In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. (x − 2)² + y² = 4

Key Concepts

Rectangular to Polar Coordinates

In trigonometry, rectangular coordinates (x, y) can be converted to polar coordinates (r, θ) using the relationships x = r cos(θ) and y = r sin(θ). This conversion is essential for expressing equations in polar form, where r represents the distance from the origin and θ the angle from the positive x-axis.

Equation of a Circle

The given equation (x − 2)² + y² = 4 represents a circle centered at (2, 0) with a radius of 2. Understanding the geometric properties of circles is crucial when converting to polar coordinates, as it helps in identifying the relationship between r and θ that describes the same shape in a different coordinate system.

Trigonometric Identities

Trigonometric identities, such as sin²(θ) + cos²(θ) = 1, are fundamental in manipulating and simplifying expressions when converting equations. These identities allow for the substitution of polar coordinates into the rectangular equation, facilitating the transformation into a polar equation that expresses r in terms of θ.