Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. We can now convert coordinates between polar and rectangular form. Transforming Equations between Polar and Rectangular Forms The point ( 3 2, − 7 π 4 ) ( 3 2, − 7 π 4 ) is a move further clockwise by − 7 π 4, − 7 π 4, from π 4. However, the angle 5 π 4 5 π 4 is located in the third quadrant and, as r r is negative, we extend the directed line segment in the opposite direction, into the first quadrant. The point ( − 3 2, 5 π 4 ) ( − 3 2, 5 π 4 ) indicates a move further counterclockwise by π, π, which is directly opposite π 4. For example, the points ( − 3 2, 5 π 4 ) ( − 3 2, 5 π 4 ) and ( 3 2, − 7 π 4 ) ( 3 2, − 7 π 4 ) will coincide with the original solution of ( 3 2, π 4 ). There are other sets of polar coordinates that will be the same as our first solution. This point is plotted on the grid in Figure 2. For example, to plot the point ( 2, π 4 ), ( 2, π 4 ), we would move π 4 π 4 units in the counterclockwise direction and then a length of 2 from the pole. Even though we measure θ θ first and then r, r, the polar point is written with the r-coordinate first. We move counterclockwise from the polar axis by an angle of θ, θ, and measure a directed line segment the length of r r in the direction of θ. The angle θ, θ, measured in radians, indicates the direction of r. The first coordinate r r is the radius or length of the directed line segment from the pole. 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 polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane. In this section, we introduce to polar coordinates, which are points labeled ( r, θ ) ( r, θ ) and plotted on a polar grid. However, there are other ways of writing a coordinate pair and other types of grid systems. When we think about plotting points in the plane, we usually think of rectangular coordinates ( x, y ) ( x, y ) in the Cartesian coordinate plane. We account for this on the graph by sketching a picture of a graph suggested by the points plotted.Figure 1 Plotting Points Using Polar Coordinates Recall that when a function is defined by an equation, we have a lot of inputs for \(x\) to choose from. Draw the function by connecting the dots.Use the ordered pairs to plot the graph of the function.Create ordered pairs from the inputs and their outputs add to table. ![]()
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