Cartesian to Polar Equations

Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.
(x + 2)² + (y − 5)² = 16"

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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 \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given equation \((x + 2)^2 + (y - 5)^2 = 16\) to rewrite it in terms of \(r\) and \(\theta\).

Expand the terms inside the parentheses: \((r \cos{\theta} + 2)^2 + (r \sin{\theta} - 5)^2 = 16\).

Use algebraic expansion to write the equation as \(r^2 \cos^2{\theta} + 4r \cos{\theta} + 4 + r^2 \sin^2{\theta} - 10r \sin{\theta} + 25 = 16\).

Combine like terms and use the Pythagorean identity \(\cos^2{\theta} + \sin^2{\theta} = 1\) to simplify the equation to a form involving \(r\) and \(\theta\) only.

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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 how these systems describe points differently is essential for converting 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 x and y in Cartesian equations with expressions in terms of r and θ, enabling the rewriting of Cartesian equations into polar form.

Equation of a Circle in Cartesian and Polar Forms

A circle centered at (h, k) with radius r satisfies (x - h)² + (y - k)² = r² in Cartesian form. To express this in polar coordinates, substitute x and y with r cos(θ) and r sin(θ), then simplify to find an equation involving r and θ.

Cartesian to Polar EquationsReplace the Cartesian equations in Exercises 53–66 with equivalent polar equations.(x + 2)² + (y − 5)² = 16"