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The first step in converting an equation to polar coordinates involves replacing the Cartesian coordinates \(x\) and \(y\) with their polar equivalents. In polar coordinates, any point in the plane can be described by a distance from the origin, \(r\), and an angle, \(θ\), from the positive x-axis.

• To convert \(x\), use the relation \(x = r \cdot \cos(θ)\).
• To convert \(y\), use the relation \(y = r \cdot \sin(θ)\).

This means, in this problem, every \(x\) in the equation \(x^2 + 4y^2 = 16\) becomes \(r \cdot \cos(θ)\) and every \(y\) becomes \(r \cdot \sin(θ)\).
This substitution transforms the entire equation into the polar coordinate form, setting the stage for further simplification. Converting to polar coordinates is a fundamental skill in mathematics that allows for the exploration of shapes and curves in a new way, making certain problems simpler to handle.

After converting to polar coordinates, understanding trigonometric identities becomes crucial. These identities can simplify complex expressions involving sine and cosine.

• Pythagorean Identity: \(\cos^2(θ) + \sin^2(θ) = 1\). Although not directly used to simplify this exercise, this identity is foundational in understanding the relationship between \(\sin\) and \(\cos\).

In this exercise, the trigonometric function terms were squared and needed to be handled carefully for simplification.
Applying identities helps to unify these terms, particularly when a common factor is present, as in \(r^2\cdot[\cos^2(θ) + 4 \cdot \sin^2(θ)]\). This understanding paves the way for the algebraic manipulation required in the following steps.

Algebraic manipulation is the process of using algebraic techniques to simplify or rearrange expressions and equations. This important final step is essential to craft an understandable polar form of a Cartesian equation.

• First, notice the common factor \(r^2\) in \(r^2 \cdot \cos^2(θ) + 4 \cdot r^2 \cdot \sin^2(θ)\). Factor it out to simplify: \(r^2 \cdot (\cos^2(θ) + 4 \cdot \sin^2(θ)) = 16\).
• To solve for \(r^2\), divide both sides by the trigonometric expression \(\cos^2(θ) + 4 \cdot \sin^2(θ)\).

By doing so, you isolate \(r^2\), giving you the final polar form: \[ r^2 = \frac{16}{\cos^2(θ) + 4 \cdot \sin^2(θ)}. \]This result reveals the relationship between \(r\), \(θ\), and the original Cartesian coordinates. Understanding these algebraic steps is incredibly valuable for identifying the geometric implications of an equation.