NCERT Solutions Class 10 Maths Chapter 9 teaches you how to apply trigonometry to solve real-world height and distance problems. You’ll learn how to calculate the height of buildings, towers, and mountains using angle of elevation and depression, understand line of sight concepts, and master all 15 exercise questions with detailed solutions. These practical applications help you visualize trigonometric ratios in everyday situations and prepare you for word problems in board exams.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 10 Maths Chapter 9 Some Applications of Trigonometry – Complete Guide
NCERT Class 10 Chapter 9 – Some Applications of Trigonometry brings mathematics into the real world by showing you how trigonometric ratios solve practical problems. This chapter builds directly on your knowledge from Chapter 8 (Introduction to Trigonometry) and demonstrates why trigonometry is essential in fields like architecture, engineering, navigation, and surveying.
π CBSE Class 10 Maths Chapter 9 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 6 Marks |
| Unit Name | Trigonometry |
| Difficulty Level | Hard |
| Importance | High |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 2-3 questions |
You’ll explore two fundamental concepts: angles of elevation and angles of depression. Through these angles, you’ll learn to calculate heights of objects like towers, buildings, cliffs, and trees without physically measuring them. You’ll also determine horizontal distances across rivers, between ships, or to inaccessible locations. The chapter includes numerous real-life scenarios where you’ll apply sine, cosine, and tangent ratios, particularly focusing on tan ΞΈ = height/distance relationships.
For CBSE board exams, this chapter carries 6 marks and is considered high difficulty with significant importance. Expect 2-3 questions including one long answer question (4 marks) and one short answer question (2 marks). Questions typically involve multi-step problems requiring diagram drawing, identifying appropriate trigonometric ratios, and accurate calculations. Common scenarios include problems with two angles of elevation, distances between two points at different heights, or combined height-distance problems.
Quick Facts – Class 10 Chapter 9
| π Chapter Number | Chapter 9 |
| π Chapter Name | Some Applications of Trigonometry |
| βοΈ Total Exercises | 1 Exercises |
| β Total Questions | 15 Questions |
| π Updated For | CBSE Session 2025-26 |
Mastering this chapter not only helps you score well in boards but also develops spatial reasoning and problem-solving skills applicable in competitive exams like JEE and real-world professions. With consistent practice of NCERT solutions and understanding the logic behind each problem type, you’ll confidently tackle any heights-and-distances question in your examination.
NCERT Solutions Class 10 Maths Chapter 9 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 9.1 | Complete step-by-step solutions for 15 questions | π₯ Download PDF |
Some Applications of Trigonometry – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Tangent of Angle of Elevation \(\\tan \\theta = \\frac{Perpendicular}{Base} = \\frac{Height}{Distance}\) | Relates the angle of elevation to the height of an object and its distance from the observer. Note: Make sure you are using the correct angle of elevation. It’s the angle formed by the line of sight with the horizontal. | Finding the height of a tower, building, or tree when the angle of elevation and distance are known. Also, finding the distance when the height and angle of elevation are known. |
| Tangent of Angle of Depression \(\\tan \\theta = \\frac{Perpendicular}{Base} = \\frac{Height}{Distance}\) | Relates the angle of depression to the height of an object and its distance from the observer. Note: Remember that the angle of depression is equal to the angle of elevation from the object to the observer (alternate interior angles). | Finding the distance of an object from a point when the angle of depression and height are known (e.g., a ship from a lighthouse). Also, finding the height when the distance and angle of depression are known. |
| Sine of Angle of Elevation/Depression \(\\sin \\theta = \\frac{Perpendicular}{Hypotenuse} = \\frac{Height}{Line \\ of \\ Sight}\) | Relates the angle of elevation/depression to the height of an object and the length of the line of sight. Note: Useful when you don’t have the distance (base). | When the hypotenuse (line of sight) is given or needs to be calculated along with height or angle. |
| Cosine of Angle of Elevation/Depression \(\\cos \\theta = \\frac{Base}{Hypotenuse} = \\frac{Distance}{Line \\ of \\ Sight}\) | Relates the angle of elevation/depression to the distance of an object and the length of the line of sight. Note: Useful when you don’t have the height (perpendicular). | When the hypotenuse (line of sight) is given or needs to be calculated along with the distance or angle. |
| Relationship between tan, sin, cos \(\\tan \\theta = \\frac{\\sin \\theta}{\\cos \\theta}\) | Connects tangent with sine and cosine of an angle. Note: Remember this identity, it can simplify calculations. | If you know sine and cosine of an angle, you can easily find tangent. |
| Angles of Elevation/Depression Relationship \(\\angle \\ of \\ Elevation = \\angle \\ of \\ Depression\) | The angle of elevation from one point to another is equal to the angle of depression from the second point to the first, assuming they are on the same vertical plane. Note: This is a direct consequence of alternate interior angles being equal when two parallel lines are intersected by a transversal. | In problems where both angle of elevation and depression are involved, and you need to relate them. |
| 30-60-90 Triangle Ratios \(\\sin 30^\\circ = \\frac{1}{2}, \\cos 30^\\circ = \\frac{\\sqrt{3}}{2}, \\tan 30^\\circ = \\frac{1}{\\sqrt{3}}\) | Values of trigonometric ratios for a 30-degree angle. Note: Memorize these values! Rationalize the denominator if needed. | In problems involving 30-degree angles of elevation or depression. |
| 45-45-90 Triangle Ratios \(\\sin 45^\\circ = \\frac{1}{\\sqrt{2}}, \\cos 45^\\circ = \\frac{1}{\\sqrt{2}}, \\tan 45^\\circ = 1\) | Values of trigonometric ratios for a 45-degree angle. Note: Memorize these values! Rationalize the denominator if needed. | In problems involving 45-degree angles of elevation or depression. |
| 60-30-90 Triangle Ratios \(\\sin 60^\\circ = \\frac{\\sqrt{3}}{2}, \\cos 60^\\circ = \\frac{1}{2}, \\tan 60^\\circ = \\sqrt{3}\) | Values of trigonometric ratios for a 60-degree angle. Note: Memorize these values! Rationalize the denominator if needed. | In problems involving 60-degree angles of elevation or depression. |
| Pythagorean Theorem \(a^2 + b^2 = c^2\) | Relates the sides of a right-angled triangle. Note: c is always the hypotenuse (the side opposite the right angle). | Whenever you need to find a missing side of a right-angled triangle when the other two sides are known. |
Frequently Asked Questions – NCERT Class 10 Maths Chapter 9
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