Polar to Cartesian Equations

Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.

r = 3 cos θ

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Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\), and also \(r^2 = x^2 + y^2\).

Start with the given polar equation: \(r = 3 \cos \theta\).

Multiply both sides of the equation by \(r\) to eliminate the \(\cos \theta\) term: \(r \cdot r = 3r \cos \theta\), which gives \(r^2 = 3r \cos \theta\).

Substitute the Cartesian equivalents: replace \(r^2\) with \(x^2 + y^2\) and \(r \cos \theta\) with \(x\), resulting in the equation \(x^2 + y^2 = 3x\).

Rewrite the equation in standard Cartesian form by moving all terms to one side: \(x^2 - 3x + y^2 = 0\). Then, complete the square for the \(x\) terms to identify the graph type.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Polar and Cartesian Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use (x, y) positions on a plane. Understanding how these systems relate is essential for converting equations between them.

Conversion Formulas Between Polar and Cartesian Coordinates

The key formulas are x = r cos θ and y = r sin θ, with r = √(x² + y²) and θ = arctan(y/x). These allow rewriting polar equations in terms of x and y, enabling analysis in the Cartesian plane.

Graphing and Identifying Curves from Equations

After converting to Cartesian form, recognizing the type of curve (circle, line, etc.) involves analyzing the equation's structure. For example, r = 3 cos θ corresponds to a circle in Cartesian coordinates.

Polar to Cartesian EquationsReplace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.r = 3 cos θ