To determine the graph of a polar equation. Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of r ) r ) By performing three tests, we will see how to apply the properties of symmetry to polar equations. If an equation has a graph that is symmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on one side would coincide with the portion on the other side. Symmetry is a property that helps us recognize and plot the graph of any equation. All points that satisfy the polar equation are on the graph. Recall that the coordinate pair ( r, θ ) ( r, θ ) indicates that we move counterclockwise from the polar axis (positive x-axis) by an angle of θ, θ, and extend a ray from the pole (origin) r r units in the direction of θ. Just as a rectangular equation such as y = x 2 y = x 2 describes the relationship between x x and y y on a Cartesian grid, a polar equation describes a relationship between r r and θ θ on a polar grid. (credit: modification of work by NASA/JPL-Caltech) Testing Polar Equations for Symmetry Figure 1 Planets follow elliptical paths as they orbit around the Sun. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation. 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. Figure 1 Plotting Points Using Polar Coordinates
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