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

Verified step by step guidance

1. Recognize that the equation x² + y² = 9 is a circle centered at the origin with radius 3.
2. Recall the polar coordinate relationships: x = r \cos(\theta) and y = r \sin(\theta).
3. Substitute x = r \cos(\theta) and y = r \sin(\theta) into the equation x² + y² = 9.
4. This substitution gives (r \cos(\theta))^2 + (r \sin(\theta))^2 = 9.
5. Factor out r² from the equation: r²(\cos^2(\theta) + \sin^2(\theta)) = 9.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points in a two-dimensional space using pairs of values (x, y). In this system, 'x' denotes the horizontal distance from the origin, while 'y' indicates the vertical distance. Understanding how to manipulate these coordinates is essential for converting equations from rectangular to polar form.

Polar Coordinates

Polar coordinates describe points in a plane using a distance 'r' from the origin and an angle 'θ' from the positive x-axis. The relationship between rectangular and polar coordinates is defined by the equations x = r cos(θ) and y = r sin(θ). This conversion is crucial for expressing equations in polar form, particularly when dealing with circular or radial symmetry.

Conversion Formulas

To convert a rectangular equation to polar form, specific formulas are used: r² = x² + y² and θ = arctan(y/x). These formulas allow for the substitution of x and y with their polar equivalents, facilitating the transformation of the equation. Mastery of these formulas is vital for accurately expressing relationships in polar coordinates.