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 $0 \leq m^{\circ}(\theta)<360$.
$(0,-4)$

Key Concepts

Negative Radius Representation

Polar coordinates allow for multiple representations of a point, including the use of a negative radius. When r is negative, the point is located in the opposite direction indicated by ?, effectively corresponding to the point (|r|, ? + 180°). This flexibility means that a point can be represented in different ways while still describing the same location.

Angle Normalization

In polar coordinate problems, angles are often restricted to a standard interval, such as 0° ? ? < 360° (or 0 ? ? < 2? radians), to ensure a unique representation of the point. This process, known as angle normalization, adjusts any computed angle to fall within the specified range, which is essential for consistency and clarity in communication.

Polar Coordinates

Polar coordinates provide a way to represent points in the plane by a radius and an angle. Instead of using (x, y) coordinates, a point is expressed as (r, ?) where r is the distance from the origin and ? is the angle measured from the positive x?axis. This system is particularly useful in contexts involving circular or rotational symmetry.

Conversion from Cartesian to Polar Coordinates

To convert from Cartesian coordinates (x, y) to polar coordinates (r, ?), the radius is calculated using the distance formula r = ?(x² + y²) and the angle is determined using ? = arctan(y/x) with appropriate adjustments based on the quadrant. This conversion requires understanding both the geometric interpretation of the coordinates and the trigonometric relationships between them.

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