Class 9 Maths Exercise 6.5: Trigonometry & Bearings

Class 9 Mathematics Exercise 6.5 – Trigonometry and Bearings (Complete Step-by-Step Guide)

Exercise 6.5 from Trigonometry and Bearings is one of the most practical and concept-based topics in Class 9 Mathematics. This exercise focuses on solving real-life problems involving heights, distances, and angles using trigonometric ratios. Many students find these questions challenging because they require both diagram understanding and correct formula application.

In this detailed guide, you will learn how to approach every type of question in Exercise 6.5 using clear steps, proper reasoning, and exam-focused techniques.

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Understanding the Core Idea

Exercise 6.5 is mainly based on right-angled triangles formed in real-life situations such as buildings, poles, trees, and distances between objects. The key to solving these questions is converting the given information into a proper diagram.

From experience, students who skip diagram drawing often make mistakes even when they know the formulas. Therefore, always start by drawing a neat and labeled figure.

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Important Trigonometric Ratios

You will repeatedly use the following ratios:

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

Each ratio is used depending on what values are given in the question.

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Worked Example 1: Finding Height Using Tangent

Problem: A triangle has a base of 20 m and an angle of elevation of 30°. Find the height.

Step 1: Identify known values
Base = 20 m, Angle = 30°

Step 2: Choose correct ratio
tan θ = Perpendicular / Base

Step 3: Apply formula
tan 30° = h / 20

Step 4: Solve
h = 20 × tan 30° ≈ 20 × 0.577 = 11.54 m

Final Answer: Height ≈ 11.54 m

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Worked Example 2: Finding Angle Using Inverse Tangent

Problem: A triangle has height = 10 and base = 10√3. Find the angle.

Step 1: Use tangent
tan θ = 10 / (10√3)

Step 2: Simplify
tan θ = 1 / √3

Step 3: Apply inverse
θ = 30°

Final Answer: Angle = 30°

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Worked Example 3: Using Pythagoras Theorem

In many problems, you need to find the hypotenuse using:

a² + b² = c²

Example:
Base = 5 m, Height = 12 m

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

This method is especially useful when two sides are given and you need the third side.

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Worked Example 4: Two Triangles with Common Height

This is one of the most important patterns in Exercise 6.5.

Concept: When two triangles share the same height, form two equations.

Let total base = 120 m

First triangle:
h = x tan 60°

Second triangle:
h = (120 − x) tan 30°

Equate both:

x tan 60° = (120 − x) tan 30°

Solve to get:

x = 30 m

Then height:
h = 30√3 ≈ 52 m

This type of question appears frequently in exams.

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Worked Example 5: Speed Application

Some questions combine trigonometry with motion.

Formula:
Speed = Distance / Time

Example:
Distance = 87.32 m, Time = 120 s

Speed = 87.32 / 120 ≈ 0.72 m/s

This shows how trigonometry connects with real-life situations.

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Common Mistakes Students Make

• Not drawing diagrams properly
• Using the wrong trigonometric ratio
• Forgetting to simplify values
• Not using inverse functions for angles
• Skipping steps in multi-part questions

From experience, most errors occur due to incorrect diagram interpretation rather than weak formulas.

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Exam Tips for High Marks

• Always draw a clear diagram first
• Label all sides and angles properly
• Write the formula before solving
• Show complete steps
• Round answers correctly

Examiners give marks for method, so even partial solutions can score marks if steps are correct.