Point G And Point H Are 58 Meters Apart. Point H Is At A Bearing Of 063° From Point G. How Far East Of Point G Is Point H? Give Your Answer To 2 Decimal Places.

Mar 8, 2025 by ADMIN 161 views

**Calculating Distance and Direction: A Mathematical Approach**

Introduction

In mathematics, particularly in trigonometry, we often encounter problems involving distances, directions, and angles. These problems can be solved using various mathematical concepts and formulas. In this article, we will explore how to calculate the distance and direction between two points on a coordinate plane.

Understanding the Problem

We are given two points, G and H, which are 58 meters apart. Point H is at a bearing of 063° from point G. Our goal is to find out how far east of point G is point H. To solve this problem, we need to understand the concept of bearing and how it relates to the coordinate plane.

Bearing and Coordinate Plane

A bearing is an angle measured clockwise from true north. In this case, point H is at a bearing of 063° from point G, which means it is 63° east of north. To visualize this, imagine a compass with true north at the top. Point H would be located 63° east of the true north direction.

Converting Bearing to Cartesian Coordinates

To solve this problem, we need to convert the bearing to Cartesian coordinates. We can do this by using the following formulas:

x = r * cos(θ) y = r * sin(θ)

where x and y are the Cartesian coordinates, r is the distance from the origin, and θ is the angle in radians.

Converting Bearing to Radians

To convert the bearing from degrees to radians, we can use the following formula:

θ (radians) = θ (degrees) * π / 180

In this case, the bearing is 063°, which is equivalent to:

θ (radians) = 63 * π / 180 ≈ 1.102

Calculating Cartesian Coordinates

Now that we have the bearing in radians, we can calculate the Cartesian coordinates of point H. We know that the distance from point G to point H is 58 meters, which is the value of r.

x = 58 * cos(1.102) ≈ 46.19 y = 58 * sin(1.102) ≈ 31.45

Finding the Eastward Distance

To find the eastward distance from point G to point H, we need to find the x-coordinate of point H. Since the x-coordinate represents the eastward distance, we can simply take the value of x.

Eastward distance = 46.19 meters

Conclusion

In this article, we have demonstrated how to calculate the distance and direction between two points on a coordinate plane. We used the concept of bearing and converted it to Cartesian coordinates using the formulas for cosine and sine. We then calculated the eastward distance from point G to point H and found it to be approximately 46.19 meters.

Final Answer

The final answer is: 46.19

Additional Information

The bearing of 063° from point G means that point H is 63° east of north.
The Cartesian coordinates of point H are approximately (46.19, 31.45).
The eastward distance from point G to point H is approximately 46.19 meters.

Mathematical Formulas

x = r * cos(θ)
y = r * sin(θ)
θ (radians) = θ (degrees) * π / 180

References

[1] Trigonometry: A Unit Circle Approach. By Charles P. McKeague and Mark D. Turner.
[2] Calculus: Early Transcendentals. By James Stewart.

Glossary

Bearing: An angle measured clockwise from true north.
Cartesian coordinates: A system of coordinates that uses the x-axis and y-axis to locate points in a plane.
Cosine: A trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right triangle.
Sine: A trigonometric function that represents the ratio of the opposite side to the hypotenuse in a right triangle.

Introduction

In our previous article, we explored the concept of calculating distance and direction between two points on a coordinate plane. We used the concept of bearing and converted it to Cartesian coordinates using the formulas for cosine and sine. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the difference between bearing and direction?

A: Bearing and direction are often used interchangeably, but they have slightly different meanings. Bearing refers to the angle measured clockwise from true north, while direction refers to the general direction in which an object is moving or pointing.

Q: How do I convert a bearing from degrees to radians?

A: To convert a bearing from degrees to radians, you can use the following formula:

θ (radians) = θ (degrees) * π / 180

For example, if the bearing is 063°, you would convert it to radians as follows:

θ (radians) = 63 * π / 180 ≈ 1.102

Q: What is the formula for calculating the Cartesian coordinates of a point?

A: The formula for calculating the Cartesian coordinates of a point is:

x = r * cos(θ) y = r * sin(θ)

where x and y are the Cartesian coordinates, r is the distance from the origin, and θ is the angle in radians.

Q: How do I calculate the eastward distance from point G to point H?

A: To calculate the eastward distance from point G to point H, you need to find the x-coordinate of point H. Since the x-coordinate represents the eastward distance, you can simply take the value of x.

Eastward distance = 46.19 meters

Q: What is the significance of the bearing in this problem?

A: The bearing of 063° from point G means that point H is 63° east of north. This information is crucial in determining the Cartesian coordinates of point H.

Q: Can I use this method to calculate the distance and direction between any two points?

A: Yes, you can use this method to calculate the distance and direction between any two points on a coordinate plane. However, you need to ensure that you have the correct bearing and distance values.

Q: What are some real-world applications of this concept?

A: This concept has numerous real-world applications, including:

Navigation: Calculating distance and direction is essential in navigation, particularly in aviation and maritime industries.
Surveying: Surveyors use this concept to calculate distances and directions between landmarks and reference points.
Geographic Information Systems (GIS): GIS software uses this concept to calculate distances and directions between geographic locations.