From The Top Of House 8 Meters High,you Can See A Dog On The Ground.The Angle Of Depression Of The Dog 60°.How Far Is The Dog From The House
Introduction
In this article, we will explore a classic trigonometric problem that involves finding the distance to an object using the angle of depression. The problem states that from the top of a house 8 meters high, we can see a dog on the ground, and the angle of depression of the dog is 60°. Our goal is to find the distance from the house to the dog.
Understanding the Problem
Before we dive into the solution, let's break down the problem and understand the key concepts involved. The angle of depression is the angle between the horizontal and the line of sight to the object. In this case, the angle of depression is 60°, which means that the line of sight to the dog makes an angle of 60° with the horizontal.
Visualizing the Problem
To better understand the problem, let's visualize it. Imagine a right-angled triangle with the house as the vertical leg, the ground as the horizontal leg, and the line of sight to the dog as the hypotenuse. The angle of depression is 60°, which means that the line of sight to the dog makes an angle of 60° with the horizontal.
Using Trigonometry to Solve the Problem
Now that we have a clear understanding of the problem, let's use trigonometry to solve it. We can use the tangent function to relate the angle of depression to the ratio of the opposite side (the height of the house) to the adjacent side (the distance from the house to the dog).
The Tangent Function
The tangent function is defined as:
tan(θ) = opposite side / adjacent side
In this case, the opposite side is the height of the house (8 meters), and the adjacent side is the distance from the house to the dog (which we want to find). The angle of depression is 60°, so we can plug in the values to get:
tan(60°) = 8 / x
Solving for x
To solve for x, we can rearrange the equation to get:
x = 8 / tan(60°)
Evaluating the Tangent Function
The tangent function is a periodic function that oscillates between positive and negative values. However, in this case, we are only interested in the positive value of the tangent function, since the angle of depression is between 0° and 90°.
Using a Calculator to Evaluate the Tangent Function
To evaluate the tangent function, we can use a calculator. The tangent of 60° is approximately 1.732. So, we can plug in the value to get:
x = 8 / 1.732
Simplifying the Expression
To simplify the expression, we can divide 8 by 1.732 to get:
x ≈ 4.59
Conclusion
In this article, we used trigonometry to solve a classic problem involving the angle of depression. We found that the distance from the house to the dog is approximately 4.59 meters. This problem illustrates the importance of trigonometry in solving real-world problems involving right-angled triangles.
Real-World Applications
Trigonometry has many real-world applications, including:
- Surveying: Trigonometry is used in surveying to measure distances and angles between landmarks.
- Navigation: Trigonometry is used in navigation to determine the position and course of a ship or aircraft.
- Physics: Trigonometry is used in physics to describe the motion of objects and the behavior of waves.
Final Thoughts
Introduction
In our previous article, we explored a classic trigonometric problem involving the angle of depression. We used the tangent function to find the distance from the house to the dog. In this article, we will answer some common questions related to trigonometry and solving problems involving right-angled triangles.
Q&A
Q: What is the tangent function?
A: The tangent function is a trigonometric function that relates the angle of a right-angled triangle to the ratio of the opposite side to the adjacent side. It is defined as:
tan(θ) = opposite side / adjacent side
Q: How do I use the tangent function to solve problems involving right-angled triangles?
A: To use the tangent function, you need to know the angle of the triangle and the length of one of the sides. You can then use the tangent function to find the length of the other side.
Q: What is the difference between the tangent function and the sine and cosine functions?
A: The tangent function is a ratio of the opposite side to the adjacent side, while the sine and cosine functions are ratios of the opposite side to the hypotenuse and the adjacent side to the hypotenuse, respectively.
Q: How do I evaluate the tangent function?
A: You can evaluate the tangent function using a calculator or by using a trigonometric table. You can also use the unit circle to evaluate the tangent function.
Q: What are some common applications of trigonometry?
A: Trigonometry has many real-world applications, including:
- Surveying: Trigonometry is used in surveying to measure distances and angles between landmarks.
- Navigation: Trigonometry is used in navigation to determine the position and course of a ship or aircraft.
- Physics: Trigonometry is used in physics to describe the motion of objects and the behavior of waves.
Q: How do I use trigonometry to solve problems involving right-angled triangles with obtuse angles?
A: To use trigonometry to solve problems involving right-angled triangles with obtuse angles, you need to use the tangent function and the Pythagorean theorem. You can also use the law of sines and the law of cosines to solve problems involving right-angled triangles with obtuse angles.
Q: What are some common mistakes to avoid when using trigonometry?
A: Some common mistakes to avoid when using trigonometry include:
- Not using the correct trigonometric function: Make sure to use the correct trigonometric function for the problem you are trying to solve.
- Not evaluating the trigonometric function correctly: Make sure to evaluate the trigonometric function correctly using a calculator or a trigonometric table.
- Not using the correct units: Make sure to use the correct units for the problem you are trying to solve.
Conclusion
In this article, we answered some common questions related to trigonometry and solving problems involving right-angled triangles. We hope that this article has been helpful in clarifying some of the concepts and techniques involved in trigonometry. If you have any further questions or need help with a specific problem, please don't hesitate to ask.
Additional Resources
For more information on trigonometry and solving problems involving right-angled triangles, please see the following resources:
- Trigonometry textbooks: There are many excellent textbooks on trigonometry that can provide a comprehensive introduction to the subject.
- Online resources: There are many online resources available that can provide additional help and support with trigonometry, including video tutorials, practice problems, and interactive simulations.
- Mathematical software: There are many mathematical software packages available that can provide additional help and support with trigonometry, including graphing calculators and computer algebra systems.