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Rectangular coordinates are a fundamental aspect of our mathematical toolkit. They locate a point in a two-dimensional space using two values: the x-coordinate and the y-coordinate. These coordinates are usually denoted as (x, y). Think of the x-coordinate as the position along the horizontal axis, and the y-coordinate as the position along the vertical axis. Together, they allow us to describe points, lines, and other geometrical shapes on a plane.

For problem-solving, rectangular coordinates can be very convenient, especially when we are working with linear equations or simple geometrical configurations. They are straightforward and provide a clear visual reference. We generally start with equations in rectangular form before exploring other systems like polar coordinates. In this particular exercise, we are dealing with the equation \(x^2 + y^2 = x\), which is expressed in these rectangular coordinates.

The real utility of moving to other coordinate systems, like polar coordinates, often comes up when dealing with circular or radial problems.

Trigonometrical substitution is a powerful technique used to simplify the conversion process from rectangular coordinates to polar coordinates. When moving from the x-y plane of rectangular coordinates to the radial-angle plane of polar coordinates, we use trigonometric identities to make the translation smooth and understandable.

In polar coordinates, any point on a plane is described by the radius \(r\) and the angle \(θ\). To convert, we use the relationships:

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

For our exercise, we substituted \(x\) with \(r \cos(θ)\) and \(y\) with \(r \sin(θ)\) into the rectangular equation \(x^2 + y^2 = x\). This substitution helps us transition into a form amenable to polar representation.

These substitutions open us to the simplified forms in trigonometry, such as \(\cos^2(θ) + \sin^2(θ) = 1\), which we used here to simplify the expression into \(r^2 = r \cos(θ)\). This simplification is crucial for identifying the relationship in the polar coordinate system.

Converting equations between coordinate systems involves manipulating and simplifying expressions to fit the characteristics of the target system. In our context, the goal was to convert a rectangular equation into its polar form.

After performing the trigonometrical substitution, we arrived at \(r^2 = r \cos(θ)\). The next step involved simplifying the equation further by isolating \(r\). We did this by dividing both sides by \(r\), resulting in \(r = \cos(θ)\). Thus, what started as a simple circle equation in rectangular coordinates becomes a straightforward representation in polar form.

This form \(r = \cos(θ)\) characterizes the equation in polar terms. It is an elegant and often simpler representation when dealing with curves that are naturally circular or radial. This conversion process highlights the versatility of different coordinate systems and the importance of selecting the most appropriate one for the type of problem at hand.