CBSE Class 10 Mathematics (2026–27)
Chapter 9: Some Applications of Trigonometry
20 Important Questions and Answers
This chapter mainly deals with heights and distances, line of sight, angle of elevation, and angle of depression, using trigonometric ratios to solve real-life problems.
Q1. What is meant by the line of sight?
Answer:
The line of sight is the straight line drawn from the observer’s eye to the object being viewed. It plays an important role in solving problems related to heights and distances. Whenever a person looks at the top or bottom of an object, an imaginary line joins the eye and the object. This line is called the line of sight. Trigonometric ratios are applied to the right triangle formed by the line of sight, the horizontal line, and the vertical object. Understanding the line of sight helps in determining unknown heights and distances in practical situations such as observing towers, buildings, mountains, and airplanes.
Q2. Define angle of elevation with an example.
Answer:
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. It is always measured above the horizontal level. For example, when a student standing on the ground looks at the top of a flagpole, the angle formed between the horizontal ground level and the line joining the eye to the top of the flagpole is called the angle of elevation. This concept is widely used in trigonometry to calculate the height of tall objects such as towers, trees, and buildings when the distance from the observer is known.
Q3. Define angle of depression with an example.
Answer:
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. It is measured below the horizontal level. For example, if a person standing on the roof of a building looks at a car on the road, the angle between the horizontal line from the observer’s eye and the line of sight to the car is called the angle of depression. This angle is useful in finding distances and heights in real-life situations involving bridges, buildings, hills, and observation towers.
Q4. Why is trigonometry useful in daily life?
Answer:
Trigonometry helps in measuring heights and distances that cannot be measured directly. It is used in engineering, architecture, navigation, astronomy, surveying, and map-making. For example, engineers use trigonometry to determine the height of towers and buildings. Pilots and sailors use it for navigation and finding positions. Astronomers use trigonometric principles to calculate distances between celestial bodies. The chapter “Some Applications of Trigonometry” introduces students to practical situations where trigonometric ratios such as sine, cosine, and tangent help solve real-world measurement problems accurately and efficiently.
Q5. A tower casts a shadow equal to its height. Find the angle of elevation of the Sun.
Answer:
Let the height of the tower be (h) and the length of its shadow also be (h).
Using the trigonometric ratio:
[\tan \theta = \frac{\text{Height}}{\text{Shadow}}]
[\tan \theta = \frac{h}{h}=1]
Since,
[\tan 45^\circ =1]
Therefore,
[\theta =45^\circ]
Hence, the angle of elevation of the Sun is 45°. This result shows that whenever the height of an object and the length of its shadow are equal, the angle formed by the Sun’s rays with the ground is 45°. Such problems are commonly asked in CBSE examinations to test understanding of trigonometric applications.
Q6. What information is generally required to find the height of an object using trigonometry?
Answer:
To determine the height of an object using trigonometry, two important pieces of information are generally needed. First, the horizontal distance between the observer and the object must be known. Second, the angle of elevation from the observer’s position to the top of the object should be measured. Using these values, an appropriate trigonometric ratio such as tangent can be applied. Since
[\tan\theta=\frac{\text{Height}}{\text{Distance}}]
the unknown height can easily be calculated. This method is useful for finding the heights of buildings, towers, trees, mountains, and other tall structures without directly measuring them.
Q7. Explain the term “heights and distances”.
Answer:
“Heights and Distances” refers to the practical application of trigonometric ratios to determine the height of an object or the distance between objects. In such problems, direct measurement may be difficult or impossible. By using angles of elevation or depression along with known distances, the required values can be calculated accurately. Heights and distances form the core topic of Chapter 9 and help students understand how mathematics is applied in real-life situations. Examples include finding the height of a tower, the width of a river, or the distance of an airplane from the ground.
Q8. Why should a diagram be drawn before solving trigonometry problems?
Answer:
A diagram helps in understanding the situation clearly and identifying the right triangle involved in the problem. It allows students to mark known and unknown quantities correctly and determine which trigonometric ratio should be applied. Drawing a labelled diagram reduces mistakes and improves accuracy in calculations. In CBSE board examinations, diagrams often help in presenting the solution systematically and can earn method marks. A clear diagram also makes it easier to visualize angles of elevation, angles of depression, heights, and distances, leading to better understanding and faster problem-solving.
Q9. What is the relationship between angle of elevation and angle of depression?
Answer:
The angle of elevation and angle of depression are equal when measured between the same two points because they are alternate interior angles formed by parallel horizontal lines. For example, if a person standing on a building observes a car on the ground, the angle of depression from the building to the car is equal to the angle of elevation from the car to the top of the building. This property is frequently used in trigonometry problems to simplify calculations and establish relationships between different parts of the figure.
Q10. A ladder makes an angle of 60° with the ground. What type of angle is it?
Answer:
The angle formed between the ladder and the ground is an angle of elevation. Since the ladder rises upward from the ground, the angle is measured above the horizontal surface. In this case, the angle of elevation is 60°. Such situations are commonly represented using right triangles, where the ladder forms the hypotenuse, the wall represents the vertical side, and the ground represents the horizontal side. Using trigonometric ratios, the height reached by the ladder or its length can be calculated if sufficient information is given.
Q11. State one practical use of angle of elevation.
Answer:
One practical use of the angle of elevation is finding the height of tall structures such as towers, buildings, trees, and mountains. Surveyors measure the angle of elevation from a known distance and then apply trigonometric ratios to calculate the height. This method avoids direct measurement, which may be difficult or dangerous. Engineers and architects frequently use this technique during construction and land surveying. It provides accurate results and demonstrates the practical importance of trigonometry in everyday life and professional fields.
Q12. State one practical use of angle of depression.
Answer:
The angle of depression is commonly used in navigation and surveying. For example, a lighthouse keeper may observe a ship in the sea and measure the angle of depression. Knowing the height of the lighthouse and the angle of depression, the distance of the ship from the shore can be calculated using trigonometric ratios. Similar applications are found in aviation, military operations, and construction projects. Thus, the angle of depression helps determine distances and positions of objects located below the observer’s level.
Q13. What trigonometric ratio is most commonly used in heights and distances problems?
Answer:
The tangent ratio is the most commonly used trigonometric ratio in heights and distances problems. It is defined as:
[
\tan\theta=\frac{\text{Opposite Side}}{\text{Adjacent Side}}
]
In practical situations, the opposite side usually represents the height of an object, while the adjacent side represents the horizontal distance from the observer. Since height and distance are generally the quantities involved, tangent becomes the most convenient ratio. It helps in finding either the height or distance when the angle and one side are known.
Q14. What happens to the angle of elevation as the observer moves closer to an object?
Answer:
As the observer moves closer to an object, the angle of elevation increases. This is because the horizontal distance between the observer and the object decreases while the height of the object remains constant. According to trigonometric principles, a smaller adjacent side leads to a larger tangent value and hence a larger angle. For example, when a person walks toward a tower, the angle formed between the line of sight and the ground becomes steeper. Therefore, the angle of elevation continuously increases as the observer approaches the object.
Q15. What happens to the angle of elevation as the observer moves away from an object?
Answer:
As the observer moves farther away from an object, the angle of elevation decreases. The height of the object remains constant, but the horizontal distance increases. Since the tangent ratio depends on the ratio of height to distance, a larger distance results in a smaller tangent value and consequently a smaller angle. For example, a nearby building appears taller than the same building viewed from a greater distance. This principle is frequently applied in solving practical trigonometric problems involving heights and distances.
Q16. Why are horizontal lines important in trigonometry applications?
Answer:
Horizontal lines serve as reference lines for measuring angles of elevation and depression. These angles are always measured with respect to the horizontal level of the observer. Without a horizontal reference, it would be difficult to define and calculate the required angles accurately. Horizontal lines also help establish parallel-line relationships that are useful in proving equal angles. Therefore, they form an essential part of diagrams and calculations in trigonometry, especially in applications involving heights and distances.
Q17. How does trigonometry help surveyors?
Answer:
Surveyors use trigonometry to measure land areas, heights of structures, and distances between locations. By measuring angles and one known side, they can calculate unknown distances and heights using trigonometric ratios. This method is highly accurate and saves time compared to direct measurements. Surveying is important in road construction, bridge building, mapping, and urban planning. Trigonometry enables surveyors to work efficiently even in places where direct access is difficult, making it one of the most valuable mathematical tools in the field.
Q18. Why is Chapter 9 called an application-based chapter?
Answer:
Chapter 9 is called an application-based chapter because it applies the trigonometric concepts learned in the previous chapter to practical situations. Instead of focusing only on formulas, it teaches students how to use trigonometric ratios to solve real-life problems involving heights and distances. Applications include measuring buildings, towers, trees, mountains, and distances in navigation and surveying. The chapter demonstrates the usefulness of mathematics in everyday life and various professions, helping students connect theoretical knowledge with practical experiences.
Q19. Mention two fields where trigonometry is widely used.
Answer:
Two important fields where trigonometry is widely used are astronomy and engineering. Astronomers use trigonometry to calculate distances between planets, stars, and other celestial objects. Engineers apply trigonometric principles in designing buildings, bridges, roads, and communication towers. Trigonometry is also used in navigation, architecture, surveying, aviation, and map-making. These applications show that trigonometry is not limited to classroom learning but plays a significant role in scientific and technological advancements worldwide.
Q20. Why is understanding trigonometric ratios important before studying this chapter?
Answer:
Understanding trigonometric ratios is essential because Chapter 9 relies entirely on their application. Ratios such as sine, cosine, and tangent are used to relate angles with sides of right triangles. Without knowledge of these ratios and their standard values, solving heights and distances problems becomes difficult. The chapter assumes that students can identify opposite, adjacent, and hypotenuse sides and apply the appropriate ratio correctly. Therefore, a strong foundation in trigonometric ratios helps students solve application-based questions accurately and confidently.