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 $0 \leq m^{\circ}(\theta)<360$.
$(6,-2 \sqrt{3})$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Polar Coordinates

Polar coordinates represent points in the plane using a distance from the origin (r) and an angle (?) from the positive x-axis. This system is especially useful for problems involving circular symmetry and rotations, providing an alternative to Cartesian coordinates.

Conversion between Cartesian and Polar Coordinates

The conversion between Cartesian (x, y) and polar (r, ?) coordinates is done using the formulas r = ?(x² + y²) and ? = arctan(y/x), with modifications according to the quadrant where the point lies. This conversion is fundamental for expressing points in different coordinate systems.

Angle Measurement and Normalization

Angles in polar coordinates are typically normalized within a specific interval, such as 0° ? ? < 360° (or 0 to 2? in radians), to ensure consistency. Normalization involves adjusting calculated angles, which may initially be negative or exceed the interval, to fall within the desired range.

Representation with Negative Radius

In polar coordinates, a negative radius can be used to represent points by taking the positive radius in the opposite direction. This is achieved by adding 180° (or ? radians) to the angle, while ensuring the angle remains within the standard interval, thereby giving an equivalent point representation.

Transcript

Please give Ace some feedback