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. Step 1: Recall the relationship between rectangular coordinates \((x, y)\) and polar coordinates \((r, \theta)\). The conversions are given by \(x = r \cos \theta\) and \(y = r \sin \theta\).
2. Step 2: Substitute the polar coordinate expressions for \(x\) and \(y\) into the given rectangular equation. This gives \((r \cos \theta)^2 + (r \sin \theta + 3)^2 = 9\).
3. Step 3: Expand the equation. First, expand \((r \cos \theta)^2\) to get \(r^2 \cos^2 \theta\). Then, expand \((r \sin \theta + 3)^2\) to get \(r^2 \sin^2 \theta + 6r \sin \theta + 9\).
4. Step 4: Combine the terms. The equation becomes \(r^2 \cos^2 \theta + r^2 \sin^2 \theta + 6r \sin \theta + 9 = 9\).
5. Step 5: Use the Pythagorean identity \(\cos^2 \theta + \sin^2 \theta = 1\) to simplify the equation to \(r^2 + 6r \sin \theta = 0\). Solve for \(r\) in terms of \(\theta\).

Key Concepts

Rectangular to Polar Coordinates

In polar coordinates, points are represented by a radius (r) and an angle (θ) rather than x and y coordinates. The conversion from rectangular to polar coordinates involves using the relationships x = r cos(θ) and y = r sin(θ). Understanding these relationships is essential for transforming equations from one coordinate system to another.

Equation of a Circle

The given equation x² + (y + 3)² = 9 represents a circle in rectangular coordinates, centered at (0, -3) with a radius of 3. Recognizing the standard form of a circle's equation helps in identifying its geometric properties, which can be useful when converting to polar form.

Trigonometric Identities

Trigonometric identities, such as sin²(θ) + cos²(θ) = 1, are crucial when manipulating equations in polar coordinates. These identities allow for the simplification and transformation of expressions involving r, θ, and their relationships, facilitating the conversion of the original rectangular equation into a polar form.