1-2 Plot the point whose polar coordinates are given. Then...

1-2 Plot the point whose polar coordinates are given. Then find
two other pairs of polar coordinates of this point, one with $r>0$
and one with $r<0 .$
$$(a)(2, \pi / 3) \quad \text { (b) }(1,-3 \pi / 4) \quad \text { (c) }(-1, \pi / 2)$$

Key Concepts

Polar Coordinates

Polar coordinates describe the location of a point in the plane using a radius and an angle. The radius, r, represents the distance of the point from the origin, while the angle, usually measured in radians, indicates the direction from the positive x-axis. This system provides an alternative to Cartesian (rectangular) coordinates and is particularly useful for problems involving circular or rotational symmetry.

Equivalence of Polar Representations

A single point in the plane can have infinitely many different polar coordinate representations. This is due to the periodicity of angles, where adding or subtracting multiples of 2? results in the same direction. Additionally, a negative radius can be used along with an adjusted angle (by adding ?) to represent the same point. Understanding these equivalences is crucial for converting between different representations of the same point, especially when specific conditions on the radius, such as r > 0 or r < 0, are required.

Conversion between Polar and Rectangular Coordinates

Transforming polar coordinates to rectangular (Cartesian) coordinates involves using the relationships x = r cos(?) and y = r sin(?). This conversion is essential when plotting points on the Cartesian plane or when comparing locations represented in different coordinate systems. It helps to visually verify that different polar coordinate representations indeed refer to the same point in the plane.

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