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Polar coordinates are a two-dimensional coordinate system for specifying points in a plane. Unlike the familiar rectangular (Cartesian) coordinate system, polar coordinates use a distance from a reference point and an angle from a reference direction. The reference point is typically called the pole (much like the origin in rectangular coordinates), and the reference direction is usually the positive x-axis.

In polar coordinates, each point is represented by a pair \(r, \theta\), where:

• \(r\) is the radial distance from the pole.
• \(\theta\) is the angle measured from the reference direction.

When working with polar coordinates, it's crucial to remember that multiple sets of values can represent the same point. For example, increasing an angle by \(2\pi\) (full rotation) keeps the location unchanged.

This system is especially useful for dealing with scenarios where relationships are naturally circular or angular, as seen with cyclical patterns in trigonometry. Polar equations, like \(r = 3\), describe all the points at a fixed distance from the pole, forming a circle in the plane.

Rectangular coordinates, commonly known as Cartesian coordinates, use two perpendicular axes to specify points in a plane. These axes often referred to as the x-axis and y-axis, meet at a point called the origin. Points in this system are given by a pair \(x, y\), reflecting their horizontal and vertical distances from the origin.

Conversion between polar and rectangular coordinates is a common task. The conversion formulas are:

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

Once in rectangular form, equations often become easier to graph and interpret. For instance, converting the polar equation \(r = 3\) yields the circular equation \(x^2 + y^2 = 9\). This describes a circle with a center at the origin and a radius of 3 units, showing how versatile rectangular coordinates can be for visualizing geometric shapes.

An equation of a circle in rectangular coordinates takes a specific form. If a circle is centered at the origin, its equation is:

• \(x^2 + y^2 = r^2\)

Here, \(r\) represents the radius of the circle.

For instance, the problem at hand transforms the polar equation \(r = 3\) into \(x^2 + y^2 = 9\). This expression represents all the points that are exactly 3 units from the origin, forming a perfect circle.

Circles centered at points other than the origin have a modified form:

• \((x - h)^2 + (y - k)^2 = r^2\)

where \(h\) and \(k\) are the coordinates of the circle's center. This general formula helps include translations of the circle along the plane, providing further flexibility in describing circular shapes in the rectangular coordinate system.