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When converting from Cartesian coordinates to polar coordinates, it's important to understand the relationship between these two systems. In the Cartesian system, any point on the plane is defined by the coordinates \(x, y\). Meanwhile, in the polar system, points are defined by \(r\) and \(\theta\). Here, \(r\) is the radial distance from the origin, and \(\theta\) is the angle measured from the positive x-axis.
To convert a Cartesian equation to a polar equation, we use the formulas:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

These equations link \(x\) and \(y\) to \(r\) and \(\theta\). For instance, with the equation \(x = 3\), we substitute the formula for \(x\): \(r \cos \theta = 3\). This gives us the polar equation, showcasing the relationship in terms of polar coordinates.

A polar equation is an expression that represents a relationship between the polar coordinates of points on a plane. In our example, we started with the Cartesian equation \(x = 3\). After conversion, the result was \(r \cos \theta = 3\).
This polar equation reveals that for any angle \(\theta\), the distance \(r\) from the origin along a line with angle \(\theta\) is constant, provided \(\cos \theta\) yields the correct value to satisfy \(r \cos \theta = 3\). This shows a vertical line parallel to the y-axis at \(x = 3\) in Cartesian coordinates, now expressed in polar terms.

Coordinate systems, like Cartesian and polar, serve as methods to specify locations and describe geometric figures. Each system has its advantages and can aid in solving different types of problems:

• In the Cartesian system, we use \(x\) and \(y\) to locate points, making it ideal for problems involving lines and rectangular shapes.
• The polar coordinate system uses \(r\) and \(\theta\), making it useful for circular and rotational problems.

Understanding both systems allows for greater versatility. For example, while Cartesian coordinates are straightforward for defining rectangles, the polar system is simpler for dealing with circles and arcs.
This skill is beneficial when translating equations like \(x = 3\) into a polar context, where it simplifies to \(r \cos \theta = 3\). By interchanging between systems, we unlock different perspectives and solutions to geometric and algebraic problems.