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 $0 \leq m^{\circ}(\theta)<360$.
$(2,-2)$

Key Concepts

Angle Normalization

Angle normalization refers to adjusting the angle ? so that it falls within a specified interval, commonly [0, 360°) or [0, 2?) radians. This ensures a standard representation of the angle regardless of how many full rotations are involved.

Negative Radius in Polar Coordinates

Representing a point with a negative radius means that the direction indicated by the angle ? actually points in the opposite direction from the origin. To reconcile this with the standard polar coordinate system, one can add or subtract 180° (or ? radians) to the angle, providing an alternative but equivalent representation of the same point.

Conversion Between Coordinate Systems

Conversion between Cartesian and polar coordinates involves the relationships x = r cos ? and y = r sin ?, along with the computation of r = ?(x² + y²) to measure the distance from the origin, and determining ? using inverse trigonometric functions while taking into account the proper quadrant.

Cartesian Coordinates

Cartesian coordinates represent points in a plane using an ordered pair (x, y), where x corresponds to the horizontal displacement and y corresponds to the vertical displacement from the origin. This system is fundamental in describing positions in a two-dimensional space.

Polar Coordinates

Polar coordinates express the location of a point using a distance (r) from the origin and an angle (?) measured from the positive x-axis. This system is particularly useful when dealing with problems involving circular motion or symmetry about a point.

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