In Exercises 15-22, use an algebraic method to find the...

In Exercises $15-22,$ use an algebraic method to find the rectangular coordinates of the point with given polar coordinates. Approximate the exact solution values with a calculator when appropriate.
$$
(-2, \pi)
$$

Key Concepts

Negative Radius in Polar Coordinates

In polar coordinates, a negative radius indicates that the point is located in the direction opposite to the angle provided. Understanding this concept is crucial when performing conversions, as it may require additional consideration, such as adjusting the angle by ? radians, to ensure the point is correctly represented in the rectangular coordinate system.

Conversion between Polar and Rectangular Coordinates

Converting a point from polar to rectangular coordinates involves using the algebraic relationships x = r cos(?) and y = r sin(?), where r is the radial distance and ? is the angle. These formulas allow one to translate the point's position between the two coordinate systems, ensuring consistency in representation regardless of the system used.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, specify a point’s location by its horizontal (x) and vertical (y) distances from a fixed origin. This coordinate system is widely used in algebra and geometry because of its straightforward representation of linear relationships and ease in performing algebraic operations.

Polar Coordinates

Polar coordinates represent points in the plane using a distance from a reference point (the pole, typically the origin) and an angle from a reference direction (usually the positive x-axis). This system is particularly useful for situations involving rotation or circular motion and provides an alternative perspective to the standard rectangular coordinate system.

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