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Unformatted text preview: UNIT 3 POLAR COORDINATES, GRAPHING INTRO: We will study polar coordinates, a system for labeling points in a plane in a manner different from Cartesian coordinates. Polar coordinates play an essential role in problems which have a sense of circular symmetry, or where the angles between lines play a central role. Coincidentally, the Cartesian coordinate system and the polar coordinate system were first formally developed at about the same time in the first half of the seventeenth century, the former by the French mathematician and philosopher Ren e Descartes (of Cogito ergo sum fame) and the latter independently by Belgian and Italian mathematicians. 1. Polar Coordinates (2 dimensions) Cartesian coordinates are the coordinate system with which we are most familiar, where points (on the plane, say) are located by specifying their positions related to the x and y axes, i.e., their x and y coordinates. However, Cartesian coordinates are not always the most convenient coordinate system, especially if the functions under study have symmetries observed by other coordinate systems but not the Cartesian system. For example, finding the area of a circle of radius 1 by integration is a lot easier by evaluating the polar integral 2 R 1 r dr than by evaluating the Cartesian integral R 1- 1 1- x 2 dx The coordinates of a point in the plane in polar coordinates are specified by giving the distance r from the origin and the angle in radians which a line segment from the origin to the point makes with the positive x-axis, measured in a counter-clockwise direction. Thus the coordinate r can take on any nonnegative real number as its value, but the coordinate should ordinarily take on a value between 0 and 2 . The phrase ordinarily has to be used, because considerable latitude is allowed with this coordinate, representing the fact that a rotation of 2 radians takes you back to the same location. Thus, multiples of 2 can be added or subtracted to the proper polar angle, and is frequently also called the polar angle of the point. The polar coordinates of a point are sometimes given in an application as the ordered pair ( r, ), although the discussion must make it clear that this is not the Cartesian coordinates of the point. An engineer or scientist must be proficient in converting back and forth between Cartesian coordinates and polar coordinates. The transformation between coordinates systems is given by the following formulas. x = r cos r = p x 2 + y 2 y = r sin tan = y x VECTOR GEOMETRY 31 GREENBERG-2-1 1 2-2-1 1 2 x y r LParen1 x,y RParen1 Cartesian coordinates (a,b) or polar coordinates (r, ) The last transformation has not been written as = arctan y x , even though this would have to be done to find , because there are complications. Review the section of Unit 1 on the arctan function, and you will recognize the problem. You must deduce what quadrant the point lies in before the polar angle can be deduced from the transformation for...
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