Example 6

9.1.6 Example 6

Description

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.

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.

Memory Aids

🎵 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.

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.

Glossary of 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.