Class 10 Maths Chapter 9 Some Applications of Trigonometry: NCERT Solutions, Important Questions & Notes PDF

CBSE Class 10 Notes | Maths Chapter 9 – Applications of Trigonometry

Master CBSE Class 10 Maths Notes for Chapter 9 Some Applications of Trigonometry. NCERT-based Heights and Distances concepts, formulas, solved examples and board questions.

CBSE Class 10 Maths Notes – Chapter 9 Some Applications of Trigonometry

1. Introduction

Welcome to comprehensive CBSE Class 10 Notes for Mathematics Chapter 9 – Some Applications of Trigonometry. This chapter is a crucial component of the Trigonometry unit, carrying significant weightage in your board examinations.

CBSE Class 10 Maths Notes for this chapter focus on the practical applications of trigonometric ratios in solving real-world problems involving heights and distances. Unlike the previous chapter that dealt with theoretical trigonometric identities, Chapter 9 demonstrates how sin θ, cos θ, and tan θ help calculate inaccessible heights and distances.

• Marks Weightage: Chapter 9 carries 5 marks in the CBSE Class 10 Board Exam 2025
• Question Type: Long Answer Type (5 marks) or Case-Based Questions (4 marks)
• Difficulty Level: Moderate to High – requires strong visualization skills
• NCERT Alignment: 100% based on NCERT Book Chapter 9

2. Chapter Overview

CBSE Class 10 Maths Notes for Chapter 9 revolve around these core concepts:

Concept	Description	Exam Importance
Line of Sight	The straight line from observer's eye to the object	Foundation concept
Angle of Elevation	Angle above horizontal when object is above eye level	Very High – Always asked
Angle of Depression	Angle below horizontal when object is below eye level	Very High – Always asked
Right Triangle Formation	Converting word problems into right-angled triangles	Critical skill
Word Problems	Real-life scenarios requiring height/distance calculation	5-mark questions

3. Key Concepts (100% NCERT Accurate)

Line of Sight

The line of sight is the straight line drawn from the observer's eye to the point on the object being viewed. It forms the hypotenuse of the right-angled triangle in most problems.

Angle of Elevation

Definition: The angle of elevation is the angle formed between the horizontal line and the line of sight when the observer looks upward at an object above their eye level.

• Object is above the horizontal level
• Observer looks upwards
• Angle is measured from horizontal upwards to line of sight
• Always taken as acute angle (0° < θ < 90°)

Angle of Depression

Definition: The angle of depression is the angle formed between the horizontal line and the line of sight when the observer looks downward at an object below their eye level.

Important Property: Angle of depression from the observer = Angle of elevation from the object (Alternate interior angles)

Use of Trigonometric Ratios

Ratio	Formula	When to Use
sin θ	Perpendicular Hypotenuse	When height and line of sight are known
cos θ	Base Hypotenuse	When distance and line of sight are known
tan θ	Perpendicular Base	Most commonly used – When height and distance are involved

Why tan θ is Most Important:
In heights and distances problems, we usually know the distance from observer to object (Base) and the angle of elevation/depression (θ), and we need to find the height (Perpendicular). Since tan θ = Perpendicular Base , it directly connects the known values to the unknown height.

4. Important Formulas

Essential Formulas for Heights and Distances

Basic Trigonometric Ratios:

• sin θ = P H = Perpendicular Hypotenuse
• cos θ = B H = Base Hypotenuse
• tan θ = P B = Perpendicular Base

Derived Formulas:

• Height = Distance × tan θ
• Distance = Height tan θ
• Height = Hypotenuse × sin θ
• Distance = Hypotenuse × cos θ

Special Angle Values:

Angle	sin θ	cos θ	tan θ
0°	0	1	0
30°	1 2	√3 2	1 √3
45°	1 √2	1 √2	1
60°	√3 2	1 2	√3
90°	1	0	Not defined

5. Solved Examples (CBSE Pattern)

* Example 1: Easy (Angle of Elevation)

Question: The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower. (Use √3 = 1.732)

Solution:

Step 1: Let AB be the tower (height = h), C be the point on ground 30m from foot B. Angle ACB = 30°.

Step 2: In right triangle ABC (right-angled at B): Base BC = 30 m, Perpendicular AB = h, Angle at C = 30°

Step 3: Using tan θ = Perpendicular Base :

tan 30° = AB BC = h 30

Step 4: Substituting value:

1 √3 = h 30

h = 30 √3 = 30√3 3 = 10√3 = 10 × 1.732 = 17.32 m

Height of tower = 17.32 m

* Example 2: Moderate (Angle of Depression)

Question: From the top of a 60 m high building, the angles of depression of the top and bottom of a tower are 45° and 60° respectively. Find the height of the tower. (Use √3 = 1.73)

Solution:

Step 1: Let AB = building = 60 m, CD = tower = h m. Let distance BD = x m.

Step 2: For bottom of tower (point D): Angle of elevation from D = 60° (alternate to angle of depression 60°)

tan 60° = 60 x => √3 = 60 x => x = 60 √3 = 20√3 m

Step 3: For top of tower (point C): Height above tower = (60 − h) m, Angle = 45°

tan 45° = 60 − h x => 1 = 60 − h 20√3 => 60 − h = 20√3 => h = 60 − 34.6 = 25.4 m

Height of tower = 25.4 m

* * * Example 3: Board-Level (5 Marks)

Question: Two poles of equal heights are standing opposite to each other on either side of a road, which is 100 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively. Find the height of the poles.

Solution:

Step 1: Let height of each pole = h m. Let point C be at distance x from first pole, so (100−x) from second pole.

Step 2: For first pole (angle 60°):

tan 60° = h x => √3 = h x => h = x√3 ... (Equation 1)

Step 3: For second pole (angle 30°):

tan 30° = h 100 − x => 1 √3 = h 100 − x => h = 100 − x √3 ... (Equation 2)

Step 4: From Equation 1 and 2:

x√3 = 100 − x √3 => 3x = 100 − x => 4x = 100 => x = 25 m

Step 5: Substituting back:

h = 25√3 = 25 × 1.732 = 43.3 m

Height of each pole = 43.3 m | Distances from point C: 25 m and 75 m

6. Smart Tricks Section

Diagram Drawing Shortcut

The "L-Rule":

• Always draw the horizontal line first
• Draw the vertical object perpendicular to it
• Connect with hypotenuse (line of sight)
• Mark the angle at the observer's position

Memory Tip: "Horizontal first, vertical second, angle at observer"

How to Quickly Decide Which Ratio to Use

Given	Find	Use
Angle + Distance	Height	tan θ
Angle + Height	Distance	tan θ or cot θ
Angle + Line of sight	Height/Distance	sin θ or cos θ

30°-45°-60° Quick Value Trick

Remember pattern for sin: √1 2 , √2 2 , √3 2

• sin 30° = 1 2 (half)
• sin 45° = 1 √2
• sin 60° = √3 2
• cos is reverse of sin

7. Most Important Board Questions

1 Mark Questions

When the length of the shadow of a pole is equal to the height of the pole, the angle of elevation of the sun is:

Answer: 45° (Since tan θ = 1, therefore θ = 45°)

2-3 Mark Questions

A ladder 15 m long makes an angle of 60° with the wall. Find the distance of the foot of the ladder from the wall.

Solution: cos 60° = Base 15 => 1 2 = Base 15 => Base = 7.5 m

4-5 Mark Questions (Most Important)

1. Two poles of equal heights are standing opposite each other on either side of a 100 m wide road. From a point between them, angles of elevation are 60° and 30°. Find the height of the poles. ***
2. From the top of a 60 m high building, angles of depression of the top and bottom of a tower are 45° and 60°. Find the height of the tower. ***
3. A man on the deck of a ship (10 m above water level) observes angle of elevation of hill top as 60° and angle of depression of hill base as 30°. Find distance of hill and height of hill. ***

8. Common Mistakes to Avoid

Mistake 1: Drawing the angle at the object instead of at the observer.
Correction: Always draw the angle at the observer's position (eye level).

Mistake 2: Using sin θ when tan θ should be used.
Correction: Remember "TOA" - Tangent = Opposite/Adjacent.

Mistake 3: Confusing Angle of Elevation and Depression.
Correction: Elevation = looking UP, Depression = looking DOWN.

Mistake 4: Ignoring units (mixing meters and kilometers).
Correction: Convert all measurements to the same unit before calculating.

9. Frequently Asked Questions (FAQ)

Is Chapter 9 important for CBSE Class 10 board exam?

Yes, absolutely. Chapter 9 carries 5 marks in the CBSE Class 10 Board Exam. It appears as 5-mark long answer questions or 4-mark case-study based questions.

Which trigonometric ratio is mostly used in heights and distances?

tan θ (tangent) is the most frequently used because most problems involve finding height when distance and angle are known using: Height = Distance × tan θ

Are these CBSE Class 10 Maths Notes based on NCERT?

Yes, 100%. These notes are strictly aligned with NCERT Class 10 Mathematics textbook Chapter 9. All concepts, formulas, and examples follow the NCERT pattern.

What is the best strategy to solve case-study based questions?

1. Read the passage carefully – Identify all numerical values
2. Draw a detailed diagram – Mark all points mentioned
3. Identify multiple right triangles – Usually 2-3 triangles involved
4. Solve part by part – Each sub-question relates to different triangles
5. Show all calculations – Case studies carry 4 marks

10. Conclusion

Mastering CBSE Class 10 Maths Notes for Chapter 9 requires consistent practice and clear conceptual understanding. This chapter bridges theoretical trigonometry with real-world applications, making it both interesting and scoring.

Key Takeaways:

• Focus on diagram drawing – it is half the battle won
• tan θ is your best friend for most problems
• Practice previous year board questions – patterns repeat frequently
• Do not forget the alternate angle property for depression problems
• Manage time efficiently – 6-7 minutes max for 5-mark questions

Revision Checklist:

• Memorize sin, cos, tan values for 0°, 30°, 45°, 60°, 90°
• Practice 5 diagram-based questions daily
• Solve last 5 years' board questions from this chapter
• Review case-study question patterns from sample papers

Next Chapter: Continue your CBSE Class 10 Notes preparation with Chapter 10 – Circles for comprehensive board exam readiness.

Best of Luck for Your CBSE Class 10 Board Exams 2025