668 Chapter 11 Parametric Equatins and Polar Coordinates

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1 668 Chapter Parametric Equatins and Polar Coordinates 5. sin ( sin Á r and sin ( sin Á r Ê not symmetric about the x-axis; sin ( sin r Ê symmetric about the y-axis; therefore not symmetric about the origin 6. sin ( sin Á r and sin ( sin Á r Ê not symmetric about the x-axis; sin ( sin r Ê symmetric about the y-axis; therefore not symmetric about the origin 7. sin ˆ sin ˆ r Ê symmetric about the y-axis; sin ˆ sin ˆ, so the graph is symmetric about the x-axis, and hence the origin. 8. cos ˆ cos ˆ r Ê symmetric about the x-axis; cos ˆ cos ˆ, so the graph is symmetric about the y-axis, and hence the origin. 9. cos ( cos r Ê (r ß and ( r ß are on the graph hen (r ß is on the graph Ê symmetric about the x-axis and the y-axis; therefore symmetric about the origin Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

2 . sin ( sin r Ê (r ß and ( r ß are on the graph hen (r ß is on the graph Ê symmetric about the y-axis and the x-axis; therefore symmetric about the origin Section. Graphing in Polar Coordinates 669. sin ( sin r Ê (r ß and ( r ß are on the graph hen (r ß is on the graph Ê symmetric about the y-axis and the x-axis; therefore symmetric about the origin. cos ( cos r Ê (r ß and ( r ß are on the graph hen (r ß is on the graph Ê symmetric about the x-axis and the y-axis; therefore symmetric about the origin. Since a r ß b are on the graph hen (r ß is on the graph ˆ a rb cos ( Ê r cos, the graph is symmetric about the x-axis and the y-axis Ê the graph is symmetric about the origin. Since (r ß on the graph Ê ( r ß is on the graph ˆ a rb sin Ê r sin, the graph is symmetric about the origin. But sin ( sin Á r and sin ( sin ( sin ( sin Ár Ê the graph is not symmetric about the x-axis; therefore the graph is not symmetric about the y-axis 5. Since (r ß on the graph Ê ( r ß is on the graph ˆ a rb sin Ê r sin, the graph is symmetric about the origin. But sin ( ( sin sin Á r and sin ( sin ( sin ( ( sin sin Ár Ê the graph is not symmetric about the x-axis; therefore the graph is not symmetric about the y-axis Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

3 67 Chapter Parametric Equatins and Polar Coordinates 6. Sincea r ß b are on the graph hen (r ß is on the graph ˆ a rb cos ( Ê r cos, the graph is symmetric about the x-axis and the y-axis Ê the graph is symmetric about the origin. 7. Ê r Ê ˆ ß, and Ê r Ê ˆ ß dr r sin r cos ; r sin ; Slope r cos r sin sin r cos Ê Slope at ˆ sin cos r sin ß is sin ˆ ( cos ; Slope at ˆ ß is sin cos ( sin sin ˆ ( cos ˆ sin ˆ cos ˆ ( sin ˆ 8. Ê r Ê ( ß, and Ê r dr r sin r cos cos sin r cos r cos r sin cos cos r sin cos sin r cos cos sin ( cos cos r sin Slope at ( is cos ( sin cos sin ( cos ; Slope at ( is cos ( sin Ê ( ß ; r cos ; Slope Ê ß ß 9. Ê r Ê ˆ ß ; Ê r Ê ˆ ß ; Ê r Ê ˆ ß ; Ê r Ê ˆ ß ; dr r sin r cos cos sin r cos r cos r sin cos cos r sin ˆ cos ˆ sin ˆ ( cos ˆ cos ˆ cos ˆ ( sin ˆ r cos ; Slope Ê Slope at ß is ; Slope at ˆ ß is ; cos ˆ sin ˆ ( cos ˆ cos ˆ cos ˆ ( sin ˆ Slope at ˆ ß is ; cos Š sin Š ( cos Š cos Š cos Š ( sin Š Slope at ˆ ß is cos Š sin Š ( cos Š cos Š cos Š ( sin Š Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

4 . Ê r Ê (ß; Ê r Ê ˆ ß ; Ê r Ê ˆ ß ; Ê r dr r sin r cos sin sin r cos r cos r sin sin cos r sin sin sin cos sin cos sin sin ˆ sin ˆ ( cos ˆ ß sin ˆ cos ˆ ( sin ˆ Ê ( ß ; r sin ; Slope Ê Slope at (ß is, hich is undefined; Slope at ˆ is ; Slope at ˆ ß is ; sin ˆ sin ˆ ( cos ˆ sin ˆ cos ˆ ( sin ˆ sin sin cos Slope at (ß is, hich is undefined sin cos sin. (a (b Section. Graphing in Polar Coordinates 67. (a (b. (a (b. (a (b Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

5 67 Chapter Parametric Equatins and Polar Coordinates 5. cos 6. r sec Ê r Ê r cos Ê x Note that (r ß and ( r ß describe the same point in the plane. Then r cos Í cos ( (cos cos sin sin cos ( cos r; therefore (r ß is on the graph of r cos Í ( r ß is on the graph of r cos Ê the anser is (a. Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

6 Section. Graphing in Polar Coordinates 67. Note that (r ß and ( r ß describe the same point in the plane. Then r cos Í sin ˆ ( sin ˆ 5 sin ( cos ˆ 5 cos ( sin ˆ 5 cos r; therefore (r ß is on the graph of r sin ˆ Ê the anser is (a.... (a (b (c (. (a (b (c Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

7 67 Chapter Parametric Equatins and Polar Coordinates ( (e.5 AREA AND LENGTHS IN POLAR COORDINATES 6 6. A Î Î Î Î Î Î Î Î Î Î cos. A a sin b sin a cos b sin ˆ ˆ cos. A ( cos a6 6 cos cos b 8 8 cos ˆ (9 8 cos cos d 9 8 sin sin 8 cos. A [a( cos ] a a cos cos b a ˆ cos a ˆ cos cos a sin sin a Î Î cos sin Î% 8 5. A cos d cos6 Î6 Î6 Î6 Î6 6. A acos b d cos d d a cos 6 b Î6 cî6 Î6 Î6 Î sin 6 ˆ ˆ Î6 Î Î 7. A ( sin sin ccos d 8. A (6( ( sin d sin d Î6 Î6 9. r cos and r sin Ê cos sin Ê cos sin Ê ; therefore Î Î Î Î ˆ cos d Î% A ( sin sin d ( cos c sin Î cos Î Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

8 . r and r sin Ê sin Ê sin 5 Ê or ; therefore 6 6 5Î6 Î6 c A ( ( sin d 5Î 6ˆ sin Î6 5Î 6ˆ cos Î6 sin & Î Î6 Î 6 5Î 6ˆ cos ˆ 5 5 sin ˆ sin. r and r ( cos Ê ( cos Ê cos Ê ; therefore Î Î a cos cos b d ˆ ( Î ˆ cos cos Î A [( cos ] area of the circle ( 8 cos cos Î c6 8 sin sin d 58. r ( cos and r ( cos Ê cos cos Ê cos Ê or ; the graph also gives the point of intersection (ß ; therefore Î Î A [( cos ] [( cos ] a cos cos b Î a cos cos b Î Î ˆ cos cos d ˆ cos cos Î Î (6 8 cos cos d (6 8 cos cos Î c6 8 sin sin d c6 8 sin sin d 66 Î Î Section.5 Area and Lengths in Polar Coordinates 675 È. r and r 6 cos Ê 6 cos Ê cos Ê 6 (in the st quadrant; e use symmetry of the graph to find the area, so A (6 cos Š È Î6 Î6 (6 cos c sin È d Î Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

9 676 Chapter Parametric Equatins and Polar Coordinates. r a cos and r a( cos Ê a cos a( cos Ê cos cos Ê cos Ê or ; the graph also gives the point of intersection (ß ; therefore A c(a cos a ( cos d Î a Î 9a cos a a cos a cos b Î a 8a cos a cos a b Î c Î a a ( cos a cos a d a a cos a cos b Î$ ca a sin a sin d a a ˆ È a Š a Š È 5. r and r cos Ê cos Ê cos Ê in quadrant II; therefore Î Î A c( cos d a cos b [( cos ] d ( cos Î Î È sin d Î$ c 6. r 6 and r csc Ê 6 sin Ê sin 5Î Î6 Ê or ; therefore A a6 9 csc b 5Î 6ˆ 9 8 csc d 9 8 cot Î6 & Î Î 9 Š 5 È 9 Š È 9È 7. r sec and r cos Ê cos sec Ê cos Ê 5,,, or ; therefore Î a Î A 6 cos sec b a88 cos sec b Î c8 sin tan d Š 8 È È ab 8 È Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

10 8. r csc and r sin Ê sin csc Ê sin 5 Ê,,, or ; therefore Î Î a Î Î A 6 sin 9 csc b a 8 8 cos 9 csc b c8 sin 9 cot d Î Î 8 abš ÈÈ 8 È È È È È È Section.5 Area and Lengths in Polar Coordinates (a r tan and r Š csc Ê tan Š csc Ê sin Š cos Ê cos Š cos Ê cos Š cos Ê cos È or È (use the quadratic formula Ê (the solution in the first quadrant; therefore the area of R is A tan d asec b ctan d ˆ tan ; AO Š csc Î Î Î% È 8 È È È È È È and OB Š csc Ê AB Ê Š Ê the area of R is A Š Š ; therefore the area of the region shaded in the text is ˆ 8. Note: The area must be found this ay since no common interval generates the region. For example, the interval Ÿ Ÿ generates the arc OB of r tan but does not generate the segment AB of the liner È csc. Instead the interval generates the half-line from B to _ on the line r È csc. (b lim tan _ and the line x is r sec in polar coordinates; then lim (tan sec Ä Î Ä Îc lim ˆ sin lim ˆ sin lim ˆ cos Ê r tan approaches Ä Îc cos cos Ä Îc cos Ä Îc sin r sec as Ä c Ê r sec (or x is a vertical asymptote of r tan. Similarly, r sec (or x is a vertical asymptote of r tan.. It is not because the circle is generated tice from to. The area of the cardioid is A (cos acos cos b ˆ cos cos sin sin 5. The area of the circle is A the area requested is actually ˆ Ê È5 È5. r, Ÿ Ÿ È dr 5 Ê ; therefore Length Éa b ( È % È5 È5 kk È d (since È d ; u Ê du d ; Ê u, 9 È 5 Ê u 9 Ä È u du u $Î * 9 % e dr e e e e È È È È. r, Ÿ Ÿ Ê ; therefore Length ÊŠ Š d Ê Š e e e Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

11 678 Chapter Parametric Equatins and Polar Coordinates. r cos Ê sin ; therefore Length ( cos ( sin dr È È ( cos É cos É ˆ cos d d d cos d sin 8. r a sin, Ÿ Ÿ dr, a Ê a sin cos ; therefore Length Ɉ a sin ˆ a sin cos d É % É a cos a a sin a sin cos a sin sin cos d (since Ÿ Ÿ a sin ˆ 6 dr 6 sin Î 6 6 sin cos d (cos cos (cos 5. r, Ÿ Ÿ Ê ; therefore Length ʈ Š Î Î 6 6 sin sin ( cos a cos b cos ( cos É d 6 % É ˆ since on Ÿ Ÿ 6 ˆ cos cos sin É cos cos (cos Î Î Î Î Î 6 ˆ cos É d 6È 6È $ sec cos ( cos ( cos $Î $Î ˆ cos Î% Î Î Î $ $ sec 6 sec u du (use tables 6 Œ sec u tan u sec u du 6 Š Î% ln ksec u tan uk È ln Š È È dr sin sin 6. r cos, Ÿ Ÿ Ê d ( cos ; therefore Length ʈ cos Š (cos Î sin ( cos sin Î ( cos a cos b Î cos ( cos Ê Š d É ˆ since cos on Ÿ Ÿ ˆ cos cos sin É Î cos (cos ˆ cos É d È È $ csc Î cos ( cos Î( cos $Î $ΠΈ sin Î Î $ csc ˆ d ˆ since csc on $ csc u du (use tables Î Î Ÿ Ÿ Î Œ csc u cot u Î csc u du Î Š ln kcsc u cot uk ln Š È Î% Î È Î% È ÈlnŠ È Î Î Î Î É % É $ dr 7. r cos Ê sin cos ; therefore Length Ɉ cos $ ˆ sin cos cos ˆ sin ˆ cos ˆ d ˆ cos cos ˆ sin ˆ d cos ˆ Î cosˆ Î% 8 8 d sin 8. r È sin, Ÿ Ÿ È dr Î Î Ê ( sin ( cos (cos ( sin ; therefore È È Length cos É( sin d sin sin cos É ( sin sin È È È sin É È È sin Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

12 Section.6 Conic Sections 679 dy d dy Š c d c d dx 9. Let r f(. Then x f( cos Ê f ( cos f( sin Ê ˆ dx cf ( cos f( sin d cf ( cos f ( f( sin cos [f( ] sin ; y f( sin Ê f ( sin f( cos Ê f ( sin f( cos f ( sin f ( f( sin cos [f( ] cos. Therefore ˆ dx dy Š cf( d acos sin b [f( ] acos sin b cf( d [f( ] r ˆ dr. Thus, L dx ʈ dy Š Ér ˆ dr d. dr. (a r a Ê ; Length Èa kk a ca d a dr (b r a cos Ê a sin ; Length È(a cos ( a sin Èa acos sin b kk a d c a d a dr (c r a sin Ê a cos ; Length È(a cos (a sin Èa acos sin b kk a d c a d a. (a r av a a( cos sin a c d (b r a ca d a av Î Î a ˆ ˆ Î c Î c d (c r a cos d a sin av dr. r f(, Ÿ Ÿ Ê f ( Ê r ˆ dr [f(] cf ( d Ê Length É[f( ] cf ( d É[f( ] cf ( d d hich is tice the length of the curve r f( for Ÿ Ÿ..6 CONIC SECTIONS y 8. x Ê p 8 Ê p ; focus is (ß, directrix is x y. x Ê p Ê p ; focus is ( ß, directrix is x. y Ê p 6 Ê p ; focus is ˆ ß, directrix is y x 6. y Ê p Ê p ; focus is ˆ ß, directrix is y x x y 5. 9 Ê c È 9 È Ê foci are Š È ß ; vertices are a ß b ; asymptotes are y x x y 6. Ê c È9 È5 Ê foci are Š ß È5 ; vertices are aß b 9 x 7. y Ê c È Ê foci are a ß b; vertices are Š Èß y 8. x Ê c È È5 Ê foci are Š ß È5 ; vertices are aß b; asymptotes are y x Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

13 68 Chapter Parametric Equatins and Polar Coordinates y x 9. y x Ê x Ê p Ê p ;. x 6y Ê y 6 Ê p 6 Ê p ; focus is ( $ß, directrix is x focus is ˆ ß, directrix is y x. x 8y Ê y 8 Ê p 8 Ê p ;. y x Ê x Ê p Ê p ; focus is (ß, directrix is y focus is ˆ ß, directrix is x y x x ˆ 6 ˆ 8 8 ˆ ß ˆ ß 6 6. y x Ê y Ê p Ê p ;. y 8x Ê y Ê p Ê p ; focus is, directrix is y focus is, directrix is y y y ˆ ˆ 8 ß 8ß 8 5. x y Ê x Ê p Ê p ; 6. x y Ê x Ê p Ê p ; focus is ˆ, directrix is x focus is ˆ, directrix is x Copyright Pearson Education, Inc. Publishing as Addison-Wesley.

14 x y 7. 6x 5y Ê x 6y Ê 6 7 Ê c Èa b È56 Ê c Èa b È67 Section.6 Conic Sections 68 x y y 9. x y Ê x. x y Ê Ê c Èa b È Ê c Èa b È È x y x y. x y 6 Ê. 9x y 9 Ê 9 Ê c Èa b È Ê c Èa b È9 x y Copyright Pearson Education, Inc. Publishing as Addison-Wesley.