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9 - Some Applications of Trigonometry

Interactive Audio Lesson

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Understanding Line of Sight and Angles

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Teacher
Teacher

Today, we will talk about how we can use trigonometry in real life. Let's start by understanding what we mean by 'line of sight'. Does anyone know what this is?

Student 1
Student 1

Is it the line from our eyes to the object we are viewing?

Teacher
Teacher

Exactly! The line of sight is crucial when measuring things at a distance, particularly with angles of elevation and depression. Can anyone tell me what the angle of elevation is?

Student 2
Student 2

It's the angle formed with the horizontal when we look up!

Teacher
Teacher

Great! And what about the angle of depression?

Student 3
Student 3

That's when we look down, forming an angle below the horizontal.

Teacher
Teacher

Exactly right! To remember these terms, think of 'elevation' as 'up' and 'depression' as 'down'. Now, let’s look at how we can find heights and distances using these angles.

Calculating Heights Using Trigonometry

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Teacher
Teacher

Let’s consider an example: A tower is 15 meters away, and the angle of elevation to its top is 60 degrees. How can we find the height of the tower?

Student 4
Student 4

We could use the tangent ratio because we have the opposite side and the adjacent side.

Teacher
Teacher

Exactly! So we set it up as tan(60) = height/15. Now, who can calculate the height for us?

Student 1
Student 1

I can! The height will be 15 * tan(60), which is 15 * √3.

Teacher
Teacher

Correct! And that gives us a height of approximately 25.98 meters. Remember, tan helps us connect angles and side lengths. Let's recap: when using the tangent ratio, we are essentially looking at the relationship between the angle and the sides of the triangle formed.

Real-life Applications of Trigonometry

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Teacher
Teacher

Trigonometry has many applications! An electrician, for instance, needs to reach a point on a pole 1.3 meters below its top. If the pole is 5 meters tall, how do we figure out how long her ladder should be at a specific angle?

Student 2
Student 2

We can create a right triangle where the height to reach is 3.7 meters.

Teacher
Teacher

Right! So now, what trigonometric function do we use?

Student 3
Student 3

We should use sine since we want the length of the hypotenuse.

Teacher
Teacher

Fantastic! And remember, sine = opposite/hypotenuse. Let’s determine the ladder's length together!

Problems on Angles and Heights

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Teacher
Teacher

Here's a challenge: An observer standing 28.5m away from a chimney sees it at a 45-degree angle of elevation. How can we find the chimney's height?

Student 4
Student 4

Using tan(45) since it equals 1, we can simply state the height equals the distance!

Teacher
Teacher

Correct! It means the chimney is 28.5 + 1.5 meters tall. Never underestimate the power of angles. They are everywhere around us!

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section explores practical applications of trigonometry, particularly in measuring heights and distances.

Standard

In this section, we delve into how trigonometric ratios can be applied in real-world scenarios like measuring the height of buildings or towers without direct measurement. The concepts of angles of elevation and depression are highlighted with detailed examples and applications.

Detailed

Some Applications of Trigonometry

In this section, we investigate the practical applications of trigonometry, particularly concerning heights and distances. The importance of trigonometric ratios in real-world scenarios becomes evident as we explore how to calculate heights that are difficult to measure directly.

Key Concepts Introduced:

  • Line of Sight: The line from the observer's eye to the object being viewed.
  • Angle of Elevation: The angle between the horizontal line and the line of sight when looking up at an object above the horizontal level.
  • Angle of Depression: The angle between the horizontal line and the line of sight when looking down at an object below the horizontal level.

a detailed approach on how to compute heights and distances using trigonometric functions such as tangent, sine, and cosine through various examples. For instance, the section discusses how the heights of towers can be determined using the distance from the observer and the angles of elevation or depression, along with practical examples such as electricians needing ladders or calculating the width of rivers based on angles of depression.

Understanding these concepts proves vital in various fields such as architecture, engineering, and even everyday problem-solving, revealing the ubiquitous presence of trigonometry in our world.

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Audio Book

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Understanding Angles of Elevation and Depression

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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.

Detailed Explanation

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.

Examples & Analogies

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.

Finding Heights Using Angles of Elevation

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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.

Detailed Explanation

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.

Examples & Analogies

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.

Examples of Applications in Real Situations

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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.

Detailed Explanation

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.

Examples & Analogies

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.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Line of Sight: The line from the observer's eye to the object being viewed.

  • Angle of Elevation: The angle between the horizontal line and the line of sight when looking up at an object above the horizontal level.

  • Angle of Depression: The angle between the horizontal line and the line of sight when looking down at an object below the horizontal level.

  • a detailed approach on how to compute heights and distances using trigonometric functions such as tangent, sine, and cosine through various examples. For instance, the section discusses how the heights of towers can be determined using the distance from the observer and the angles of elevation or depression, along with practical examples such as electricians needing ladders or calculating the width of rivers based on angles of depression.

  • Understanding these concepts proves vital in various fields such as architecture, engineering, and even everyday problem-solving, revealing the ubiquitous presence of trigonometry in our world.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Using the angle of elevation to determine the height of a tower when standing 15m away is practical to understand distances.

  • Understanding the ladder length required for an electrician when reaching a point below a pole incorporates angles of depression.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • When looking high, the angle shows; elevation's where the upward gaze goes.

📖 Fascinating Stories

  • Imagine a squirrel on a tree: it looks down at folks like you and me; the angle it makes with the ground so low is the angle of depression, so off it goes!

🧠 Other Memory Gems

  • E.D. - Elevation is Up, Depression is Down.

🎯 Super Acronyms

LAD - Line of sight, Angle of elevation, Angle of depression.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Line of Sight

    Definition:

    The line drawn from the observer's eye to the point being viewed.

  • Term: Angle of Elevation

    Definition:

    The angle formed between the line of sight and the horizontal when looking up.

  • Term: Angle of Depression

    Definition:

    The angle formed between the line of sight and the horizontal when looking down.

  • Term: Trigonometric Ratios

    Definition:

    Ratios that relate the angles of a right triangle to the lengths of its sides.