Find two pairs of polar coordinates, one pair with r>0 and...

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(2 \sqrt{3},-2)$

Key Concepts

Polar Coordinates

Polar coordinates represent points in the plane using a distance from a reference point and an angle from a reference direction. This system provides an alternative to the Cartesian (rectangular) coordinate system, and is particularly useful when dealing with problems that have circular or rotational symmetry.

Cartesian Coordinates

Cartesian coordinates specify points using a pair of numerical values representing horizontal and vertical distances from a fixed origin. This system is the most common for plotting points in a plane, and serves as a basis for converting to and from polar coordinates.

Conversion between Cartesian and Polar Coordinates

The process of converting between Cartesian and polar coordinates involves the relationships x = r cos(?) and y = r sin(?). Finding the corresponding polar coordinates for a given Cartesian point requires calculating the radial distance using r = ?(x² + y²) and determining the appropriate angle ? based on the arctan of the ratio y/x, while accounting for the point's quadrant.

Multiple Representations in Polar Coordinates

A key concept in polar coordinates is that a single point can be represented by infinitely many coordinate pairs. This is due to the periodicity of angles (? + 360°n represents the same direction) and the possibility of using a negative r value, which corresponds to a point diametrically opposite the direction of the angle.

Angle Normalization

Angle normalization ensures that the angle ? is chosen within a specified range, typically 0° ? ? < 360°. This is important for providing a unique or standard form of the polar coordinate representation, especially when converting from Cartesian coordinates where multiple valid angles may exist.

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