A Mother Gazes Out A Second-floor Window At Her Son Playing At The Playground. If The Mother's Eye Level Is 12.6 Meters Off The Ground And The Playground Is 20 Meters From The Base Of The Building, What Is The Angle Of Depression From The Mother's Line
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Introduction
In trigonometry, the angle of depression is a fundamental concept that helps us understand the relationship between the height of an object and the distance between the object and the observer. In this article, we will explore how to calculate the angle of depression using a real-life scenario. We will use the example of a mother gazing out a second-floor window at her son playing at the playground.
The Problem
A mother's eye level is 12.6 meters off the ground, and she is standing at a second-floor window. Her son is playing at the playground, which is 20 meters from the base of the building. We need to find the angle of depression from the mother's line of sight to the ground.
Understanding the Angle of Depression
The angle of depression is the angle between the horizontal and the line of sight from the observer to the object. In this case, the object is the ground, and the observer is the mother. The angle of depression is measured from the horizontal, and it is always less than 90 degrees.
Calculating the Angle of Depression
To calculate the angle of depression, we can use the tangent function. The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the height of the mother's eye level (12.6 meters), and the adjacent side is the distance from the base of the building to the playground (20 meters).
We can use the following formula to calculate the angle of depression:
tan(θ) = opposite side / adjacent side
where θ is the angle of depression.
Plugging in the Values
Now, let's plug in the values into the formula:
tan(θ) = 12.6 / 20
Simplifying the Expression
To simplify the expression, we can divide both sides by 20:
tan(θ) = 0.63
Finding the Angle
Now, we need to find the angle θ. We can use the inverse tangent function (arctangent) to find the angle:
θ = arctan(0.63)
Calculating the Angle
Using a calculator, we can calculate the angle:
θ ≈ 32.0°
Conclusion
In this article, we calculated the angle of depression from the mother's line of sight to the ground. We used the tangent function and the inverse tangent function to find the angle. The angle of depression is approximately 32.0°.
Real-World Applications
The angle of depression has many real-world applications, including:
- Surveying: The angle of depression is used in surveying to measure the height of objects and the distance between objects.
- Architecture: The angle of depression is used in architecture to design buildings and structures that are safe and functional.
- Engineering: The angle of depression is used in engineering to design systems that are efficient and effective.
Final Thoughts
In conclusion, the angle of depression is a fundamental concept in trigonometry that helps us understand the relationship between the height of an object and the distance between the object and the observer. We calculated the angle of depression using a real-life scenario and used the tangent function and the inverse tangent function to find the angle. The angle of depression has many real-world applications, including surveying, architecture, and engineering.
Additional Resources
For more information on the angle of depression, please refer to the following resources:
- Trigonometry textbooks: Trigonometry textbooks provide a comprehensive overview of the angle of depression and its applications.
- Online resources: Online resources, such as Khan Academy and Mathway, provide interactive lessons and exercises on the angle of depression.
- Mathematical software: Mathematical software, such as Mathematica and MATLAB, provide tools and functions for calculating the angle of depression.
References
- Trigonometry textbooks: Trigonometry textbooks provide a comprehensive overview of the angle of depression and its applications.
- Online resources: Online resources, such as Khan Academy and Mathway, provide interactive lessons and exercises on the angle of depression.
- Mathematical software: Mathematical software, such as Mathematica and MATLAB, provide tools and functions for calculating the angle of depression.
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Introduction
In our previous article, we explored how to calculate the angle of depression using a real-life scenario. We used the example of a mother gazing out a second-floor window at her son playing at the playground. In this article, we will answer some frequently asked questions about the angle of depression.
Q&A
Q: What is the angle of depression?
A: The angle of depression is the angle between the horizontal and the line of sight from the observer to the object. In this case, the object is the ground, and the observer is the mother.
Q: How do I calculate the angle of depression?
A: To calculate the angle of depression, you can use the tangent function. The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the height of the mother's eye level (12.6 meters), and the adjacent side is the distance from the base of the building to the playground (20 meters).
Q: What is the formula for calculating the angle of depression?
A: The formula for calculating the angle of depression is:
tan(θ) = opposite side / adjacent side
where θ is the angle of depression.
Q: How do I find the angle of depression?
A: To find the angle of depression, you can use the inverse tangent function (arctangent) to find the angle. Using a calculator, you can calculate the angle:
θ = arctan(0.63)
Q: What is the angle of depression in this scenario?
A: The angle of depression in this scenario is approximately 32.0°.
Q: What are some real-world applications of the angle of depression?
A: The angle of depression has many real-world applications, including:
- Surveying: The angle of depression is used in surveying to measure the height of objects and the distance between objects.
- Architecture: The angle of depression is used in architecture to design buildings and structures that are safe and functional.
- Engineering: The angle of depression is used in engineering to design systems that are efficient and effective.
Q: How do I use the angle of depression in real-world applications?
A: To use the angle of depression in real-world applications, you can use the tangent function and the inverse tangent function to find the angle. You can also use mathematical software, such as Mathematica and MATLAB, to calculate the angle.
Q: What are some common mistakes to avoid when calculating the angle of depression?
A: Some common mistakes to avoid when calculating the angle of depression include:
- Using the wrong formula: Make sure to use the correct formula for calculating the angle of depression.
- Rounding errors: Be careful when rounding numbers to avoid errors in the calculation.
- Not considering the context: Make sure to consider the context of the problem and the units of measurement.
Conclusion
In this article, we answered some frequently asked questions about the angle of depression. We provided formulas and examples to help you understand how to calculate the angle of depression. We also discussed some real-world applications of the angle of depression and some common mistakes to avoid when calculating the angle of depression.
Additional Resources
For more information on the angle of depression, please refer to the following resources:
- Trigonometry textbooks: Trigonometry textbooks provide a comprehensive overview of the angle of depression and its applications.
- Online resources: Online resources, such as Khan Academy and Mathway, provide interactive lessons and exercises on the angle of depression.
- Mathematical software: Mathematical software, such as Mathematica and MATLAB, provide tools and functions for calculating the angle of depression.
References
- Trigonometry textbooks: Trigonometry textbooks provide a comprehensive overview of the angle of depression and its applications.
- Online resources: Online resources, such as Khan Academy and Mathway, provide interactive lessons and exercises on the angle of depression.
- Mathematical software: Mathematical software, such as Mathematica and MATLAB, provide tools and functions for calculating the angle of depression.