# 18 Polar Coordinates

### Learning Objectives

In this section, you will:

- Plot points using polar coordinates.
- Convert from polar coordinates to rectangular coordinates.
- Convert from rectangular coordinates to polar coordinates.
- Transform equations between polar and rectangular forms.
- Identify and graph polar equations by converting to rectangular equations.

Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see (Figure 1)). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.

### Plotting Points Using Polar Coordinates

When we think about plotting points in the plane, we usually think of rectangular coordinates in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce polar coordinates, which are points labeled and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.

The polar grid is scaled as the unit circle with the positive *x-*axis now viewed as the polar axis and the origin as the pole. The first coordinate is the radius or length of the directed line segment from the pole. The angle measured in radians, indicates the direction of We move counterclockwise from the polar axis by an angle ofand measure a distance r along a directed line segment in the direction of Even though we measure first and then the polar point is written with the *r*-coordinate first. For example, to plot the point we would move units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in (Figure 2).

### Plotting a Point on the Polar Grid

Plot the pointon the polar grid.

## Show Solution

The angle is found by sweeping in a counterclockwise direction 90° from the polar axis. The point is located at a length of 3 units from the pole in the direction, as shown in (Figure 3).

### Try It

Plot the point in the polar grid.

## Show Solution

### Plotting a Point in the Polar Coordinate System with a Negative Component

Plot the pointon the polar grid.

## Show Solution

We know that is located in the first quadrant. However, We can approach plotting a point with a negative in two ways:

- Plot the point by moving in the counterclockwise direction and extending a directed line segment 2 units into the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the third quadrant;
- Move in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.

See (Figure 4)(a). Compare this to the graph of the polar coordinate shown in (Figure 4)(b).