MTH 253 Calculus (Other Topics)Cartesian (aka: Rectangular) CoordinatesPolar CoordinatesSlide 4Polar Graph Paper Locating and Graphing PointsConverting Coordinates Polar  CartesianExamples: Converting Coordinates Polar  CartesianExamples: Converting Coordinates Polar  CartesianSlide 9Slide 10Slide 11Converting Equations Polar  RectangularConverting Equations Polar  CartesianConverting Equations Polar  CartesianGraphing Simple Polar Equations and InequalitiesSlide 16MTH 253Calculus (Other Topics)Chapter 10 – Conic Sections and Polar CoordinatesSection 10.5 – Polar CoordinatesCopyright © 2009 by Ron Wallace, all rights reserved.Cartesian (aka: Rectangular) Coordinatespositive x-axisnegative x-axispositive y-axisnegative y-axisxy(x, y)originFor any point there is a unique ordered pair (x, y) that specifies the location of that point.Polar Coordinatespolar axis(r, )rpoleIs (r, ) unique for every point?NO!All of the following refer to the same point:(5, 120º)(5, 480º)(-5, 300º)(-5, -60º)etc ...The angle  may be expressed in degrees or radians.Polar Coordinatespolar axis(r, )rpoleFind ALL order pairs for a given point.The angle  may be expressed in degrees or radians.( ), 2r kq p�( ), (2 1)r kq p- � +Assume that r > 0and 0 ≤  < 2( ), 360r kq �o( ), (2 1)180r kq- � +oAssume that r > 0and 0 ≤  < 360° NOTE: k is a positive integer.Polar Graph PaperLocating and Graphing Points0306090180120150210240270300330(5, 150)(6, 75)(3, 300)(3, -60)(-3, 120)(-4, 30)(7, 0)(-7, 180)Converting CoordinatesPolar  Cartesian2 2 2r x y= +xy(r, )  (x, y)rRecommendation: Find (r, ) wherer > 0 and0 ≤  < 2 or 0 ≤  < 360 . Relationships between r, , x, & yC  PP  Ctanyxq =cosx r q=siny r q=Examples: Converting CoordinatesPolar  Cartesiansinry cosrx (3, 210 )o (3cos 210 ,3sin 210 )�o o--213 ,23323 ,2336 ,2 2cos , 2sin6 6p p� �� - -� �� �--212 ,232 1- ,3Examples: Converting CoordinatesPolar  Cartesian222yxr xytanQuadrant I)7 ,3(587322r8.6637tan1)8.66 ,58( )7 ,3(7tan3q =Examples: Converting CoordinatesPolar  Cartesian222yxr xytanQuadrant II)7 ,3(587)3(22r8.6637tan1)2.113 ,58()1808.66 ,58( )7 ,3(37tan )8.66 ,58( )7 ,3(ORExamples: Converting CoordinatesPolar  Cartesian222yxr xytanQuadrant III)7 ,3( 58)7()3(22r8.6637tan1)8.246 ,58()1808.66 ,58( )7 ,3(37tan )8.66 ,58( )7 ,3(ORExamples: Converting CoordinatesPolar  Cartesian222yxr xytanQuadrant IV)7 ,3( 58)7(322r8.6637tan1)93.22 ,58()3608.66 ,58( )7 ,3(37tan )8.66 ,58( )7 ,3(ORConverting EquationsPolar  RectangularUse the same identities:222yxr xytansinry cosrx Converting EquationsPolar  CartesianReplace all occurrences of xx with r cos . Replace all occurrences of yy with r sin .SimplifySolve for rr (if possible).Converting EquationsPolar  CartesianExpress the equation in terms of sine and cosine only.If possible, manipulate the equation so that all occurrences of cos  and sin  are multiplied by r.Replace all occurrences of …Simplify(solve for y if possible)r cos  with xr sin  with yr2 with x2 + y2Or, if all else fails, use:2 2cosxx yq =+22sinyxy22yxr Graphing Simple Polar Equations and InequalitiesReviewSimple Cartesian EquationsSimple Cartesian Inequalities5x =2y =-5x <2y �-Graphing Simple Polar Equations and InequalitiesSimple Polar EquationsSimple Polar Inequalities5r =4pq =5r <3 , 1 3rpq = < <3 5r� <52 6p pq�