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.
