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# Class Note for MATH 1432 at UH

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Date Created: 02/06/15

Lecture 12Section 93 Polar Coordinates Section 94 Graphing in Polar Coordinates Jiwen He 1 Polar Coordinates 11 Polar Coordinates Polar Coordinate System ray 6 0 A polar axis ray67r The purpose of the polar coordinates is to represent curves that have symmetry about a point or spiral about a point Frame of Reference In the polar coordinate system the frame of reference is a point 0 that we call the pole and a ray that emanates from it that we call the polar axis Polar Coordinates I 2 raygzr 2 7r 3 3 7r polar axrs polar ms 0 2 in ray 6 7 ray 5 7139 3 De nition A point is given polar coordinates 7 6 iff it lies at a distance lrl from the pole a long the ray 6 if r 2 O and along the ray 6 7r if r lt 0 Points in Polar Coordinates ray 6 13919ll 927rlr 647ral 276E2 6 9 polar axis 0 polar axrs 0 2 6 7 2 6 62 9 47 ray 6 7239 Points in Polar Coordinates 0 O 0 6 for all 6 0 7 6 7 6 2mr for all integers n 0 r 6 r6 7r 12 Relation to Rectangular Coordinates Relation to Rectangular Coordinates y I x y 3 polar axis A l 0 Relation to Rectangular Coordinates oxrcos6yrsin6 gt x2y2r2 tan 9 I o 7 xx2y2 6tanL Circles in Polar Coordinates rh EV Circles in Polar Coordinates r 2 14sin6 r 4 c039 In rectangular coordinates In polar coordinates 62 y2 a2 7 a x2y a2a2 r2asin6 2 x a2y2a r2acos6 2 2 2 x2y2a gt 7 a 132y a a gt x2y22ay gt r22arsin6 x a2y a gt x2y22ax gt r22arcos6 Lines in Polar Coordinates X 2 3 3 J 2 V V V Y Y X X 91 62 r 2e00 r2c506 4 4 Lines in Polar Coordinates In rectangular coordinates In polar coordinates yzmx Qzawithaztan1m x a 7 asec6 y a 7 acsc6 y y mx gt m gt tan6 m x xza gt rcoseza gt rasec6 yza gt rsin6a gt racsc0 13 Symmetry Symmetry y y 1 17r 9 1 0 9 0 OI 9 I I X r Mr 9 I39T0 symmetry about the v axis symmetry about the y axis symmetry about the origin LVA 10 10 17z 10136 r2 cos 26 lemniscate Lemniscate ribbon r2 cos 26 c0s2 6 c0s 26 cos 26 1eXgt if 736 E graph then 73 6 E graph 1eXgt symmetric about the x axis c0s27r 6 c0s27r 26 cos 26 1exgt if 736 E graph then 737T 6 E graph 1eXgt symmetric about the y axis c0s27r 6 c0s27r 26 cos 26 1eXgt if 73 6 E graph then 73 7T 6 E graph 1eXgt symmetric about the origin Lemniscates Ribbons r2 asin 26 r2 acos 26 7724sin 26 7724cos 26 Lemniscate r2 asin 26 sin27r 6 sin27r 26 sin 26 2eXgt if 736 E graph then 737T 6 E graph 2eXgt symmetric about the origin 2 Graphing in Polar Coordinates 21 Spiral Spiral of Archimedes 7 6 6 Z 0 l l Polar ms 3 3 47 7r 71quot 7t 271 211x 393 iii Em git Em 34 r26 620 spiral of Archimedes The curve is a nonending spiral Here it is shown in detail from 6 0 to 6 27r 22 Limagons Limacon Snail 7 1 2 cos6 quot39 polar axis 3 if v V polar axis 1 0303572 OSQS Polar axis 0 polar axis 03632 5 OSQSEIT e 0 714 713 712 2713 3714 71 5714 4713 3712 5713 7714 271 1 1 041 0 1 2 241 3 241 2 1 0 041 1 o 7quot 0 at 6 7r 37139 r is a local maximum at 6 07r27r 0 Sketch in 4 stages 0 7r 7r7r 7r 37139 327r o cos 6 cos6 i if 736 E graph then 73 6 E graph i symmetric about the x axis Limagons Snails r a b cos 6 3 1 r3COS6 r COS6 r1cose r cose convex Iimacon cardioid limagon with imagon with a dimple an inner loop The general shape of the curve depends on the relative magnitudes of a and W Cardioids HeartShaped r 1 i cos 6 r 1 i sin6 r1COS6 r1sin6 rl COS6 rl sin0 Each change cos 6 gt sin 6 gt cos 6 gt sin 6 represents a counterclockwise rotation by 7r radians o Rotation by 7r r 1 cos6 7r 1 sin6 o Rotation by 7r r 1 sin6 7r 1 cos 6 l 2 o Rotation by 7r r 1 cos6 7r 1 sin 6 2 3 Flowers Petal Curve r cos 26 02 1 1 3 OSQSZE OSQSE OSQSZH OSQSE 03032 OSQSEE OSHSZK OSQSZH 6 W 3 5W 7W 1 1 39 6 0 w 3quot 2 o r Oat ZTTT M 1sa oca maX1mum at 7r7 7r 0 Sketch the curve in 8 stages 0 cos2 6 cos 26 cos27r i 6 cos 26 gt symmetric about the IJ aXis the y axis and the origin Petal Curves 1 acos n6 7 asin n6 rsin36 rcos 49 o If n is odd there are n petals o If n is even there are 2n petals 24 Intersections Intersections 1 a1 cos 6 and 7 a1 cos 6 polar axis 197 ra1 cost9 ral COSt9 937 a n10 a o ra1 cos6 andra1cos9 gtraandcos60gtraand 6 g mt gt a g mr E intersection gt 71 even7 a g WI 2 a n odd7 a g WI 2 a 0 Two intersection points a 07 a and a 07 a o The intersection third point the origin but the two cardioids pass through the origin at different times Outline Contents 1 Polar Coordinates 1 11 Polar Coordinates 1 12 Relation to Rectangular Coordinates 2 13 Symmetry 4 2 Graphing in Polar Coordinates 6 21 Spiral 6 22 Limacons 6 23 Flowers 7 24 Intersections 8

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