Cartesian to Polar Equations

Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.

x - y = 3

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

Substitute \(x\) and \(y\) in the given Cartesian equation \(x - y = 3\) with their polar equivalents: \(r \cos{\theta} - r \sin{\theta} = 3\).

Factor out \(r\) from the left side to get \(r (\cos{\theta} - \sin{\theta}) = 3\).

Solve for \(r\) by dividing both sides by \((\cos{\theta} - \sin{\theta})\), yielding \(r = \frac{3}{\cos{\theta} - \sin{\theta}}\).

This expression represents the equivalent polar equation for the given Cartesian equation.

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주요 개념

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Cartesian and Polar Coordinate Systems

Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding both systems is essential to convert equations between them.

Conversion Formulas Between Cartesian and Polar Coordinates

The key formulas for conversion are x = r cos(θ) and y = r sin(θ). These allow substitution of Cartesian variables with polar expressions, enabling the rewriting of Cartesian equations in terms of r and θ.

Algebraic Manipulation for Equation Conversion

After substituting x and y with their polar equivalents, algebraic manipulation is required to simplify and express the equation purely in terms of r and θ. This may involve factoring, isolating r, or using trigonometric identities.

Cartesian to Polar EquationsReplace the Cartesian equations in Exercises 53–66 with equivalent polar equations.x - y = 3