An Airplane Is Flying At A Height Of 2 Miles Above The Ground. The Distance Along The Ground From The Airplane To The Airport Is 5 Miles. What Is The Angle Of Depression From The Airplane To The Airport? (Round To The Nearest Tenth.)

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Introduction

When an airplane is flying at a certain height above the ground, the distance between the airplane and the airport on the ground can be calculated using trigonometric functions. In this article, we will explore how to find the angle of depression from the airplane to the airport using the given information. We will use the tangent function to solve this problem and provide a step-by-step solution.

Understanding the Problem

The problem states that an airplane is flying at a height of 2 miles above the ground. The distance along the ground from the airplane to the airport is 5 miles. We need to find the angle of depression from the airplane to the airport. The angle of depression is the angle between the horizontal and the line of sight from the airplane to the airport.

Drawing a Diagram

To visualize the problem, let's draw a diagram. We can draw a right triangle with the airplane at the top, the airport at the bottom, and the line of sight from the airplane to the airport as the hypotenuse. The height of the airplane above the ground is 2 miles, and the distance from the airplane to the airport is 5 miles.

  +---------------+
  |              |
  |  2 miles    |
  |  (height)  |
  |              |
  +---------------+
  |              |
  |  5 miles    |
  |  (distance) |
  |              |
  +---------------+

Using Trigonometry to Solve the Problem

We can use the tangent function to solve this problem. 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 airplane (2 miles), and the adjacent side is the distance from the airplane to the airport (5 miles).

tan(angle) = opposite side / adjacent side
tan(angle) = 2 miles / 5 miles

Solving for the Angle

To solve for the angle, we can take the inverse tangent (arctangent) of both sides of the equation.

angle = arctan(2 miles / 5 miles)

Calculating the Angle

Using a calculator, we can calculate the angle.

angle ≈ 21.8°

Conclusion

In this article, we used the tangent function to find the angle of depression from an airplane to an airport. We drew a diagram to visualize the problem and used the arctangent function to solve for the angle. The final answer is approximately 21.8°.

Additional Information

  • The angle of depression is an important concept in aviation and navigation.
  • The tangent function is a fundamental concept in trigonometry and is used to solve a wide range of problems.
  • The arctangent function is the inverse of the tangent function and is used to find the angle corresponding to a given ratio.

Real-World Applications

  • The angle of depression is used in aviation to determine the distance and direction of an object on the ground.
  • The tangent function is used in navigation to determine the direction and distance of a ship or aircraft.
  • The arctangent function is used in computer graphics to determine the angle of a line or a curve.

Future Research Directions

  • Investigating the use of the tangent function in other fields, such as physics and engineering.
  • Developing new algorithms and techniques for solving trigonometric problems.
  • Exploring the applications of the arctangent function in computer science and data analysis.

References

  • [1] "Trigonometry" by Michael Corral, 2013.
  • [2] "Calculus" by Michael Spivak, 2008.
  • [3] "Geometry" by I.M. Gelfand, 1996.

Note: The references provided are for illustrative purposes only and are not actual references used in this article.

Introduction

In our previous article, we explored how to find the angle of depression from an airplane to an airport using the tangent function. In this article, we will answer some frequently asked questions related to the angle of depression and provide additional information to help you better understand this concept.

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 airplane to the airport.

Q: Why is the angle of depression important?

A: The angle of depression is important in aviation and navigation because it helps pilots and navigators determine the distance and direction of an object on the ground.

Q: How do you 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 airplane, and the adjacent side is the distance from the airplane to the airport.

Q: What is the formula for calculating the angle of depression?

A: The formula for calculating the angle of depression is:

tan(angle) = opposite side / adjacent side

Q: How do you solve for the angle?

A: To solve for the angle, you can take the inverse tangent (arctangent) of both sides of the equation.

Q: What is the inverse tangent function?

A: The inverse tangent function is the inverse of the tangent function. It is used to find the angle corresponding to a given ratio.

Q: How do you use the inverse tangent function to solve for the angle?

A: To use the inverse tangent function, you can plug in the ratio of the opposite side to the adjacent side into the function and solve for the angle.

Q: What is the unit of measurement for the angle of depression?

A: The unit of measurement for the angle of depression is typically degrees.

Q: Can the angle of depression be negative?

A: No, the angle of depression cannot be negative. The angle of depression is always a positive value.

Q: Can the angle of depression be greater than 90 degrees?

A: No, the angle of depression cannot be greater than 90 degrees. The angle of depression is always less than 90 degrees.

Q: What are some real-world applications of the angle of depression?

A: Some real-world applications of the angle of depression include:

  • Aviation: The angle of depression is used in aviation to determine the distance and direction of an object on the ground.
  • Navigation: The tangent function is used in navigation to determine the direction and distance of a ship or aircraft.
  • Computer graphics: The arctangent function is used in computer graphics to determine the angle of a line or a curve.

Conclusion

In this article, we answered some frequently asked questions related to the angle of depression and provided additional information to help you better understand this concept. We hope this article has been helpful in clarifying any confusion you may have had about the angle of depression.

Additional Resources

  • [1] "Trigonometry" by Michael Corral, 2013.
  • [2] "Calculus" by Michael Spivak, 2008.
  • [3] "Geometry" by I.M. Gelfand, 1996.

Note: The references provided are for illustrative purposes only and are not actual references used in this article.

Frequently Asked Questions

  • Q: What is the difference between the angle of depression and the angle of elevation? A: The angle of depression is the angle between the horizontal and the line of sight from the airplane to the airport, while the angle of elevation is the angle between the horizontal and the line of sight from the airport to the airplane.
  • Q: Can the angle of depression be used to determine the height of an object? A: Yes, the angle of depression can be used to determine the height of an object.
  • Q: Can the angle of depression be used to determine the distance of an object? A: Yes, the angle of depression can be used to determine the distance of an object.

Real-World Examples

  • Example 1: An airplane is flying at a height of 2 miles above the ground. The distance along the ground from the airplane to the airport is 5 miles. What is the angle of depression from the airplane to the airport?
  • Example 2: A ship is sailing at a distance of 10 miles from the shore. The height of the ship above the water is 3 miles. What is the angle of depression from the ship to the shore?

Note: The real-world examples provided are for illustrative purposes only and are not actual examples used in this article.