Polar to Cartesian Equations

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

r² = 4r sin θ

검증된 단계별 안내

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

Start with the given polar equation: \(r^2 = 4r \sin \theta\).

Divide both sides of the equation by \(r\) (noting that \(r \neq 0\)) to simplify: \(r = 4 \sin \theta\).

Substitute \(r = \sqrt{x^2 + y^2}\) and \(\sin \theta = \frac{y}{r}\) into the simplified equation: \(\sqrt{x^2 + y^2} = 4 \cdot \frac{y}{\sqrt{x^2 + y^2}}\).

Multiply both sides by \(\sqrt{x^2 + y^2}\) to eliminate the denominator, then simplify and rearrange the resulting equation to express it purely in terms of \(x\) and \(y\). This will give the Cartesian form of the curve.

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주요 개념

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Polar to Cartesian Coordinate Conversion

Polar coordinates (r, θ) relate to Cartesian coordinates (x, y) through the formulas x = r cos θ and y = r sin θ. Understanding these relationships allows one to rewrite polar equations in terms of x and y, facilitating analysis using Cartesian methods.

Algebraic Manipulation of Equations

After substituting polar expressions with Cartesian equivalents, algebraic skills are essential to simplify and rearrange the equation. This process helps in identifying the standard form of the curve and makes it easier to classify the graph.

Graph Identification from Equations

Recognizing the type of graph from its Cartesian equation involves knowledge of common curves like circles, lines, parabolas, and ellipses. Identifying these shapes helps in visualizing and describing the graph represented by the original polar equation.

Polar to Cartesian EquationsReplace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.r² = 4r sin θ