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9.1.6 - Example 6

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Understanding Angles of Depression

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Teacher
Teacher

Today we're going to learn about angles of depression. Can anyone tell me what an angle of depression is?

Student 1
Student 1

Is it the angle formed between a horizontal line and the line of sight looking down?

Teacher
Teacher

Exactly, great definition! Remember, as we look down from a height, we create an angle with the horizontal line. We can find some important relationships using these angles.

Student 2
Student 2

How do we use those angles to find heights or distances?

Teacher
Teacher

We can use trigonometric functions, especially the tangent function which relates angles to opposite and adjacent sides in right-angled triangles. Let's explore how with an example.

Defining the Problem

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Teacher
Teacher

In our example, we have two buildings. The first is 8 m tall, and we need to find the height of the multi-storeyed building. What are the angles we are given?

Student 3
Student 3

The angles are 30° to the top of the building and 45° to the bottom.

Teacher
Teacher

Correct! So, let’s represent these in a diagram to better visualize the problem. Remember, the angle at the top relates to the top of our building and the angle at the bottom relates to the base.

Applying Tangent Ratios

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Teacher
Teacher

Now, how can we use these angles? Who can set up the equation for the angle of depression of 30° first?

Student 4
Student 4

For triangle PBD, we can say tan(30°) = PD/BD, which implies BD = PD√3.

Teacher
Teacher

Perfect! Now what about the other triangle, PAC?

Student 1
Student 1

Since tan(45°) = 1, PC must equal AC.

Teacher
Teacher

Well done! By linking these two triangles, we can solve for both height and distance here.

Final Calculations

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Teacher
Teacher

Now we have two equations. If PD + 8 = BD and BD = PD√3, we can solve for PD. Can anyone help with the calculations?

Student 3
Student 3

So, PD + 8 = PD√3, right? Is that rearranging to find PD?

Teacher
Teacher

Exactly! What do we get when we isolate PD?

Student 4
Student 4

We get PD as a function of 8 and √3!

Teacher
Teacher

Right! Don't forget to plug back in to find the height of the multi-storeyed building and the distance. Get ready to wrap this up!

Summary and Key Takeaways

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Teacher
Teacher

Let's recap. We defined angles of depression, set up our problem, and applied tangent ratios to find our height and distances. What should we remember about using trigonometry in real-world examples?

Student 2
Student 2

We can effectively solve for unknown heights and distances given angles!

Teacher
Teacher

Exactly! Trigonometry is powerful for solving problems in various fields, from architecture to navigation. Keep practicing!

Introduction & Overview

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Quick Overview

This section explores the angles of depression from the top of a multi-storeyed building to an 8 m tall building, using trigonometry to determine the height of the multi-storeyed building and the distance between the two.

Standard

In this example, we analyze angles of depression from the top of a multi-storeyed building to both the bottom and the top of an adjacent 8 m tall building. By applying tangent function properties from right triangles, we derive the heights and distances involved, leading to practical applications of trigonometric concepts.

Detailed

In Example 6, we are tasked with finding the height of a multi-storeyed building (denoted as PC) and the horizontal distance to an 8 m tall building (denoted as AB). The angles of depression to the top and bottom of building AB from the top of building PC are 30° and 45° respectively. Using properties of alternate angles and tangent ratios for both right triangles (PBD and PAC), we established relationships between the height and distances involved.

From the triangle PBD with angle PBD = 30°, we use the tangent property: tan(30°) = PD/BD, thus establishing that BD = PD√3. For triangle PAC with angle PAC = 45°, we have PC = AC due to the properties of 45° triangles (where the lengths of opposite and adjacent sides are equal).

By combining these relationships, we conclude that the height of the multi-storeyed building is approximately (4√3 + 3) m, and the distance between the two buildings also measures (4√3 + 3) m. This example effectively demonstrates the application of trigonometric functions in solving real-world geometric problems.

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Audio Book

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Problem Statement

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The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-storeyed building are 30° and 45°, respectively. Find the height of the multistoreyed building and the distance between the two buildings.

Detailed Explanation

In this problem, we are given a scenario involving two buildings: an 8-meter tall building and a multi-storeyed building. We need to determine the height of the multi-storeyed building and the horizontal distance between the two buildings based on the angles of depression measured from the top of the taller building.

Examples & Analogies

Imagine standing on the roof of a tall building and looking down at a shorter building. The angle at which you look down to see the top of the shorter building is one angle of depression, while the angle to see the bottom is another. These angles help us figure out not just how high the building is, but also how far away it is.

Understanding the Angles

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In the right triangle formed, the angle at point PBD, which is the point of observation at the top of the multi-storeyed building, corresponds to the angle of depression to the top of the 8m building that is 30°. Similarly, the angle PAC (the angle to the bottom) is 45°.

Detailed Explanation

The angles of depression are important because they are related to the angles in the triangles we create when we draw lines from the observer to the points on the lower building. The angle of 30° gives us one triangle, and 45° gives us another triangle. This allows us to set up equations based on the properties of these triangles.

Examples & Analogies

Think of a slide at a playground. The angle you slide down is like the angle of depression. If the slide is steep, like 30°, you'd go down quickly (indicating that the bottom of the building is further away), but if it’s more gradual, like 45°, you would go down slower, showing that the building is closer.

Using Trigonometric Ratios

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In triangle PBD, we have PD/BD = tan 30° = 1/sqrt(3) or BD = PD * sqrt(3). In triangle PAC, the relationship is given by PC/AC = tan 45° = 1, which leads to PC = AC.

Detailed Explanation

We apply trigonometric ratios to relate the angles we have to the sides of our triangles. For the triangle with angle 30°, we find the relation between one side (PD) and the opposite side (BD). For the triangle with angle 45°, since tan 45° equals 1, we know the two sides are equal, leading us to the relationship PC = AC.

Examples & Analogies

Think about using a ladder against a wall: If you know the angle of the ladder and the height you want to reach, you can use trigonometry to find out how far the base of the ladder needs to be placed away from the wall.

Setting Up the Equations

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We conclude that PC = PD + DC, and from our earlier findings, we know PD + 8 = BD = PD * sqrt(3).

Detailed Explanation

This section combines the height of the taller building and the height relationships we've established through trigonometry. We set up the equation based on the relationships of PD, the height of the two buildings, and the distances derived earlier.

Examples & Analogies

Imagine two people building towers of blocks. The first person builds a tower of 8 blocks high while the second person has an unknown number above that. The total height of the second tower needs to equal the first tower plus some extra blocks. Similarly, we find the total height of the multi-storeyed building by figuring out these component heights.

Solving for Heights and Distances

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From the earlier relationship, we derive PD = (8 * (3 - sqrt(3))) / (3 - 1). Thus, after calculations, we find the height of the multi-storeyed building is 8 + PD and the distance AC is also calculated as 4 * sqrt(3) m.

Detailed Explanation

At this point, we solve our equation to find the height of the multi-storeyed building. By determining PD and then adding the 8 m height of the smaller building, we finally find out how tall the larger building is and the specific distance separating them.

Examples & Analogies

Think of it like measuring the height of a tree you cannot directly measure. You see the top from a distance and use your measuring tools (the angles) to compute both how tall it is and how far away you are standing. In this case, through calculation, we can determine the 'height' of our invisible tree (the multi-storeyed building)!

Definitions & Key Concepts

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

Key Concepts

  • Angle of Depression: The angle between the horizontal line and the line of sight down to an object.

  • Tangent Function: A primary trigonometric function used to relate opposite sides and adjacent sides of a triangle.

  • Height and Distance Calculation: Utilizing angles and tangent to find unknown measurements in geometric figures.

Examples & Real-Life Applications

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

Examples

  • If you are on top of a 10 m tall building, and you see another building that is 5 m tall, using angles of depression can help calculate their distance apart.

  • When measuring the height of a tower using the angle of depression from a certain height, you apply tangent ratios to solve for the tower's height.

Memory Aids

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

🎵 Rhymes Time

  • When looking down with sight so keen, the angle shows the height unseen.

📖 Fascinating Stories

  • Imagine you are on top of a tall building, peering down at the street below. You measure the angle to the edge of the sidewalk and to the tallest tree nearby. Each angle gives you a clue about how far apart everything is and how high it stands.

🧠 Other Memory Gems

  • Remember 'To Opposite A' which stands for Tangent = Opposite/Adjacent.

🎯 Super Acronyms

HAD = Height and Distance from Angles.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Angle of Depression

    Definition:

    The angle formed between a horizontal line from an observer's eye and the line of sight to an object below the horizontal level.

  • Term: Tangent Function (tan)

    Definition:

    In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

  • Term: Right Triangle

    Definition:

    A triangle that has one angle equal to 90 degrees.

  • Term: Height

    Definition:

    The measurement of an object from its base to its top.

  • Term: Distance

    Definition:

    The space between two points measured along the horizontal plane.