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When studying mathematics, you'll find that different coordinate systems can be used to describe the location of points in a plane. One such system is the polar coordinate system, which is particularly handy in scenarios involving circular or angular motion, or when dealing with periodic functions.

Polar coordinates are defined by a pair \( (r, \theta) \) where \( r \) is the distance from the origin to the point, known as the radius, and \( \theta \) is the angle measured in radians between the positive x-axis and the line segment connecting the origin to the point.

To convert from rectangular coordinates \( (x, y) \) to polar coordinates, we use the formulas \( x = r\cos{\theta} \) and \( y = r\sin{\theta} \). These allow us to move freely between the two systems and leverage the strengths of each where appropriate.

Trigonometric substitution is a powerful mathematical technique used when simplifying equations or integrals that contain radical expressions. It involves substituting expressions with trigonometric functions that correspond to a right-angled triangle's sides. The purpose is to simplify the algebraic manipulation and to utilize the known trigonometric identities.

In the context of converting rectangular equations to polar, we use trigonometric substitution to replace \( x \) and \( y \) with their polar equivalents, \( r\cos{\theta} \) and \( r\sin{\theta} \) respectively. This substitution is based directly on the Pythagorean identity \( r^2 = x^2 + y^2 \) and the definitions of the cosine and sine functions. This step simplifies the equation and helps us express it using \( r \) and \( \theta \) only, which is needed to complete the conversion to polar coordinates.

Equation transformation involves changing the form of an equation without altering its solutions or the set of points it represents. This can be done through various operations such as factoring, expanding, substituting terms, or changing the coordinate system as we have seen with polar coordinates.

In transforming the given rectangular equation to a polar one, \( x + 2y = 3 \), we first substitute \( x \) and \( y \) with the polar forms involving \( r \) and \( \theta \) through trigonometric substitution. Once the terms involving \( r \) are combined, we factor \( r \) out to simplify the expression. Finally, to isolate \( r \) and obtain the polar equation, we divide both sides by the remaining trigonometric expression. The result is a transformed equation in polar form, representing the same set of points in a different coordinate system.

Equation transformation is not limited to coordinate changes—it’s a fundamental tool for solving algebraic and calculus problems, where manipulating the form of an equation can reveal properties or solutions that are not readily apparent in its original form.