Given three polar coordinate representations for the origin.

Verified step by step guidance

1. Understand that the origin in polar coordinates is represented by the point where the radius \(r = 0\). This means the distance from the origin to the point is zero, regardless of the angle \(\theta\).
2. Recall that in polar coordinates, a point is given by \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
3. Since the radius \(r = 0\) at the origin, the angle \(\theta\) can be any real number because the point is at the center and the direction does not affect its position.
4. Therefore, the three polar coordinate representations for the origin can be written as \((0, \theta_1)\), \((0, \theta_2)\), and \((0, \theta_3)\), where \(\theta_1\), \(\theta_2\), and \(\theta_3\) are any angles.
5. In summary, the key concept is that the origin in polar coordinates is always represented by \(r = 0\), and the angle \(\theta\) is arbitrary.

Key Concepts

Polar Coordinates System

Polar coordinates represent points in a plane using a radius and an angle relative to a fixed direction, typically the positive x-axis. Each point is described by (r, θ), where r is the distance from the origin and θ is the angle measured in radians or degrees.

Multiple Representations of the Same Point

In polar coordinates, a single point can have multiple representations because adding or subtracting full rotations (2π radians) to the angle or using negative radius values can yield equivalent positions. For example, the origin (r=0) is represented by any angle θ.

The Origin in Polar Coordinates

The origin in polar coordinates is unique because its radius r is zero, making the angle θ arbitrary. This means the origin can be represented by infinitely many coordinate pairs (0, θ), highlighting the flexibility and special nature of the origin in this system.