**Introduction to Polar Coordinates**

Cartesian coordinates do not represent the only way that points can be located on a plane or in 3D space. In this chapter we’ll look at polar, latitude/longitude, cylindrical, and spherical schemes.

Two versions of the *polar coordinate plane* are shown in Figs. 10-1 and 10-2. 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 a distance (called the *radius* ) *r* from the origin. A coordinate point is thus denoted in the form of an ordered pair ( *θ, r* ).

**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. 10-1 and 10-2, the circles are not labeled in units. You can do that for yourself. 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.

**Fig. 10-1** . The polar coordinate plane. The angle *θ* is in degrees, and the radius *r* is in uniform increments.

**Fig. 10-2** . Another form of the polar coordinate plane. The angle *θ* is in radians, and the radius *r* is in uniform increments.

**Direction, Negative Radii, and Non-Standard Directions**

**The Direction**

Direction can be expressed in degrees or radians counterclockwise from a reference axis pointing to the right or “east.” In Fig. 10-1, the direction *θ* is in degrees. Figure 10-2 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. The physical angle on the graph is directly proportional to the value of *θ* .

**Fig. 10-1** . The polar coordinate plane. The angle *θ* is in degrees, and the radius *r* is in uniform increments.

**Fig. 10-2** . Another form of the polar coordinate plane. The angle *θ* is in radians, and the radius *r* is in uniform increments.

**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, “Go 10 kilometers east” instead of “Go minus 10 kilometers west.” Negative radii must be allowed in order to graph figures that represent functions whose ranges can attain negative values.

**Non-standard Directions**

It’s okay to have non-standard direction angles in polar coordinates. If the value of *θ* is 360° (2 *π* 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 must be allowed in order to graph figures that represent functions whose domains go outside the standard angle range.

**Polar Coordinate Practice Problem**

**PROBLEM 1**

Provide an example of a graphical object that can be represented as a function in polar coordinates, but not in Cartesian coordinates.

**SOLUTION 1**

Recall the definitions of the terms *relation* and *function* from Chapter 6. When we talk about a function *f* , we can say that *r = f* ( *θ* ). A simple function of *θ* in polar coordinates is a *constant function* such as this:

*f* ( *θ* ) = 3

Because *f* ( *θ* ) is just another way of denoting *r* , the radius, this function tells us that *r* = 3. This is a circle with a radius of 3 units.

In Cartesian coordinates, the equation of the circle with radius of 3 units is more complicated. It looks like this:

x ^{2} + *y* ^{2} = 9

(Note that 9 = 3 ^{2} , the square of the radius.) If we let *y* be the dependent variable and *x* be the independent variable, we can rearrange the equation of the circle to get:

*y* = ±(9 − *x* ^{2} ) ^{1/2}

If we say that *y* = *g* ( *x* ) where *g* is a function of *x* in this case, we are mistaken. There are values of *x* (the independent variable) that produce two values of *y* (the dependent variable). For example, when *x* = 0, *y* = ± 3. If we want to say that *g* is a relation, that’s fine, but we cannot call it a function.

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