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I. Chapter Summary:

This chapter focuses on applying trigonometric concepts to real-world situations, especially in measuring the heights and distances of objects without physically measuring them. Students learn how to use angles of elevation and depression and apply trigonometric ratios (sine, cosine, tangent) to solve practical problems. These concepts are critical in fields like architecture, engineering, and navigation.

II. Key Concepts Covered:

Concept	Explanation
Line of Sight	An imaginary straight line between the observer’s eye and the object viewed.
Angle of Elevation	The angle between the horizontal line and the line of sight when looking up at an object.
Angle of Depression	The angle between the horizontal line and the line of sight when looking down at an object.
Trigonometric Ratios	Use of sin, cos, and tan functions in right-angled triangles to relate angles and sides.
Height and Distance Problems	Applying trigonometry to calculate unknown heights and distances using given angles and sides.

III. Important Questions:

(A) Multiple Choice Questions (1 Mark):

1. The angle of elevation of the top of a tower from a point on the ground is 30°. If the height of the tower is 50 m, the distance of the point from the tower is:
   a) $50sqrt{3} , text{m}$
   b) $frac{50}{sqrt{3}} , text{m}$✅
   c) 25 m
   d) 100 m
   (PYQ 2020)
2. The angle of depression from the top of a lighthouse of height 60 m to a boat is 30°. The distance of the boat from the base of the lighthouse is:
   a) $60sqrt{3} , text{m}$✅
   b) 30 m
   c) $20sqrt{3} , text{m}$
   d) 90 m
3. If the angle of elevation of the sun is 60°, then the length of the shadow of a 10 m pole is:
   a) $10sqrt{3} , text{m}$
   b) $frac{10}{sqrt{3}} , text{m}$✅
   c) 5 m
   d) 20 m
4. Which of the following statements is true?
   a) Angle of depression is always greater than angle of elevation
   b) Angle of elevation is measured below the horizontal
   c) Trigonometric ratios are not used in height and distance
   d) Angle of depression is measured below the horizontal ✅

(B) Short Answer Questions (2/3 Marks):

1. A pole 10 m high casts a shadow 10√3 m long. Find the angle of elevation of the sun.
   (PYQ 2019)
2. The angle of elevation of a ladder leaning against a wall is 60°, and the foot of the ladder is 2 m from the wall. Find the length of the ladder.
3. A man on the top of a building finds the angle of depression of a point on the ground to be 45°. If the height of the building is 20 m, find the distance of the point from the base of the building.
4. From a point on the ground, the angle of elevation of the top of a tower is 60°. From another point 10 m closer to the tower, the angle of elevation is 45°. Find the height of the tower.
   (PYQ 2022)

(C) Long Answer Questions (5 Marks):

1. Two poles of equal height are standing 20 m apart. From the midpoint of the line joining their feet, the angles of elevation of their tops are 60° and 30°. Find the height of each pole.
   (PYQ 2018)
2. From the top of a 7 m high building, the angle of elevation of the top of a tower is 45° and the angle of depression of the foot of the tower is 30°. Find the height of the tower.
3. An observer 1.5 m tall is 28.5 m away from a tower. The angle of elevation from his eye to the top of the tower is 45°. Find the height of the tower.
4. A balloon is flying at a height of 50 m. The angle of elevation of the balloon from a point on the ground is 60°. Find the distance of the balloon from the observer.

(D) HOTS (Higher Order Thinking Skills):

1. A person standing on a 60 m tall tower observes a car moving away from the tower. The angles of depression of the car at two instances 5 seconds apart are 45° and 30°. Find the speed of the car.
   (Use tan values and time-speed-distance concepts.)
2. A vertical pole and a tower are standing opposite each other across a road 40 m wide. From a point midway between them, angles of elevation of their tops are 60° and 30°, respectively. Find the height of the tower and the pole.

IV. Key Formulas/Concepts:

Formula/Concept	Description
$tan theta = frac{text{Perpendicular}}{text{Base}}$	Used to find height/distance in right-angled triangles.
$sin theta = frac{text{Perpendicular}}{text{Hypotenuse}}$	Useful when height and hypotenuse are given.
$cos theta = frac{text{Base}}{text{Hypotenuse}}$	Useful to find the base when hypotenuse and angle are known.
Angle of Elevation/Depression	Always measured from the horizontal line of sight.
Use of identities	trigonometric ratios for 30°, 45°, and 60°

V. Deleted Portions (CBSE 2025–26):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–26):

Unit/Chapter	Estimated Marks	Type of Questions Typically Asked
Some Applications of Trigonometry	4–5 Marks	1 Short Answer (3M) + 1 Long Answer (5M) or HOTS

VII. Previous Year Questions (PYQs):

Year	Marks	Question
2018	5M	Two poles of equal height standing 20 m apart with different angles of elevation – Find height.
2019	3M	Pole and shadow-based angle of elevation question.
2020	1M	Find distance using angle of elevation and height of tower (MCQ).
2022	3M	Two angles of elevation from different distances – Find height of tower.

VIII. Real-World Application Examples:

Real-Life Scenario	Concept Used
Determining the height of a mountain from ground level	Angle of elevation & trigonometry
Navigation and mapping for ships and aircraft	Trigonometric calculation of direction and distance
Building architecture and design	Height and angle measurements
Sports (e.g., cricket – angle of ball trajectory)	Application of angle of elevation

IX. Student Tips & Strategies for Success:

 Time Management:

• Allocate 30–40 minutes per day to Maths practice.
• Solve 1–2 trigonometry questions daily.

 Exam Preparation:

• Memorize standard trigonometric ratios (30°, 45°, 60°).
• Practice drawing diagrams correctly.
• Label all sides and angles carefully.
• Always write units (m, cm, etc.).

 Stress Management:

• Do breathing exercises before maths tests.
• Practice questions in a relaxed state to avoid silly errors.

X. Career Guidance & Exploration (Class-Specific):

For Class 10 Students:

Stream	Example Careers	Entrance Exams
Science	Engineer, Architect, Pilot	NTSE, Olympiads, JEE (later)
Commerce	Chartered Accountant, Data Analyst	Commerce Olympiads
Arts	Designer, Historian, Lawyer	NTSE, Creative Aptitude Tests

Recommended:

• Participate in Math Olympiads to boost logical thinking.
• Consider careers like Surveying, Architecture, Astronomy, where trigonometry is used extensively.

XI. Important Notes:

•  Always refer to https://cbseacademic.nic.in and https://ncert.nic.in for official updates.
•  Build concepts before solving problems.
•  Practice mixed questions from different difficulty levels.
•  Don’t just memorize – understand “why” a formula works.