key term - Pole

Definition

The pole is a special point in a polar coordinate system that serves as the origin, around which all other points are defined by their distance and angle. It is the fixed reference point from which the coordinates of any other point in the plane are measured.

5 Must Know Facts For Your Next Test

1. The pole is the origin or starting point of a polar coordinate system, from which all other points are measured.
2. The polar axis is the fixed reference line that extends from the pole and serves as the 0-degree angle in the polar coordinate system.
3. The polar radius, or radial coordinate, is the distance from the pole to a point in the polar coordinate system.
4. The polar coordinates of a point are represented as (r, θ), where r is the polar radius and θ is the angle from the polar axis.
5. Conic sections, such as circles, ellipses, parabolas, and hyperbolas, can be represented in polar coordinates by using the pole as the reference point.

Review Questions

• Explain the role of the pole in a polar coordinate system and how it differs from the origin in a Cartesian coordinate system.
  • In a polar coordinate system, the pole serves as the fixed reference point from which all other points are measured. It is the origin or starting point of the coordinate system, similar to the origin (0, 0) in a Cartesian coordinate system. However, the key difference is that in polar coordinates, points are defined by their distance (polar radius) and angle (polar angle) from the pole, rather than their x and y coordinates. This makes polar coordinates particularly useful for representing shapes and functions that have a natural radial or circular symmetry.
• Describe how the pole is used in the graphing of conic sections in polar coordinates.
  • When graphing conic sections, such as circles, ellipses, parabolas, and hyperbolas, in polar coordinates, the pole serves as the reference point. The equation of the conic section is expressed in terms of the polar radius (r) and the polar angle (θ), with the pole as the origin. This allows for the representation of conic sections that may not be easily expressed in Cartesian coordinates, particularly those with a radial or circular symmetry. The pole provides a natural and intuitive way to define the shape and position of the conic section within the polar coordinate system.
• Analyze how the concept of the pole in polar coordinates is essential for understanding and working with graphs of polar functions.
  • The pole is a fundamental concept in polar coordinates that is essential for understanding and working with graphs of polar functions. The pole serves as the fixed reference point from which the polar radius and polar angle are measured for any point on the graph. Without the pole, it would not be possible to accurately represent the shape and position of polar functions, which often exhibit radial or circular symmetry. The pole allows for the precise definition of the polar function's domain, range, and critical points, as well as the identification of important features such as cusps, loops, and spirals. Understanding the role of the pole is crucial for interpreting and sketching the graphs of polar functions, as well as for solving problems involving polar coordinates.

Related terms

Polar Coordinates:

A coordinate system that uses the distance from a fixed point (the pole) and the angle from a fixed direction (the polar axis) to locate a point in a plane.

Polar Axis:

The fixed reference line that extends from the pole and serves as the 0-degree angle in a polar coordinate system.

Polar Radius: The distance from the pole to a point in a polar coordinate system, also known as the radial coordinate.