Polar coordinate system

Definition

A polar coordinate system represents points in a plane using a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the reference direction is usually the positive x-axis.

5 Must Know Facts For Your Next Test

1. A point in polar coordinates is written as $(r, \theta)$ where $r$ is the radius (distance from the pole) and $\theta$ is the angle.
2. The conversion formulas between Cartesian coordinates $(x, y)$ and polar coordinates $(r, \theta)$ are $x = r \cos(\theta)$ and $y = r \sin(\theta)$. Conversely, $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan(\frac{y}{x})$.
3. The graphs of equations in polar coordinates can produce unique shapes like spirals and roses that are not easily represented in Cartesian coordinates.
4. Polar coordinates can describe curves using parametric equations where both $r$ and $\theta$ are functions of a parameter (usually time).
5. To integrate over regions described by polar coordinates, one uses double integrals with the area element given by $dA = r \, dr \, d\theta$.

Review Questions

• What are the Cartesian coordinate equivalents of the polar coordinates $(5, \frac{\pi}{4})$?
• How do you convert from Cartesian to polar coordinates for a point $(3, 4)$?
• What is the area element in polar coordinates when setting up double integrals?

Related terms

Cartesian Coordinate System: A coordinate system that specifies each point uniquely by a pair of numerical coordinates representing horizontal (x) and vertical (y) distances.

Parametric Equations: Equations that express sets of quantities as explicit functions of one or more independent variables called parameters.

Double Integral: An integral used to compute volumes under surfaces or areas on planes, often used in converting between different coordinate systems like Cartesian and polar.