ma067 - Finding Points of Intersection of Polar-Coordinate...

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Unformatted text preview: Finding Points of Intersection of Polar-Coordinate Graphs Warren W. Esty Mathematics Teacher, September 1991, Volume 84, Number 6, pp. 472–78. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). More than 200 books, videos, software, posters, and research reports are available through NCTM’S publication program. Individual members receive a 20% reduction off the list price. For more information on membership in the NCTM, please call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (707) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright September 1991 by the National Council of Teachers of Mathematics. All rights reserved. S tudents studying polar coordinates may be required to find the points of intersection of the polar-coordinate graphs of two functions, f and g . Their experiences with rectangular coordinates may lead them to expect that all the points of intersection can be found by solving the equation They may be distressed to discover that points of intersection of the graphs can occur that do not correspond to solutions of that equation. Example 1. Consider the polar-coordinate graphs of and a circle, given in figure 1. Two points of intersection, B and C, obviously occur in the second and third quadrants. A natural approach to find these points is to set equal to and solve for For the given functions this approach yields cos s u d 5 0. 3 cos s u d 5 0, 1 2 3 cos s u d 5 1, u . g s u d f s u d g s u d 5 1, f s u d 5 1 2 3 cos s u d f s u d 5 g s u d . The solutions in the interval are given by and from which we obtain the points and in figure 1. The solutions of correspond to the two points of intersection on the y-axis, but none of the solutions corresponds to the points B and C in the second and third quadrants. Fig. 1. and Points B and C do not solve the equation The points of intersection B and C occur when the two functions yield opposite functional values at angles that differ by since the point is the same as the point in polar coordinates. Solving we have with solutions in given by and from which we obtain the points and What the graphics display shows so well is that the curves and pass through the points B and C at “different times,” that is, with different angles, and therefore these points of intersection will not be found by solving the equation that equates the two functions with the same angle, (Example 1 is discussed further after theorem 2.) If any two functions are graphed in rectangular coordinates, all the points of intersection can be found by setting the functions equal to each other and solving. (See theorem 1 that follows.) The purpose of this article is to help students understand why the method of solving for points of intersection in polar coordinates is not precisely parallel to the rectangular-coordinate method....
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ma067 - Finding Points of Intersection of Polar-Coordinate...

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