NCERT Solutions for Class 10 Maths Chapter 9: Some Applications of Trigonometry
If Chapter 8 introduced you to trigonometric ratios in the abstract, Chapter 9 of Class 10 NCERT Maths brings them to life in the real world. This chapter is entirely about heights and distances — using trigonometry to calculate how tall a building is, how far a ship is from a lighthouse, or at what angle you would need to look up to see the top of a tower. The two central concepts here are the Angle of Elevation and the Angle of Depression. The angle of elevation is the angle formed when you look upward from horizontal to an object above you, while the angle of depression is formed when you look downward from horizontal to an object below. These might sound simple, but the NCERT questions in this chapter are cleverly designed, often involving two observers, two angles, or moving objects.
Chapter 9 has only one exercise — Exercise 9.1 — but it contains 16 questions ranging from straightforward to genuinely challenging. What makes this chapter special is how directly it connects mathematics to everyday life. Surveyors, pilots, sailors, and architects all use exactly these techniques. Myclass24's NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry are solved with proper diagrams and clearly labelled triangles, which is crucial since incomplete diagrams can cost marks in board exams. For other chapters of NCERT Solutions for Class 10 Maths, check NCERT Solutions for Class 10 and NCERT Solutions.
Download PDF: NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry
Chapter 9 Heights & Distances – Concepts, Question Types & Key Facts
Chapter 9 is the practical extension of Chapter 8, and the same standard angle values you memorised there (30°, 45°, 60°) are used constantly here. The chapter relies on constructing the right triangle from the word problem and then applying the correct trigonometric ratio.
Angle of Elevation vs Angle of Depression
| Term | Definition | Observer Position | Object Position |
|---|---|---|---|
| Angle of Elevation | Angle formed between the horizontal line and line of sight going upward | Below the object | Above the observer |
| Angle of Depression | Angle formed between the horizontal line and line of sight going downward | Above the object | Below the observer |
Standard Trigonometric Values Used in Chapter 9 Problems
| Angle | tan value | sin value | cos value | Commonly Used In |
|---|---|---|---|---|
| 30° | 1/√3 | 1/2 | √3/2 | Shorter heights, gentle slopes |
| 45° | 1 | 1/√2 | 1/√2 | Height = horizontal distance problems |
| 60° | √3 | √3/2 | 1/2 | Taller structures, steeper angles |
Question Patterns in Exercise 9.1
| Question Type | Typical Setup | Ratio Used |
|---|---|---|
| Single observer, one angle | Person on ground looking up at building | tan θ = height / distance |
| Two angles, same observer | Top and bottom of a tower seen from one point | Two tan equations, solve simultaneously |
| Two observers, same object | Two people looking at tower from opposite sides | Two tan equations, add distances |
| Moving observer | Ship approaching a lighthouse; angle changes | tan of each angle; find speed or distance |
| Depression from height | Person in building looking down at car | Angle of depression = angle of elevation (alternate angles) |
Key Facts & Tips for Chapter 9
- There is only one exercise in Chapter 9 — Exercise 9.1 — but it contains 16 questions, some of which are multi-step problems that take 5–6 lines of working.
- The angle of depression from point A to point B is always equal to the angle of elevation from point B to point A — this is because of alternate interior angles formed by the horizontal lines and line of sight.
- Always draw a neat, labelled diagram before solving. CBSE examiners award a mark specifically for the diagram in many questions.
- The most commonly tested angles in this chapter are 30°, 45°, and 60° — memorise their tan, sin, and cos values thoroughly.
- When the problem involves two different angles from two different points, set up two separate equations and solve them as a system — do not try to combine them prematurely.
- Rationalize your final answer (remove surds from the denominator) to match the expected NCERT format.
Myclass24 provides NCERT Solutions for Class 10 Maths Chapter 9 with fully drawn triangles for each question — a key feature, because students who skip the diagram often make errors in identifying which side is opposite, adjacent, or hypotenuse relative to the given angle.
NCERT Solutions for Class 10 Maths Chapter 9: Some Applications of Trigonometry
This chapter focuses on using trigonometric ratios to solve practical problems involving heights and distances. Students learn how to calculate inaccessible heights and distances using angles of elevation and depression. The chapter demonstrates how trigonometry can be applied outside the classroom to solve real-world measurement problems. Understanding these applications helps students appreciate the practical value of mathematics. The concepts learned here are important for both board examinations and future studies involving engineering, architecture, and surveying.
An angle of elevation is the angle formed between the horizontal line and the line of sight when an observer looks upward at an object. Examples include looking at the top of a building, tower, or mountain. This concept is important because it helps calculate heights using trigonometric ratios. Students frequently encounter questions involving angles of elevation in examinations. Understanding how to draw diagrams and identify relevant sides correctly is essential for solving such problems accurately.
An angle of depression is the angle formed between the horizontal line and the line of sight when an observer looks downward at an object. It is commonly used in problems involving observation from elevated positions. Understanding angles of depression helps students solve practical measurement questions effectively. These problems often require careful interpretation of diagrams and correct application of trigonometric ratios. Board examination questions frequently test students’ understanding of this important concept.
Heights and distances are calculated by forming right-angled triangles and applying appropriate trigonometric ratios. By knowing an angle and one side of a triangle, students can determine unknown measurements. This method allows indirect measurement of objects that cannot be measured directly. Such calculations are widely used in surveying and construction. Understanding the process helps students solve practical problems systematically and accurately. Examination questions often require both diagram interpretation and numerical calculations.
Diagrams help visualize the situation and identify the correct trigonometric relationships. They allow students to understand the position of the observer, object, and angles involved. A well-drawn diagram simplifies problem-solving and reduces the chances of errors. Many mistakes occur because students misinterpret the given information. Creating accurate diagrams improves conceptual clarity and makes calculations easier. This skill is particularly important in examination questions involving heights and distances.
The concepts of heights and distances are used in surveying, navigation, aviation, engineering, and construction. Professionals often use trigonometric methods to measure objects that are difficult to access directly. For example, engineers determine the height of structures, while surveyors calculate land measurements. Learning these applications helps students understand the practical significance of mathematics. It also demonstrates how theoretical concepts can solve real-world challenges efficiently and accurately.
Students often confuse angles of elevation with angles of depression or choose incorrect trigonometric ratios. Errors in drawing diagrams and identifying triangle sides are also common. Some students perform calculations correctly but misinterpret the final answer. To avoid these mistakes, it is important to analyze the question carefully and draw clear figures. Regular practice helps students improve accuracy and confidence. A systematic approach ensures better performance in application-based trigonometry questions.