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


x² + y² = 16

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 position, while 'y' indicates the vertical position. 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 from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The coordinates are expressed as (r, θ), where 'r' is the radial distance and 'θ' is the angle. Converting rectangular equations to polar form involves expressing 'x' and 'y' in terms of 'r' and 'θ' using the relationships x = r cos(θ) and y = r sin(θ).

Conversion Formulas

The conversion from rectangular to polar coordinates relies on specific formulas that relate the two systems. The key formulas are x = r cos(θ) and y = r sin(θ), which allow for the substitution of 'x' and 'y' in a rectangular equation with their polar equivalents. Mastery of these formulas is crucial for successfully transforming equations like x² + y² = 16 into their polar counterparts.