Class 9 Maths Exercise 6.6: Trigonometry & Bearings

Class 9 Mathematics Exercise 6.6 – Trigonometry and Bearings (Complete Guide with Solved Examples)

Exercise 6.6 from Trigonometry and Bearings is one of the most important and practical topics in Class 9 Mathematics. Unlike basic trigonometry, this exercise focuses on real-life problem solving using directions, angles, and distances. Many students find this topic difficult at first, but once you understand the logic behind diagrams and angle relationships, it becomes much easier.

In this detailed guide, you will learn how to solve bearings questions step by step using clear concepts, proper diagrams, and correct trigonometric methods.

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Understanding Bearings

A bearing is a way of describing direction. It is always measured in degrees, clockwise from the North direction. This is a very important rule and must always be followed in exams.

• North = 0° or 360°
• East = 90°
• South = 180°
• West = 270°

Key Concept: Always draw a North line at every point when solving a bearings problem. This is where most students lose marks.

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Trigonometric Ratios Used

To solve distance and angle problems, you will use basic trigonometric ratios:

• sin θ = Perpendicular / Hypotenuse
• cos θ = Base / Hypotenuse
• tan θ = Perpendicular / Base

You must first identify the right-angled triangle correctly before applying any formula.

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Important Geometry Rule (Very Frequently Used)

Alternate Angles Rule: When a transversal cuts two parallel lines (North lines), alternate angles are equal.

This rule is extremely important in bearings. In most questions, you will use this concept to transfer angles from one point to another.

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Worked Example 1: Basic Bearings Triangle

Problem: A ship B is at a bearing of 45° from A. The distance AB = 10 km. Find the horizontal and vertical distances.

Step 1: Draw Diagram
Draw point A and a North line. From A, draw a line at 45° to locate B.

Step 2: Form Right Triangle
Drop a perpendicular from B to form a right triangle.

Step 3: Apply Trigonometry

Base (horizontal distance):
cos 45° = Base / 10
Base = 10 × cos 45° = 10 × 0.707 = 7.07 km

Perpendicular (vertical distance):
sin 45° = Height / 10
Height = 10 × sin 45° = 7.07 km

Conclusion: Both distances are equal because the angle is 45°.

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Worked Example 2: Multi-Step Bearings Problem

Problem: A person walks 3 km at a bearing of 50° and then continues in another direction. Find the final position.

Step 1: Draw First Movement
Start from point P, draw North, then mark 50° clockwise and draw line PQ = 3 km.

Step 2: Draw Second Movement
At point Q, draw another North line. This is very important.

Step 3: Use Angle Relationships
Use alternate angles or straight-line angle rules to find missing angles.

Step 4: Apply tan θ or sin θ depending on given values

This type of question requires careful diagram work. In exams, most mistakes happen due to poor diagrams, not wrong formulas.

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Worked Example 3: Using Pythagoras Theorem

In many questions, you will form a right triangle and use:

a² + b² = c²

Example:

If one side = 5 km and another = 12 km

c = √(5² + 12²)
c = √(25 + 144)
c = √169 = 13 km

This method is commonly used to find the shortest distance between two points.

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Real Exam-Level Example

Problem: A car travels 13 km at a bearing of 40°. Find how far it travels North and East.

Step 1: Convert Angle
Angle with horizontal = 50°

Step 2: Find North Component
sin 50° = North / 13
North = 13 × 0.766 = 9.96 km

Step 3: Find East Component
cos 50° = East / 13
East = 13 × 0.643 = 8.36 km

Final Answer:
North distance ≈ 9.96 km
East distance ≈ 8.36 km

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Common Mistakes Students Make

• Forgetting to draw the North line at each point
• Measuring angle anticlockwise instead of clockwise
• Using wrong trigonometric ratio
• Skipping diagram (very risky in this chapter)
• Not labeling angles clearly

From experience, students who draw neat diagrams score significantly higher in this exercise.

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Exam Tips for Full Marks

• Always start with a clear diagram
• Label all points and directions properly
• Write formula before calculation
• Show each step clearly
• Round answers properly where required

Examiners give marks for method, not just the final answer. Even if your final answer is slightly off, correct steps can still earn marks.

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Conclusion

Exercise 6.6 is not difficult if you approach it correctly. Focus on diagrams, understand angle relationships, and apply trigonometry step by step. With practice, you will start recognizing patterns in questions and solve them much faster.

This topic is extremely important for board exams, so make sure you practice different types of problems and revise the key concepts regularly.