Plotting points in the polar plane

 Goal: plot points in the polar plane

Plotting a point in the polar plane

The polar representation of a point is not unique. For example, the polar coordinates (2, π/3) and (2, 7π/3) both represent the point (1, ⎷3). Also the value of r can be negative. Aa a result the polar coordinate (-2. 2𝛑/3) represents also the point (1, ⎷3) in the cartesian plane as demonstrated here:

Every point in the plane has an infinite number of representations in polar coordinates. The polar representation of a point in the plane has a visual interpretation. First, we have r that represents the distance of the segment that joins the origin to the point. Then Ө is the angle made of that segment and the positive direction of the x-axis. Positive angles are measured in counterclockwise direction and negative angles are measured in clockwise direction. Here is a representation of the polar coordinate system.

The  positive x-axis in the cartesian plane is the polar axis, The origin of the coordinate system is the pole and corresponds to r = 0. The innermost circle of the figure above contains all the points with r = 1 meaning all the points of which the distance from the pole is 1. Then the figure progresses from r = 2 and so on. To plot a point in the coordinate system start from the angle. If the angle is positive, move in counterclockwise direction from the polar axis. If the angle is negative move in clockwise direction of the polar axis. If the value of r is positive consider the ray that is terminal to the angle. If r is negative, consider the opposite ray terminal to the angle.                                                                                       

 

Example                                                                                                                                                 

                                                                                                                                                  

Plot each of the following points in the polar plane                                                                                  

 

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Solution                                                                                                                                               

Practice