Introduction to Polar Coordinates

Polar coordinates are a coordinate system used to represent points in a plane. Instead of using the x and y-axis like in Cartesian coordinates, polar coordinates use distance (r) from a reference point called the pole and an angle (θ) to determine the position of a point.

Polar Coordinates and Formulas

A point in polar coordinates is represented as (r, θ), where r is the distance from the pole to the point, and θ is the angle measured in radians from a reference line known as the polar axis.

Converting from polar coordinates to Cartesian coordinates can be done using the formulas:

• x = r * cos(θ)
• y = r * sin(θ)

Conversely, converting from Cartesian coordinates to polar coordinates involves using the following formulas:

• r = √(x² + y²)
• θ = arctan(y / x)

The Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin (0, 0). It is an essential tool for understanding polar coordinates and their relationship with Cartesian coordinates.

Angles in the unit circle are measured counterclockwise from the positive x-axis, called the polar axis. The point (1, 0) on the unit circle corresponds to an angle of 0 radians, and going in the counterclockwise direction, angles increase. Similarly, going clockwise from (1, 0), angles are measured as negative values.

Example: Converting between Polar and Cartesian Coordinates

Let's convert the point (3, π/4) in polar coordinates to Cartesian coordinates.

Using the conversion formulas:

• x = r * cos(θ) = 3 * cos(π/4) = 3 * (√2 / 2) = 3√2 / 2
• y = r * sin(θ) = 3 * sin(π/4) = 3 * (√2 / 2) = 3√2 / 2

So, the point (3, π/4) in polar coordinates is equivalent to the Cartesian coordinates (3√2 / 2, 3√2 / 2).

Example: Converting between Cartesian and Polar Coordinates

Let's convert the point (-2, -2√3) in Cartesian coordinates to polar coordinates.

Using the conversion formulas:

• r = √(x² + y²) = √((-2)² + (-2√3)²) = √(4 + 12) = √16 = 4
• θ = arctan(y / x) = arctan((-2√3) / (-2)) = arctan(√3) = π/3

Thus, the point (-2, -2√3) in Cartesian coordinates is equivalent to the polar coordinates (4, π/3).

Polar coordinates provide an alternative way to represent points in a plane, particularly useful in describing curves and angles. Understanding how to convert between Cartesian and polar coordinates, as well as the role of the unit circle, forms the foundation for further exploration of calculus concepts in polar coordinates.