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 $0 \leq m^{\circ}(\theta)<360$.
$(-4,-4 \sqrt{3})$

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Key Concepts

Multiple Representations in Polar Coordinates

A single point in the plane can be represented by infinite pairs of polar coordinates. Specifically, a point can be described using a positive radius r with a corresponding angle, or a negative radius with an angle that is 180° shifted from the original. Understanding this provides flexibility in expressing the same point in different, yet equivalent, polar forms.

Normalization of Angular Measure

This concept ensures that the angle ? is chosen within a specified interval, typically 0° ? ? < 360°. Adjusting ? to fall within this range is crucial for standardizing polar coordinate representations.

Cartesian to Polar Conversion

This concept involves converting a point given by its Cartesian coordinates (x, y) to polar coordinates (r, ?) using the formulas r = ?(x² + y²) and tan(?) = y/x. It is essential to compute the radial distance and the corresponding angle that locates the point in the plane.

Determining the Reference Angle and Quadrant

After calculating the value of tan(?) from the absolute values of the coordinates to find a reference angle, it is important to determine the correct quadrant in which the point lies. This step ensures that the angle ?, after appropriate adjustment (such as adding 180° for points in the third quadrant), accurately represents the point's position.

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