Converting Rectangular Coordinates to Polar Coordinates

Oct 13, 2025 by wbrum.org 19 views

Introduction

In mathematics, rectangular coordinates and polar coordinates are two different ways to represent the position of a point in a two-dimensional plane. Rectangular coordinates are given as (x, y), where x is the horizontal distance from the origin and y is the vertical distance from the origin. Polar coordinates, on the other hand, are given as (r, θ), where r is the distance from the origin to the point and θ is the angle between the positive x-axis and the line connecting the origin to the point. In this article, we will learn how to convert rectangular coordinates to polar coordinates.

What are Rectangular Coordinates?

Rectangular coordinates are a way to represent the position of a point in a two-dimensional plane using a pair of numbers (x, y). The x-coordinate represents the horizontal distance from the origin, and the y-coordinate represents the vertical distance from the origin. For example, the point (3, 4) has an x-coordinate of 3 and a y-coordinate of 4.

What are Polar Coordinates?

Polar coordinates are a way to represent the position of a point in a two-dimensional plane using a pair of numbers (r, θ). The r-coordinate represents the distance from the origin to the point, and the θ-coordinate represents the angle between the positive x-axis and the line connecting the origin to the point. For example, the point (3, 4) can be represented in polar coordinates as (5, 53.13°), where 5 is the distance from the origin to the point and 53.13° is the angle between the positive x-axis and the line connecting the origin to the point.

Converting Rectangular Coordinates to Polar Coordinates

To convert rectangular coordinates to polar coordinates, we need to use the following formulas:

r = √(x² + y²)
θ = arctan(y/x)

where r is the distance from the origin to the point, θ is the angle between the positive x-axis and the line connecting the origin to the point, x is the x-coordinate of the point, and y is the y-coordinate of the point.

Step-by-Step Procedure

Here is a step-by-step procedure to convert rectangular coordinates to polar coordinates:

Calculate the distance from the origin to the point: Use the formula r = √(x² + y²) to calculate the distance from the origin to the point.
Calculate the angle between the positive x-axis and the line connecting the origin to the point: Use the formula θ = arctan(y/x) to calculate the angle between the positive x-axis and the line connecting the origin to the point.
Round the answer to the nearest hundredth: Round the values of r and θ to the nearest hundredth.

Example

Let's convert the rectangular coordinates (-2, 5) to polar coordinates.

Calculate the distance from the origin to the point: r = √((-2)² + 5²) = √(4 + 25) = √29 ≈ 5.39
Calculate the angle between the positive x-axis and the line connecting the origin to the point: θ = arctan(5/(-2)) = arctan(-2.5) ≈ -68.20°
Round the answer to the nearest hundredth: r ≈ 5.39, θ ≈ -68.20°

Therefore, the polar coordinates of the point (-2, 5) are (5.39, -68.20°).

Conclusion

In this article, we learned how to convert rectangular coordinates to polar coordinates. We used the formulas r = √(x² + y²) and θ = arctan(y/x) to calculate the distance from the origin to the point and the angle between the positive x-axis and the line connecting the origin to the point. We also provided a step-by-step procedure to convert rectangular coordinates to polar coordinates. We hope this article has been helpful in understanding the concept of converting rectangular coordinates to polar coordinates.

Common Mistakes to Avoid

When converting rectangular coordinates to polar coordinates, there are several common mistakes to avoid:

Not using the correct formulas: Make sure to use the correct formulas r = √(x² + y²) and θ = arctan(y/x) to calculate the distance from the origin to the point and the angle between the positive x-axis and the line connecting the origin to the point.
Not rounding the answer to the nearest hundredth: Make sure to round the values of r and θ to the nearest hundredth.
Not considering the quadrant: Make sure to consider the quadrant in which the point lies when calculating the angle θ.

Real-World Applications

Converting rectangular coordinates to polar coordinates has several real-world applications:

Navigation: Polar coordinates are used in navigation to represent the position of a ship or an aircraft.
Engineering: Polar coordinates are used in engineering to represent the position of a point in a two-dimensional plane.
Computer Graphics: Polar coordinates are used in computer graphics to represent the position of a point in a two-dimensional plane.

Practice Problems

Here are some practice problems to help you understand the concept of converting rectangular coordinates to polar coordinates:

Convert the rectangular coordinates (3, 4) to polar coordinates.
Convert the rectangular coordinates (-2, 5) to polar coordinates.
Convert the rectangular coordinates (1, 1) to polar coordinates.

Answer Key

Here is the answer key to the practice problems:

(3, 4) = (5, 53.13°)
(-2, 5) = (5.39, -68.20°)
(1, 1) = (1.41, 45.00°)