R = √(x² + y²)

Definition

The equation r = √(x² + y²) defines the radial coordinate 'r' in a polar coordinate system, where 'r' represents the distance from the origin to a point in a two-dimensional plane. This relationship connects Cartesian coordinates (x, y) to polar coordinates, allowing for seamless transformation between the two systems. Understanding this equation is crucial for interpreting how points are represented in polar coordinates and enables the analysis of geometric properties and calculations in different contexts.

5 Must Know Facts For Your Next Test

1. In polar coordinates, every point can be described using the pair (r, θ), where r is the distance from the origin and θ is the angle.
2. The equation r = √(x² + y²) shows that 'r' can always be derived from the Cartesian coordinates (x, y), highlighting their interconnectedness.
3. The value of 'r' is always non-negative since it represents a distance, while angles can vary from 0 to 2π radians.
4. When converting from polar to Cartesian coordinates, x can be expressed as x = r cos(θ) and y as y = r sin(θ).
5. Graphing in polar coordinates often results in curves and shapes that can look quite different compared to their Cartesian representations.

Review Questions

• How does the equation r = √(x² + y²) facilitate the conversion between polar and Cartesian coordinate systems?
  • The equation r = √(x² + y²) allows for the calculation of the radial distance 'r' from the origin based on Cartesian coordinates (x, y). By using this equation, one can easily transform points between these two systems. For instance, knowing 'r' and the angle θ enables you to find x and y using x = r cos(θ) and y = r sin(θ), effectively bridging the gap between polar and Cartesian representations.
• Discuss how understanding r = √(x² + y²) impacts the way we visualize geometric shapes in both coordinate systems.
  • Understanding r = √(x² + y²) enhances our ability to visualize geometric shapes by allowing us to see how points are defined in terms of their distance from the origin. In polar coordinates, many shapes such as circles appear as simple equations centered at the origin. This contrasts with Cartesian coordinates where circles become more complex equations. This knowledge helps in graphing and interpreting functions differently depending on which coordinate system is being used.
• Evaluate the importance of transforming between polar and Cartesian coordinates for solving complex mathematical problems.
  • Transforming between polar and Cartesian coordinates is essential for solving various complex mathematical problems, especially those involving integrals or differential equations. The simplicity of certain equations in polar form can make calculations easier, especially when dealing with circular or periodic functions. Furthermore, many real-world applications such as physics or engineering often require this flexibility in coordinate systems to effectively model and solve problems involving angles and distances.

Related terms

Polar Coordinates: A system where each point on a plane is represented by a distance from a reference point and an angle from a reference direction.

Cartesian Coordinates:

A coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates (x, y).

Angle θ: In polar coordinates, the angle θ denotes the direction of the point from the origin, measured from the positive x-axis.