Angle Of Elevation & Depression Worksheet With Answers
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Understanding the concepts of angle of elevation and angle of depression is essential in trigonometry, particularly when solving real-world problems involving heights and distances. These angles are crucial in fields such as architecture, aviation, and navigation. In this article, we will delve into these concepts, provide a worksheet with problems related to angle of elevation and depression, and offer answers to enhance your understanding.
What are Angles of Elevation and Depression?
Angle of Elevation refers to the angle formed between the horizontal line and the line of sight to an object above the horizontal. For example, when you look up at a tree or a tall building, the angle formed between your line of sight and the horizontal level (the ground) is the angle of elevation. 🌳🏢
Angle of Depression, on the other hand, is the angle formed between the horizontal line and the line of sight to an object below the horizontal. If you stand on a cliff and look down at the ocean, the angle between your line of sight and the horizontal level at the cliff’s edge is the angle of depression. 🌊
Real-World Applications
Angles of elevation and depression have significant applications in various fields. Here are a few examples:
- Architecture: When designing buildings, architects use these angles to determine sightlines and ensure visibility from different points.
- Navigation: Pilots calculate the angle of depression to determine altitude during landing.
- Surveying: Surveyors use these angles to determine distances and heights of landmarks.
Worksheet: Angle of Elevation and Depression Problems
Below is a worksheet that includes various problems related to angles of elevation and depression. You can use this to practice and enhance your skills.
Problems
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A person standing 50 meters away from a building measures the angle of elevation to the top of the building as 30 degrees. How tall is the building?
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A drone is flying at an altitude of 150 meters. If the angle of depression from the drone to a point on the ground is 45 degrees, how far is the drone from that point horizontally?
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A tower is 80 meters high. If a person is standing 60 meters from the base of the tower, what is the angle of elevation from the person’s eye level to the top of the tower? (Assume eye level is 1.5 meters above the ground)
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From the top of a hill, a person looks down at a boat on the water. If the angle of depression is 20 degrees and the hill is 100 meters tall, how far is the boat from the base of the hill?
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A firefighter is 40 meters away from a burning building. If the angle of elevation to the top of the building is 60 degrees, what is the height of the building?
Important Note
To solve the above problems, you can use the following trigonometric ratios:
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For the angle of elevation and depression, use the tangent function:
[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} ]
Where:
- Opposite is the height of the object (or the difference in height between the observer and the object).
- Adjacent is the horizontal distance from the observer to the base of the object.
Answers to the Worksheet
Below are the answers to the problems provided in the worksheet.
<table> <tr> <th>Problem</th> <th>Answer</th> </tr> <tr> <td>1. Height of the building (angle of elevation = 30°, distance = 50m)</td> <td>Height = 50 * tan(30) = 50 * (1/√3) ≈ 28.87 meters</td> </tr> <tr> <td>2. Horizontal distance (angle of depression = 45°, height = 150m)</td> <td>Distance = 150 * tan(45) = 150 * 1 = 150 meters</td> </tr> <tr> <td>3. Angle of elevation (height = 80m, distance = 60m)</td> <td>Angle = arctan((80 - 1.5) / 60) ≈ 53.13 degrees</td> </tr> <tr> <td>4. Distance to the boat (angle of depression = 20°, height = 100m)</td> <td>Distance = 100 * tan(20) ≈ 36.33 meters</td> </tr> <tr> <td>5. Height of the building (angle of elevation = 60°, distance = 40m)</td> <td>Height = 40 * tan(60) = 40 * √3 ≈ 69.28 meters</td> </tr> </table>
Conclusion
Understanding the angle of elevation and depression is crucial for solving various practical problems in mathematics and science. Whether you’re an architect, a student, or just someone interested in the applications of trigonometry, mastering these concepts will prove beneficial. By practicing with worksheets like the one provided, you can develop a solid grasp of these essential skills. 🏗️📐