In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x² + y² = 9

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Recall the relationship 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 \(x^{2} + y^{2} = 9\) to rewrite it in terms of \(r\) and \(\theta\).

After substitution, the equation becomes \((r \cos{\theta})^{2} + (r \sin{\theta})^{2} = 9\).

Simplify the left side using the Pythagorean identity \(\cos^{2}{\theta} + \sin^{2}{\theta} = 1\), resulting in \(r^{2} (\cos^{2}{\theta} + \sin^{2}{\theta}) = r^{2} = 9\).

Solve for \(r\) by taking the square root of both sides, giving \(r = \pm 3\). Since \(r\) represents a distance, consider the positive value \(r = 3\) as the polar equation.

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Rectangular and Polar Coordinate Systems

Rectangular coordinates represent points using (x, y) on a plane, 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 relate is essential for converting equations between them.

Conversion Formulas Between Coordinates

The key formulas for conversion are x = r cos(θ) and y = r sin(θ). Additionally, r² = x² + y². These relationships allow us to rewrite rectangular equations in terms of r and θ, facilitating the conversion process.

Expressing r in Terms of θ

After substituting x and y with their polar equivalents, the goal is to isolate r as a function of θ. This often involves algebraic manipulation and using trigonometric identities to rewrite the equation in the form r = f(θ).