Angle Of Elevation And Depression Worksheet Answers Explained
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Understanding the concepts of angle of elevation and depression is crucial in various fields such as mathematics, physics, and engineering. In this article, we will delve into these concepts, provide explanations, and offer worksheet answers with examples to help you better understand how to solve problems related to angles of elevation and depression. Let's get started! 📈
What is Angle of Elevation? 🌄
The angle of elevation is defined as the angle formed between the horizontal line of sight and the line of sight to an object above that horizontal line. When you look up at a tall building or a mountain, the angle you make from your eye level to the top of the object is the angle of elevation.
Example of Angle of Elevation
Imagine you are standing 50 meters away from a building that is 30 meters tall. To find the angle of elevation ((\theta)), you can use the tangent function:
[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{30}{50} ]
Calculating the angle using inverse tangent:
[ \theta = \tan^{-1}(0.6) \approx 30.96° ]
What is Angle of Depression? 📉
The angle of depression, on the other hand, is the angle formed between the horizontal line of sight and the line of sight to an object below that horizontal line. When you look down from a high point, such as a cliff, the angle you form from your eye level to the base of the object is the angle of depression.
Example of Angle of Depression
Consider you are standing at the edge of a cliff that is 100 meters high, and you see a boat in the water. If the boat is 80 meters away from the base of the cliff, you can find the angle of depression ((\phi)) using the tangent function:
[ \tan(\phi) = \frac{\text{opposite}}{\text{adjacent}} = \frac{100}{80} ]
Calculating the angle:
[ \phi = \tan^{-1}(1.25) \approx 51.34° ]
How to Solve Problems Involving Angles of Elevation and Depression
Now that we have defined both angles, let's explore how to solve problems involving angles of elevation and depression. Here are some steps to follow:
- Identify the Situation: Determine if you are looking up (angle of elevation) or down (angle of depression).
- Draw a Diagram: Visual representation can help clarify the problem.
- Use Trigonometric Functions: Apply the tangent, sine, or cosine functions as necessary, depending on the information provided.
- Calculate the Angles: Use inverse trigonometric functions to find the angles.
- Interpret Results: Make sure to interpret your answer in the context of the problem.
Example Problems and Solutions
Here are a few example problems to illustrate these concepts further.
Problem 1: Angle of Elevation
A person is standing 200 meters away from the base of a skyscraper. If the angle of elevation to the top of the skyscraper is 40°, what is the height of the skyscraper?
Solution:
Using the tangent function:
[ \tan(40°) = \frac{\text{height}}{200} ]
To find the height:
[ \text{height} = 200 \times \tan(40°) \approx 200 \times 0.8391 \approx 167.82 \text{ meters} ]
Problem 2: Angle of Depression
A drone is flying at an altitude of 150 meters. If it spots a car on the ground that is 100 meters horizontally away, what is the angle of depression from the drone to the car?
Solution:
Using the tangent function:
[ \tan(\phi) = \frac{150}{100} ]
To find the angle of depression:
[ \phi = \tan^{-1}(1.5) \approx 56.31° ]
Worksheet Table Example
Here's a simplified table that can be used as part of a worksheet for further practice with angles of elevation and depression:
<table> <tr> <th>Problem</th> <th>Angle Type</th> <th>Distance (m)</th> <th>Height (m)</th> <th>Angle (°)</th> </tr> <tr> <td>1</td> <td>Elevation</td> <td>100</td> <td>80</td> <td>38.66</td> </tr> <tr> <td>2</td> <td>Depression</td> <td>60</td> <td>45</td> <td>39.81</td> </tr> <tr> <td>3</td> <td>Elevation</td> <td>120</td> <td>100</td> <td>39.81</td> </tr> <tr> <td>4</td> <td>Depression</td> <td>150</td> <td>100</td> <td>53.13</td> </tr> </table>
Important Notes:
- Trigonometric Relationships: Remember that the tangent function is only one of several trigonometric functions used in these calculations. Depending on the information provided, you may need to apply the sine or cosine functions.
- Measurement Units: Always ensure your distance measurements are in the same units when using trigonometric calculations.
- Real-life Applications: Angles of elevation and depression are often used in fields such as architecture, navigation, and even sports.
By understanding the angles of elevation and depression, you can tackle a variety of real-world problems involving heights and distances. With enough practice, these concepts will become second nature, helping you excel in mathematics and beyond. 🏗️