Convert x² + (y + 8)² = 64 to a polar equation that expresses r in terms of θ.

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

Substitute \(x = r \cos\theta\) and \(y = r \sin\theta\) into the given equation \(x^2 + (y + 8)^2 = 64\) to rewrite it in terms of \(r\) and \(\theta\).

Expand the expression: \(x^2\) becomes \((r \cos\theta)^2 = r^2 \cos^2\theta\), and \((y + 8)^2\) becomes \((r \sin\theta + 8)^2\).

Write the equation as \(r^2 \cos^2\theta + (r \sin\theta + 8)^2 = 64\) and expand the squared term to get \(r^2 \sin^2\theta + 16r \sin\theta + 64\).

Combine like terms and simplify the equation to isolate \(r\) in terms of \(\theta\), then solve the resulting quadratic equation for \(r\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates

Polar coordinates represent points in the plane using a radius r and an angle θ from the positive x-axis. The relationship between Cartesian coordinates (x, y) and polar coordinates is given by x = r cos θ and y = r sin θ, allowing conversion between the two systems.

Equation of a Circle in Cartesian Form

A circle's equation in Cartesian coordinates is typically expressed as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Understanding this form helps identify the circle's center and radius before converting to polar form.

Substitution and Simplification in Polar Form

To convert Cartesian equations to polar form, substitute x = r cos θ and y = r sin θ into the equation. Then, simplify the resulting expression to isolate r as a function of θ, which often involves algebraic manipulation and applying trigonometric identities.

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