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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 [link] ). 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.

An illustration of a boat on the polar grid.

Plotting points using polar coordinates

When we think about plotting points in the plane, we usually think of rectangular coordinates ( x , y ) 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 to polar coordinates    , which are points labeled ( r , θ ) 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 r is the radius or length of the directed line segment from the pole. The angle θ , measured in radians, indicates the direction of r . We move counterclockwise from the polar axis by an angle of θ , and measure a directed line segment the length of r in the direction of θ . Even though we measure θ first and then r , the polar point is written with the r -coordinate first. For example, to plot the point ( 2 , π 4 ) , we would move π 4 units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in [link] .

Polar grid with point (2, pi/4) plotted.

Plotting a point on the polar grid

Plot the point ( 3 , π 2 ) on the polar grid.

The angle π 2 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 π 2 direction, as shown in [link] .

Polar grid with point (3, pi/2) plotted.
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Plot the point ( 2 , π 3 ) in the polar grid .

Polar grid with point (2, pi/3) plotted.

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Plotting a point in the polar coordinate system with a negative component

Plot the point ( 2 , π 6 ) on the polar grid.

We know that π 6 is located in the first quadrant. However, r = −2. We can approach plotting a point with a negative r in two ways:

  1. Plot the point ( 2 , π 6 ) by moving π 6 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;
  2. Move π 6 in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.

See [link] (a). Compare this to the graph of the polar coordinate ( 2 , π 6 ) shown in [link] (b).

Two polar grids. Points (2, pi/6) and (-2, pi/6) are plotted. They are reflections across the origin in Q1 and Q3.
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Plot the points ( 3 , π 6 ) and ( 2 , 9 π 4 ) on the same polar grid.

Points (2, 9pi/4) and (3, -pi/6) are plotted in the polar grid.
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Converting from polar coordinates to rectangular coordinates

When given a set of polar coordinates    , we may need to convert them to rectangular coordinates . To do so, we can recall the relationships that exist among the variables x , y , r , and θ .

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