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Rectangular coordinates, also known as Cartesian coordinates, are a system used to identify the position of a point in a plane. In this system, each point is specified with a pair of numerical values, often denoted as

• \( x \) - the horizontal coordinate or 'abscissa', which indicates the distance along the horizontal axis.
• \( y \) - the vertical coordinate or 'ordinate', indicating the distance along the vertical axis.

These axes meet at a point called the origin, which is (0,0). The rectangular coordinate system creates a grid where every location in the plane can be reached using a unique (\( x, y \)) combination.
This system is widely used in algebra and geometry because of its straightforwardness. For instance, a line equation like \( y=3 \) describes a simple horizontal line where every point has the same y-value of 3, regardless of the x-value. It is easy to visualize and allows for the mathematical analysis of geometric shapes and figures.

A polar equation describes a relationship between the distance from the origin and the angle from the positive x-axis in polar coordinates. This system represents points using:

• \( r \) - the radial coordinate, which denotes the distance of the point from the origin.
• \( \theta \) - the angular coordinate, representing the angle between the radial line and the positive x-axis.

Polar coordinates are particularly useful in scenarios where symmetry about a point is a key feature, such as with circles and spirals. In the exercise provided,converting the rectangular equation \( y = 3 \) to polar coordinates involves recognizing that y can be transformed using \( y = r \sin(\theta) \). This transformationresults in the polar equation \( r \sin(\theta) = 3 \), which still describes a horizontal line in the polar coordinate plane, analogous to the original rectangular coordinate system.

Converting between rectangular and polar coordinates involves using trigonometric relationships to express a point in terms of the other system. The primary formulas for conversion are:

• From rectangular to polar: \[ r = \sqrt{x^2 + y^2} \ \theta = \tan^{-1}\left(\frac{y}{x}\right)\]
• From polar to rectangular: \[x = r \cos(\theta) \y = r \sin(\theta)\]

These relationships allow points expressed in Cartesian form (\( x, y \)) to be transformed into polar form (\( r, \theta \)), and vice versa. This processis essential in fields like physics and engineering, where problems might be more naturally expressed in one coordinate system but easier to solve in another.
In our specific exercise, the conversion required involved substituting the expression for y, \( y = r\sin(\theta) \), which allowed the transformation from\( y = 3 \) to \( r\sin(\theta) = 3 \). Understanding these formulas enhances one's ability to switch between different approaches to describing spatial relations, making it a versatile tool for solving complex problems.