The Polar Coordinate Plane Help

Introduction to the Polar Coordinate Plane

The Cartesian scheme is not the only way that points can be located on a flat surface. Instead of moving right-left and up-down from an origin point, we can travel outward a certain distance, and in a certain direction, from that point. The outward distance is called the radius or range. It is measured in linear units, either arbitrary or specific (such as meters or kilometers). The direction is measured in angular units (either radians or degrees). It is sometimes called the azimuth, bearing, or heading.

The polar coordinate plane, as used by mathematicians and also by some engineers, is shown in Figs. 5-1 and 5-2. The independent variable is plotted as an angle θ relative to a reference axis pointing to the right (or “east”), and the dependent variable is plotted as the distance or radius r from the origin. Coordinate points are thus denoted as ordered pairs ( θ,r).

Fig. 5-1. The polar coordinate plane. The angle θ is in degrees, and the radius r is in arbitrary units.

Fig. 5-2. Another form of the polar coordinate plane. The angle θ is in radians, and the radius r is in arbitrary units.

Radius and Direction

The Radius

In any polar plane, the radii are shown by concentric circles. The larger the circle, the greater the value of r. In Figs. 5-1 and 5-2, the circles are not labeled in units. Imagine each concentric circle, working outward, as increasing by any number of units you want. For example, each radial division might represent one unit, or five units, or 10, or 100.

The Direction

Direction can be expressed in degrees or radians counterclockwise from a reference axis pointing to the right or “east.” In Fig. 5-1, the direction θ is in degrees. Figure 5-2 shows the same polar plane, using radians to express the direction. (The “rad” abbreviation is not used, because it is obvious from the fact that the angles are multiples of π.) Regardless of whether degrees or radians are used, the angular scale is linear. That is, the physical angle on the graph is directly proportional to the value of θ .

Negative Radii

In polar coordinates, it is all right to have a negative radius. If some point is specified with r < 0, we multiply r by –1 so it becomes positive, and then add or subtract 180° (π rad) to or from the direction. That’s like saying, “Proceed 10 kilometers east” instead of “Proceed negative 10 kilometers west.” Negative radii are allowed in order to graph figures that represent functions whose ranges can attain negative values.

Non-standard Directions

It’s all right to have non-standard direction angles in polar coordinates. If the value of θ is 360° (2π rad) or more, it represents more than one complete counterclockwise revolution from the 0° (0 rad) reference axis. If the direction angle is less than 0 ° (0 rad), it represents clockwise revolution instead of counterclockwise revolution. Non-standard direction angles are allowed in order to graph figures that represent functions whose domains go outside the standard angle range.

Practice problems for these concepts can be found at:  Polar Coordinates Practice Test