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


x² + (y + 3)² = 9

Verified step by step guidance

1. Recall the relationships between rectangular coordinates (x, y) and polar coordinates (r, \(\theta\)): \(x = r \cos\theta\) and \(y = r \sin\theta\).
2. Substitute \(x = r \cos\theta\) and \(y = r \sin\theta\) into the given equation \(x^2 + (y + 3)^2 = 9\) to rewrite it in terms of \(r\) and \(\theta\).
3. Expand the expression: \(x^2\) becomes \((r \cos\theta)^2 = r^2 \cos^2\theta\), and \((y + 3)^2\) becomes \((r \sin\theta + 3)^2\).
4. Write the equation as \(r^2 \cos^2\theta + (r \sin\theta + 3)^2 = 9\) and expand the squared term: \((r \sin\theta + 3)^2 = r^2 \sin^2\theta + 6r \sin\theta + 9\).
5. Combine like terms and simplify the equation to isolate \(r\) in terms of \(\theta\). Use the Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\) to simplify \(r^2\) terms.

Key Concepts

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 Rectangular and Polar Coordinates

The key formulas are x = r cos θ and y = r sin θ, which allow substitution of rectangular variables with polar expressions. These formulas enable rewriting equations from rectangular form into polar form by expressing x and y in terms of r and θ.

Equation of a Circle in Polar Coordinates

A circle centered at (h, k) with radius a in rectangular form is (x - h)² + (y - k)² = a². When converting to polar, substitute x and y with r cos θ and r sin θ, then simplify to express r as a function of θ, which may involve algebraic manipulation to isolate r.