In Problems 33-40, plot each point given in polar...

In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:
(a) $r>0, \quad-2 \pi \leq \theta<0$
(b) $r<0, \quad 0 \leq \theta<2 \pi$
(c) $r>0,2 \pi \leq \theta<4 \pi$
$\left(5, \frac{2 \pi}{3}\right)$

Key Concepts

Plotting Points in the Polar Coordinate System

Plotting involves locating the point at a distance r from the origin at an angle ? from the positive x?axis. This concept is an essential skill in transitioning between algebraic parameters and geometric visualization, ensuring a better grasp of how points are situated on the plane in polar coordinates.

Negative Radius

Using a negative radius involves moving in the direction opposite to the angle given, and it can be converted to an equivalent expression with a positive radius by adding ? to the angle. Understanding this property allows one to express points with negative radii in a standardized form with a positive radius by adjusting the angle accordingly.

Multiple Representations in Polar Coordinates

A single point in the plane can have infinitely many polar coordinate representations because angles can be adjusted by adding or subtracting multiples of 2? and because a negative radius can be used with an appropriate adjustment of the angle. This concept is fundamental to understanding that different (r, ?) pairs can denote the same point.

Polar Coordinates

Polar coordinates represent points in the plane using a distance (radius) from the origin and an angle measured from the positive x?axis. This system allows for describing points based on their direction and magnitude, which is particularly useful in situations involving circular or radial symmetry.

Angle Periodicity

Since the trigonometric functions are periodic with a period of 2?, adding (or subtracting) 2? (or any multiple thereof) to the angle ? does not change the position of the point. This periodicity is key to finding alternative valid representations for a given point in polar coordinates.

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