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When converting polar coordinates into rectangular coordinates, we use a set of straightforward formulas. Polar coordinates describe a point in terms of its distance from the origin, denoted as \( r \), and an angle \( \theta \) from the positive x-axis. This can be converted into rectangular coordinates \((x, y)\) by using:

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

These equations give us the rectangular coordinates (x, y) based on the given polar coordinates (r, θ).

For the exercise, we have \( r = 2 \). Thus, \( x = 2 \cos \theta \) and \( y = 2 \sin \theta \). These transformations allow us to rewrite polar coordinates into a more familiar Cartesian plane format, making it easier to perform further analysis and graphing tasks. This step is crucial for understanding the geometry of the equation, as rectangular coordinates give a clear depiction of shapes like circles.

Graphing a polar equation involves understanding the pattern of how points are plotted based on radius and angle from the origin. The given equation in polar coordinates is \( r = 2 \). This tells us that regardless of the angle \( \theta \), the radius \( r \) remains constant at 2.

Such equations form circles centered at the origin. Here's how to visualize it:

• Imagine the point (r, θ) moving, with r staying at 2 while θ varies around the circle.
• The value of r being constant indicates a fixed distance from the origin in all directions.
• As θ increases, the point traces out a circle.

On a polar graph, this appears clearly as a circle with radius 2 centered at the origin. When graphed, you can see a perfect circle, since all points satisfy the equation \( r = 2 \). This consistent radius is what makes polar graphing quite intuitive for circular shapes.

In the rectangular coordinate system, a circle is defined by its center and radius. The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle, and \(r\) is the radius.

For the exercise at hand, we start with the polar equation \( r = 2 \). When converting to a rectangular form, we use the identity \( r^2 = x^2 + y^2 \). Substituting \( r = 2 \) gives us \( 2^2 = x^2 + y^2 \). Simplifying, we find \( x^2 + y^2 = 4 \).

This is the equation of a circle centered at the origin (0,0) with a radius of 2. The conversion from the polar equation to this rectangular form is helpful when using Cartesian systems to analyze and graph circles. The equation \(x^2 + y^2 = 4\) succinctly tells us that every point on the circle is exactly 2 units from the center, thus defining a perfect circle on a flat plane.