Polar coordinates

Definition

Polar coordinates represent a point in the plane using a distance from the origin and an angle from the positive x-axis. They are denoted as $(r, \theta)$ where $r$ is the radial distance and $\theta$ is the angular coordinate.

5 Must Know Facts For Your Next Test

1. The relationship between polar coordinates $(r, \theta)$ and Cartesian coordinates $(x, y)$ is given by $x = r \cos(\theta)$ and $y = r \sin(\theta)$.
2. Converting from Cartesian to polar coordinates involves $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}(\frac{y}{x})$.
3. $r$ can be negative, which means the point lies in the opposite direction of the angle $\theta$.
4. The angle $\theta$ is typically measured in radians but can also be expressed in degrees.
5. Polar graphs feature unique shapes such as circles, spirals, and roses that are not easily represented in Cartesian coordinates.

Review Questions

• How do you convert Cartesian coordinates $(x, y)$ to polar coordinates $(r, \theta)$?
• What is the significance of a negative value for $r$ in polar coordinates?
• Describe how to plot a point given its polar coordinates.

Related terms

Cartesian Coordinates: $(x, y)$ coordinate system where points are defined by their horizontal and vertical distances from two perpendicular axes.

$r$ (Radial Distance): Distance from the origin to a point in polar coordinates.

$\theta$ (Angular Coordinate): Angle measured counterclockwise from the positive x-axis to the line segment connecting the origin with a point.