Angles Of Depression And Elevation Worksheet Answers Explained
Table of Contents :
Understanding angles of depression and elevation is crucial in various fields, especially in geometry and trigonometry. These concepts describe the angles formed when a line of sight looks downward (angle of depression) or upward (angle of elevation) from a horizontal line. In this article, we will explore the definitions, applications, and how to tackle worksheet problems concerning angles of depression and elevation.
What are Angles of Depression and Elevation?
Angle of Elevation 📈
The angle of elevation is the angle formed between the horizontal line and the line of sight when an observer looks up at an object. For example, if you are standing on the ground and looking at the top of a tree, the angle between the line of sight to the tree's top and the horizontal ground is the angle of elevation.
Angle of Depression 📉
Conversely, the angle of depression is the angle formed between the horizontal line and the line of sight when an observer looks down at an object. For instance, if you are standing on top of a cliff and looking down at a boat on the water, the angle between your line of sight to the boat and the horizontal cliff edge is the angle of depression.
Visual Representation
To better understand these concepts, consider the following diagram:
A
|\
| \
| \
| \
| \
| \
|______\B
In this triangle:
- A is the point of observation (e.g., the top of a building).
- B is the point being observed (e.g., a car on the ground).
- The angle formed at point A looking up or down is either the angle of elevation or depression.
Why Are Angles of Depression and Elevation Important?
Angles of depression and elevation play a significant role in various real-life applications, such as:
- Architecture: To ensure structures have the correct slope.
- Navigation: For pilots and sailors to calculate distances and heights.
- Surveying: To measure distances from a known height.
Solving Worksheet Problems
Example Problem 1
Problem: A person standing 50 meters from a tree notices that the top of the tree forms an angle of elevation of 30°. How tall is the tree?
To find the height of the tree, we can use trigonometric ratios. The tangent of the angle is defined as the opposite side (height of the tree, h) over the adjacent side (distance from the tree, 50 m).
Using the formula: [ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} ]
Calculation:
[ \tan(30°) = \frac{h}{50} ]
Using the value of (\tan(30°) = \frac{1}{\sqrt{3}}): [ \frac{1}{\sqrt{3}} = \frac{h}{50} ] [ h = 50 \times \frac{1}{\sqrt{3}} \approx 28.87 \text{ m} ]
Example Problem 2
Problem: From the top of a 100-meter tall building, an observer looks down at a point on the ground, forming an angle of depression of 45°. How far is the point from the base of the building?
Here, the height of the building is 100 m, and we can find the distance (d) using the same tangent ratio.
Using the formula: [ \tan(45°) = \frac{\text{height}}{\text{distance}} ] Since (\tan(45°) = 1): [ 1 = \frac{100}{d} ] So, [ d = 100 \text{ m} ]
Summary of Calculations
The above problems show how to apply trigonometric functions to find unknown distances and heights based on angles of elevation and depression. The table below summarizes key trigonometric values needed for these calculations:
<table> <tr> <th>Angle (Degrees)</th> <th>sin</th> <th>cos</th> <th>tan</th> </tr> <tr> <td>0°</td> <td>0</td> <td>1</td> <td>0</td> </tr> <tr> <td>30°</td> <td>0.5</td> <td>√3/2</td> <td>1/√3</td> </tr> <tr> <td>45°</td> <td>√2/2</td> <td>√2/2</td> <td>1</td> </tr> <tr> <td>60°</td> <td>√3/2</td> <td>0.5</td> <td>√3</td> </tr> <tr> <td>90°</td> <td>1</td> <td>0</td> <td>undefined</td> </tr> </table>
Important Notes
Remember: The height of the observer is not always considered in basic angle of elevation and depression problems unless specified.
Final Tips for Solving Problems
- Draw a diagram: Visualizing the problem can greatly assist in understanding the relationships between angles and distances.
- Identify known values: Clearly label your known sides and angles.
- Choose the right trigonometric function: Decide whether to use sine, cosine, or tangent based on the sides you know and the side you need to find.
- Practice regularly: The more problems you solve, the better you will understand these concepts.
With the basics of angles of depression and elevation explained and examples provided, tackling related worksheet problems becomes a more manageable and enjoyable task! Whether you're preparing for an exam or simply sharpening your skills, mastering these angles is a valuable asset in mathematics and practical applications alike. Happy studying! 📚✨