64 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)). 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[latex]\,\left(x,y\right)\,[/latex]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[latex]\,\left(r,\theta \right)\,[/latex]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[latex]\,r\,[/latex]is the radius or length of the directed line segment from the pole. The angle[latex]\,\theta ,[/latex] measured in radians, indicates the direction of[latex]\,r.\,[/latex]We move counterclockwise from the polar axis by an angle of[latex]\,\theta ,[/latex]and measure a directed line segment the length of[latex]\,r\,[/latex]in the direction of[latex]\,\theta .\,[/latex]Even though we measure[latex]\,\theta \,[/latex]first and then[latex]\,r,[/latex] the polar point is written with the r-coordinate first. For example, to plot the point[latex]\,\left(2,\frac{\pi }{4}\right),[/latex]we would move[latex]\,\frac{\pi }{4}\,[/latex]units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in (Figure).
Plotting a Point on the Polar Grid
Plot the point[latex]\,\left(3,\frac{\pi }{2}\right)\,[/latex]on the polar grid.
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The angle[latex]\,\frac{\pi }{2}\,[/latex]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[latex]\,\frac{\pi }{2}\,[/latex]direction, as shown in (Figure).
Try It
Plot the point[latex]\,\left(2,\,\frac{\pi }{3}\right)\,[/latex]in the polar grid.
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Plotting a Point in the Polar Coordinate System with a Negative Component
Plot the point[latex]\,\left(-2,\,\frac{\pi }{6}\right)\,[/latex]on the polar grid.
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We know that[latex]\,\frac{\pi }{6}\,[/latex]is located in the first quadrant. However,[latex]\,r=-