bearings (230° – 140°). - We used Pythagoras because it was a right-angled triangle.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in—tonight, before your exam, here’s what you need to remember:

1. Bearings start at North (000°) and go clockwise. 090° is East, 180° is South, 270° is West.
2. Draw the diagram first. Always. North line, bearings, distances—label everything.
3. Find the angle between bearings by subtracting. If it’s 90°, use Pythagoras. If not, use Sine Rule or Cosine Rule.
4. Back bearings? Add or subtract 180°. If it’s over 360°, take it away.
5. Check your triangle. Right angle? SOHCAHTOA. No right angle? Sine or Cosine Rule.
6. Final answer? Three-figure bearing, correct units, one decimal place.

You’ve got this. Now go practice two questions—one right-angled, one non-right-angled—and you’ll be set for exam day."

Final Note for Teachers: - Pacing: Spend 50% of time on diagrams—students struggle most here. - Visuals: Use colour (e.g., red for North lines, blue for bearings). - Common Question: "Why do we subtract bearings?" → Answer: "Because bearings are measured from the same North line, so the difference gives the angle between them." - Exam Tip: Remind students to write down every step—examiners give marks for method, not just the answer.