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Polar coordinates provide an alternative method to the more common Cartesian (or rectangular) coordinate system for describing positions on a plane. While rectangular coordinates are based on perpendicular axes (x and y), polar coordinates are based on an angle and a distance from a reference point, called the pole, typically represented as the origin in the Cartesian system.

A point in polar coordinates is represented by a pair \( (r, \Theta) \) where \( r \) is the radius or the distance from the pole (origin) and \( \Theta \) (often pronounced 'theta') is the angle formed with the positive x-axis. The angle is usually measured in radians or degrees.

To convert from rectangular to polar form, we use the relationships \( x = r \cos(\Theta) \) and \( y = r \sin(\Theta) \). This allows us to translate between the two systems. Understanding these relationships is key when needing to switch between coordinate systems to simplify the analysis of problems or to leverage different techniques of calculus and trigonometry.

In geometry and algebra, the term rectangular equation refers to the equation of a curve, line, or function which is expressed in Cartesian coordinates as x and y. For example, the equation \( 3x - y + 2 = 0 \) given in the problem is a linear rectangular equation.

When we look at such an equation, it's understood to represent a set of all the points \( (x, y) \) that satisfy the equation when plotted on a Cartesian plane. The process of converting a rectangular equation to a polar equation is a method used to transition from x and y coordinates to r and \Theta coordinates, making it easier or more insightful to solve certain problems, particularly those involving rotation or circles.

In mathematics, trigonometric substitution is a technique for simplifying integrals and equations. It involves replacing expressions containing square roots or certain other algebraic expressions with trigonometric functions, where the variable is an angle. In the context of converting rectangular to polar form, trigonometric substitution is about expressing x and y, the rectangular coordinates, in terms of r and \Theta, the polar coordinates.

The most commonly used substitutions are \( x = r \cos(\Theta) \) and \( y = r \sin(\Theta) \), as seen in the exercise. This change of variables leverages the fundamental trigonometric identities that describe the relationship between the sides of a right triangle and its angles. These substitutions are particularly useful in calculus for evaluating integrals that are complex in Cartesian coordinates but become simpler when expressed in polar coordinates.