Polar Coordinate System

Introduction to Polar Coordinates

The polar coordinate system is an alternative to the rectangular (Cartesian) coordinate system for locating points in a plane. In this system, each point is determined by a distance from a fixed point (the pole, analogous to the origin) and an angle from a fixed direction (the polar axis, analogous to the positive x-axis).

• r: The distance from the pole (origin).

• θ: The angle measured from the polar axis (usually the positive x-axis), typically in radians or degrees.

To plot points in polar coordinates, locate the angle θ, then move r units from the pole along that direction.

Rectangular vs. Polar Coordinates

• Rectangular Coordinates (x, y): Points are located by horizontal (x) and vertical (y) distances from the origin.

• Polar Coordinates (r, θ): Points are located by a distance r from the pole and an angle θ from the polar axis.

Conversion between systems:

• From polar to rectangular:

• From rectangular to polar:

Plotting Points in Polar Coordinates

To plot a point (r, θ):

1. Measure the angle θ from the polar axis (counterclockwise is positive, clockwise is negative).

2. Move r units from the pole along the direction of θ.

3. If r is negative, move |r| units in the direction opposite to θ.

Example: Plot the point (5, ). Start at the pole, measure $\frac{\pi}{2}$ radians (90°) from the polar axis, and move 5 units outward.

Determining Different Coordinates for the Same Point

In polar coordinates, a single point can be represented by multiple pairs (r, θ) due to the periodic nature of angles and the possibility of negative r values.

• Co-terminal Angles: Adding or subtracting multiples of to θ locates the same direction.

• Negative r: The point (−r, θ) is equivalent to (r, θ + ).

General formula for equivalent coordinates:

• for any integer n

• for any integer n

Example: The point (3, ) can also be written as (3, ), (−3, ), etc.

Practice and Application

• Plotting points with various r and θ values, including negative r and angles outside [0, 2π].

• Finding all possible polar coordinates for a given point.

• Identifying which sets of coordinates represent the same point.

Summary Table: Equivalent Polar Coordinates

Additional info: Mastery of polar coordinates is essential for understanding advanced trigonometric graphs, conic sections, and applications in physics and engineering.