91Ó°ÊÓ

Rectangular coordinates, also known as Cartesian coordinates, are a way to represent points in a plane using a pair of numbers. These numbers specify the position of a point relative to two perpendicular reference lines, known as the x-axis and the y-axis. Each coordinate pair (x, y) tells us:

• x-coordinate: This is the horizontal distance from the y-axis. It's also referred to as the abscissa.
• y-coordinate: This represents the vertical distance from the x-axis, often called the ordinate.

For instance, when you see the point (-6, 0), the -6 refers to six units to the left of the y-axis (since it’s negative), and 0 indicates that there's no movement up or down, placing the point directly on the x-axis. In this system, points are plotted within a grid that extends infinitely in all directions.

Calculating the angle in the context of coordinate conversion requires understanding the trigonometric function arctangent, denoted as arctan or tan^-1. This function helps determine the angle that a line forms with the positive direction of the x-axis.
When converting a point from rectangular coordinates to polar coordinates, you often use the formula:

• \( \theta = \text{arctan}(\frac{y}{x}) \)

In the current example with the point (-6, 0), you might wonder about using this formula since dividing by zero is undefined. However, you can circumvent this by considering the positional aspects:

• Since the point lies on the negative x-axis, the angle \( \theta \) is \( \pi \) radians, directly equivalent to 180 degrees. This is because the point forms a straight line along the negative x-axis, thus simplifying the angle calculation in such special cases.

To convert from rectangular to polar coordinates, you utilize two main formulas that help translate a point from one system to another.
These involve:

• Polar radius \( r \): \[ r = \sqrt{x^2 + y^2} \]This equation gives the distance from the origin (0, 0) to the point in question. It's derived from the Pythagorean theorem.
• Angle \( \theta \): \[ \theta = \text{arctan}\left(\frac{y}{x}\right) \]This calculation reveals the angle needed to reach the point when starting from the positive x-axis, measured counterclockwise.

For the point (-6, 0), applying these equations provides a clear path to obtaining polar coordinates. First, calculate \( r = \sqrt{(-6)^2 + 0^2} = 6 \), and then note that the position of the point lends \( \theta = \pi \) radians or 180 degrees.

The polar radius is a crucial concept in converting points from rectangular to polar coordinates. It essentially measures the straight-line distance from the origin of the coordinate system (the point (0, 0)) to the given point.
To find the polar radius, we use the formula:

• \( r = \sqrt{x^2 + y^2} \)

This formula arises from the Pythagorean Theorem, as it considers the point's horizontal and vertical distances (x and y coordinates) from the origin. In our example with the point (-6, 0), the calculation works like this:

• Substitute the values \((-6)^2 + 0^2\), you get \[ r = \sqrt{36} = 6 \].

Thus, the polar radius, represented as \( r = 6 \), tells us that the point is 6 units away from the origin. Understanding this concept is foundational for mastering the conversion process between coordinate systems.