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 $0 \leq m^{\circ}(\theta)<360$.
$(-\sqrt{3}, 1)$

Key Concepts

Conversion between Cartesian and Polar Coordinates

In the conversion between Cartesian coordinates (x, y) and polar coordinates (r, ?), the key relationships are x = r cos ? and y = r sin ?. This conversion is fundamental for representing points in the plane in either system, depending on which best suits the problem's requirements.

Calculating the Radius and Angle from Cartesian Coordinates

The radius in polar coordinates is determined by the distance from the origin using the formula r = ?(x² + y²). The angle ? can be found using the arctan function (with adjustments for the correct quadrant based on the signs of x and y). This process yields the polar coordinates when r is taken to be positive.

Handling Negative Values of r in Polar Coordinates

In polar coordinates, a negative radius is interpreted by reversing the direction of the angle by 180°; that is, the point (?r, ?) is equivalent to the point (r, ? + 180°). This concept allows multiple representations of the same point when the conditions on r differ (positive or negative).

Ensuring the Angle Lies Within the Specified Range

When providing polar coordinates, it is often necessary to constrain the angle ? to a specific range, such as 0° ? ? < 360°. This requirement demands adjusting computed angles by adding or subtracting full rotations (360°) to ensure the angle falls within the desired interval, without altering the point's location.

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