In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x = 7

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points in a two-dimensional space using an ordered pair (x, y). In this system, 'x' denotes the horizontal distance from the origin, while 'y' indicates the vertical distance. Understanding how these coordinates relate to polar coordinates is essential for converting equations between the two systems.

Polar Coordinates

Polar coordinates describe a point in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). A point is represented as (r, θ), where 'r' is the radial distance and 'θ' is the angle. Converting from rectangular to polar coordinates involves using the relationships r = √(x² + y²) and θ = arctan(y/x).

Conversion Formulas

To convert rectangular equations to polar form, specific formulas are used. The key relationships are x = r cos(θ) and y = r sin(θ). For the given equation x = 7, substituting the polar equivalent gives r cos(θ) = 7, which can be rearranged to express r in terms of θ, facilitating the conversion process.