In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. (x − 2)² + y² = 4

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Recall the relationships between rectangular 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 \((x - 2)^2 + y^2 = 4\) to rewrite it in terms of \(r\) and \(\theta\).

Expand the expression: \((r \cos{\theta} - 2)^2 + (r \sin{\theta})^2 = 4\).

Simplify the equation by expanding the square and combining like terms: \((r^2 \cos^2{\theta} - 4r \cos{\theta} + 4) + r^2 \sin^2{\theta} = 4\).

Use the Pythagorean identity \(\cos^2{\theta} + \sin^2{\theta} = 1\) to combine \(r^2 \cos^2{\theta} + r^2 \sin^2{\theta}\) into \(r^2\), then isolate \(r\) to express it in terms of \(\theta\).

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Rectangular and Polar Coordinate Systems

Rectangular coordinates (x, y) represent points on a plane using horizontal and vertical distances, while polar coordinates (r, θ) represent points by their distance from the origin and the angle from the positive x-axis. Understanding how to switch between these systems is essential for converting equations.

Conversion Formulas Between Coordinates

The key formulas for conversion are x = r cos(θ) and y = r sin(θ). These allow substitution of rectangular variables with polar expressions, enabling the transformation of equations from rectangular to polar form.

Manipulating Equations to Express r in Terms of θ

After substituting x and y with their polar equivalents, algebraic manipulation is required to isolate r as a function of θ. This often involves expanding, simplifying, and using trigonometric identities to achieve the desired polar form.