9. Some Applications of Trigonometry

In this chapter, you will be studying about some ways in which trigonometry is used in the life around you.

In Fig. 9.1, the line AC drawn from the eye of the student to the top of the minar is called the line of sight. The angle BAC, so formed by the line of sight with the horizontal, is called the angle of elevation of the top of the minar from the eye of the student. Thus, the angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level.

Now, consider the situation given in Fig. 9.2. The girl sitting on the balcony is looking down at a flower pot placed on a stair of the temple. In this case, the line of sight is below the horizontal level. The angle so formed by the line of sight with the horizontal is called the angle of depression.

This chunk introduces the concepts of angles of elevation and depression, which are crucial in trigonometry. The angle of elevation occurs when an observer looks upward at an object, creating an angle with the horizontal line (the eye level). Conversely, the angle of depression happens when the observer looks downward at an object, also forming an angle with the horizontal. Understanding these angles is fundamental because these geometrical relationships help in solving various real-world problems involving heights and distances.

Imagine standing on a hill and looking at a tall building. The angle you look up from your eye level to the top of the building is called the angle of elevation. Now, think about being on a tall building and looking down at a friend on the ground; the angle formed between your line of sight to your friend and your horizontal view is the angle of depression. These concepts help architects and engineers calculate structures' heights without needing to measure directly.

Let us refer to Fig. 9.1 again. If you want to find the height CD of the minar without actually measuring it, what information do you need? You would need to know the distance DE at which the student is standing from the foot of the minar, the angle of elevation, ∠BAC, of the top of the minar, and the height AE of the student.

To find BC, we will use trigonometric ratios of ∠BAC. In triangle ABC, the side BC is the opposite side in relation to the known ∠A. Therefore, tan A = BC/AB which on solving would give us BC.

In this chunk, we learn how to apply trigonometric ratios to find unknown heights, such as the height of a minar. By knowing the distance from the observer to the minar and the angle of elevation, we can use the tangent function to find the height. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. Thus, once we have the height of the observer and the height calculated from the angle and distance, we can add these to get the actual height of the minar.

Consider measuring the height of a tree without climbing it. If you stand 15 meters away from the tree and look up at a 60° angle, you can use trigonometry to calculate the tree's height. By applying the concept of the angle of elevation and the tangent ratio, you find out how tall the tree is without needing to use a measuring tape directly.

Example 1: A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower. We use tan 60° = AB/BC, leading to AB = 15√3 meters.

Example 2: An electrician needs to reach a point below the top of a 5 m pole to repair it. If the ladder is inclined at 60°, we find the required length of the ladder and its distance from the pole using sine ratios.

This chunk contains practical examples where trigonometric concepts are applied to solve problems. In the first example, we determine the height of a tower by applying the tangent function with the known distance and angle of elevation. In the second example, the scenario involves using sine ratios to calculate the length of a ladder needed to reach a specific point below the top of a pole. These examples illustrate how trigonometry offers tools for solving everyday problems related to heights and distances in different professions like construction and repair.

Think about engineers using trigonometry to build communication towers. They need to know how tall the tower should be, requiring them to calculate distances and angles. Similarly, when a firefighter needs to rescue someone from a high place, they have to use ladders at the right angle for safety, which directly involves these trigonometric principles.

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