10.2 Polar Equations and Graphs

Transcription

1 SECTIN 0. Polar Equations and Grahs 77 Elaining Concets: Discussion and Writing 85. In converting from olar coordinates to rectangular coordinates, what formulas will ou use? 86. Elain how ou roceed to convert from rectangular coordinates to olar coordinates. 87. Is the street sstem in our town based on a rectangular coordinate sstem, a olar coordinate sstem, or some other sstem? Elain. Are You Preared? Answers b. ; quadrant IV a (, ) 0. Polar Equations and Grahs PREPARING FR THIS SECTIN Before getting started, review the following: Smmetr (Section.,. 60 6) Circles (Section.,. 8 85) Even dd Proerties of Trigonometric Functions (Section 7.5,. 558) Now Work the Are You Preared? roblems on age 79. Difference Formulas for Sine and Cosine (Section 8.5,. 60 and 6) Values of the Sine and Cosine Functions at Certain Angles (Section 7., , Section 7., ) BJECTIVES Identif and Grah Polar Equations b Converting to Rectangular Equations (. 78) Test Polar Equations for Smmetr (. 7) Grah Polar Equations b Plotting Points (. 7) Just as a rectangular grid ma be used to lot oints given b rectangular coordinates, as in Figure 9(a), we can use a grid consisting of concentric circles (with centers at the ole) and ras (with vertices at the ole) to lot oints given b olar coordinates, as shown in Figure 9(b). We use such olar grids to grah olar equations. Figure 9 A (, ) B (, ) P, 0 r r r 5 5 Q, 5 7 (a) Rectangular grid (b) Polar grid

2 78 CHAPTER 0 Polar Coordinates; Vectors DEFINITIN An equation whose variables are olar coordinates is called a olar equation. The grah of a olar equation consists of all oints whose olar coordinates satisf the equation. Identif and Grah Polar Equations b Converting to Rectangular Equations ne method that we can use to grah a olar equation is to convert the equation to rectangular coordinates. In the discussion that follows,, reresent the rectangular coordinates of a oint P, and r, u reresent olar coordinates of the oint P. EXAMPLE Identifing and Grahing a Polar Equation (Circle) Identif and grah the equation: r = Convert the olar equation to a rectangular equation. r = r = 9 + = 9 Square both sides. r = + The grah of r = is a circle, with center at the ole and radius. See Figure 0. Figure 0 r = or + = Now Work PRBLEM EXAMPLE Identifing and Grahing a Polar Equation (Line) Figure Identif and grah the equation: u = Convert the olar equation to a rectangular equation. u = or = u = tan u = tan = = Take the tangent of both sides. The grah of u = is a line assing through the ole making an angle of with the olar ais. See Figure. Now Work PRBLEM 5 tan u = ; tan =

3 SECTIN 0. Polar Equations and Grahs 79 EXAMPLE Identifing and Grahing a Polar Equation (Horizontal Line) Identif and grah the equation: r sin u = Since = r sin u, equation as = we can write the We conclude that the grah of r sin u = is a horizontal line units above the ole. See Figure. Figure r sin u = or = 5 0 CMMENT A grahing utilit can be used to grah olar equations. Read Using a Grahing Utilit to Grah a Polar Equation, Aendi, Section EXAMPLE Identifing and Grahing a Polar Equation (Vertical Line) Identif and grah the equation: r cos u = - Since = r cos u, equation as = - we can write the We conclude that the grah of r cos u = - is a vertical line units to the left of the ole. See Figure. Figure r cos u = - or = Based on Eamles and, we are led to the following results. (The roofs are left as eercises. See Problems 8 and 8.) THEREM Let a be a real number. Then the grah of the equation r sin u = a is a horizontal line. It lies a units above the ole if a Ú 0 and ƒaƒ units below the ole if a 6 0. The grah of the equation r cos u = a is a vertical line. It lies a units to the right of the ole if a Ú 0 and ƒaƒ units to the left of the ole if a 6 0. Now Work PRBLEM 9

4 70 CHAPTER 0 Polar Coordinates; Vectors EXAMPLE 5 Figure r = sin u or + ( - ) = = = = = 5 = 0 Identifing and Grahing a Polar Equation (Circle) Identif and grah the equation: r = sin u To transform the equation to rectangular coordinates, multil each side b r. r = r sin u Now use the facts that r = + and = r sin u. Then + = + - = = + - = Comlete the square in. Factor. = 5 = 7 = This is the standard equation of a circle with center at coordinates and radius. See Figure. 0, in rectangular = EXAMPLE 6 Figure 5 r = - cos u or ( + ) + = = = 5 = Identifing and Grahing a Polar Equation (Circle) Identif and grah the equation: Proceed as in Eamle 5. + = = = r = - cos u r = -r cos u Multil both sides b r. r = + ; = r cos u Comlete the square in. + + = Factor. 5 = 0 This is the standard equation of a circle with center at -, 0 coordinates and radius. See Figure 5. = = = 7 Eloration in rectangular Using a square screen, grah r = sin u, r = sin u, and r = sin u. Do ou see the attern? Clear the screen and grah r = -sin u, r = - sin u, and r = - sin u. Do ou see the attern? Clear the screen and grah r = cos u, r = cos u, and r = cos u. Do ou see the attern? Clear the screen and grah r = -cos u, r = - cos u, and r = - cos u. Do ou see the attern? Based on Eamles 5 and 6 and the receding Eloration, we are led to the following results. (The roofs are left as eercises. See Problems 8 86.) THEREM Let a be a ositive real number. Then Equation Descrition (a) r = a sin u Circle: radius a; center at 0, a in rectangular coordinates (b) r =-a sin u Circle: radius a; center at 0, -a in rectangular coordinates (c) r = a cos u Circle: radius a; center at a, 0 in rectangular coordinates (d) r =-a cos u Circle: radius a; center at -a, 0 in rectangular coordinates Each circle asses through the ole. Now Work PRBLEM

5 SECTIN 0. Polar Equations and Grahs 7 The method of converting a olar equation to an identifiable rectangular equation to obtain the grah is not alwas helful, nor is it alwas necessar. Usuall, we set u a table that lists several oints on the grah. B checking for smmetr, it ma be ossible to reduce the number of oints needed to draw the grah. Test Polar Equations for Smmetr In olar coordinates, the oints r, u and r, -u are smmetric with resect to the olar ais (and to the -ais). See Figure 6(a). The oints r, u and r, - u are smmetric with resect to the line u = (the -ais). See Figure 6(b). The oints r, u and -r, u are smmetric with resect to the ole (the origin). See Figure 6(c). Figure 6 5 (a) (r, ) 5 (r, ) 7 Points smmetric with resect to the olar ais 0 5 (b) (r, ) (r, ) 5 0 Points smmetric with resect to the line 7 5 (c) (r, ) (r, ) (r, ) 7 Points smmetric with resect to the ole 5 0 The following tests are a consequence of these observations. THEREM Tests for Smmetr Smmetr with Resect to the Polar Ais (-Ais) In a olar equation, relace u b -u. If an equivalent equation results, the grah is smmetric with resect to the olar ais. Smmetr with Resect to the Line U P (-Ais) In a olar equation, relace u b - u. If an equivalent equation results, the grah is smmetric with resect to the line u =. Smmetr with Resect to the Pole (rigin) In a olar equation, relace r b -r or u b u +. If an equivalent equation results, the grah is smmetric with resect to the ole. The three tests for smmetr given here are sufficient conditions for smmetr, but the are not necessar conditions. That is, an equation ma fail these tests and still have a grah that is smmetric with resect to the olar ais, the line u = or, the ole. For eamle, the grah of r = sinu turns out to be smmetric with resect to the olar ais, the line u = and the ole, but onl the test for smmetr, with resect to the ole (relace u b u + ) works. See also Problems

6 7 CHAPTER 0 Polar Coordinates; Vectors Grah Polar Equations b Plotting Points EXAMPLE 7 Grahing a Polar Equation (Cardioid) Grah the equation: Check for smmetr first. r = - sin u Polar Ais: Relace u b -u. The result is r = - sin-u = + sin u The test fails, so the grah ma or ma not be smmetric with resect to the olar ais. The Line U P Relace u b - u. The result is : sin (-u) = -sin u r = - sin - u = - sin cos u - cos sin u = - 0 # cos u - - sin u = - sin u The test is satisfied, so the grah is smmetric with resect to the line u =. Table U r sin U - (-) = - a- b L.87 - a- b = - 0 = - = - - = 0 L 0. The Pole: Relace r b -r. Then the result is -r = - sin u, so r = - + sin u. The test fails. Relace u b u +. The result is r = - sin(u + ) = - sin u cos + cos u sin = - sin u # (-) + cos u # 0 = + sin u This test also fails. So the grah ma or ma not be smmetric with resect to the ole. Net, identif oints on the grah b assigning values to the angle u and calculating the corresonding values of r.due to the eriodicit of the sine function and the smmetr with resect to the line u = we onl need to assign values to from - to as given in Table.,, u Now lot the oints r, u from Table and trace out the grah, beginning at the oint a, - and ending at the oint a0, Then reflect this ortion of the b b. grah about the line u = (the -ais) to obtain the comlete grah. See Figure 7. Figure 7 r = - sin u (0., ) ( 6, ) (, 0) (0, ) ( 6, ) 0 Eloration Grah r = + sin u. Clear the screen and grah r = - cos u. Clear the screen and grah r = + cos u. Do ou see a attern? 5 (, ) (.87, ) 7 The curve in Figure 7 is an eamle of a cardioid (a heart-shaed curve).

7 SECTIN 0. Polar Equations and Grahs 7 DEFINITIN Cardioids are characterized b equations of the form where r = a + cos u r = a + sin u r = a - cos u r = a - sin u a 7 0. The grah of a cardioid asses through the ole. Now Work PRBLEM 7 EXAMPLE 8 Grahing a Polar Equation (Limaçon without an Inner Loo) Grah the equation: r = + cos u Check for smmetr first. Polar Ais: Relace u b -u. The result is r = + cos-u = + cos u cos (-u) = cos u The test is satisfied, so the grah is smmetric with resect to the olar ais. Table U r cos U + () = 5 + a b L.7 + a b = + (0) = + a- b = + a- b L.7 + (-) = Figure 8 r = + cos u The Line U P Relace u b - u. The result is : The test fails, so the grah ma or ma not be smmetric with resect to the line u =. r = + cos - u = + cos cos u + sin sin u = - cos u The Pole: Relace r b -r. The test fails, so the grah ma or ma not be smmetric with resect to the ole. Relace u b u +. The test fails, so the grah ma or ma not be smmetric with resect to the ole. Net, identif oints on the grah b assigning values to the angle u and calculating the corresonding values of r.due to the eriodicit of the cosine function and the smmetr with resect to the olar ais, we onl need to assign values to u from 0 to, as given in Table. Now lot the oints r, u from Table and trace out the grah, beginning at the oint 5, 0 and ending at the oint,. Then reflect this ortion of the grah about the olar ais (the -ais) to obtain the comlete grah. See Figure 8. = = = = (, ) (, ) (, ) (.7, 6 ) 5 (.7, 6 ) (, ) (5, 0) 5 = 0 Eloration Grah r = - cos u. Clear the screen and grah r = + sin u. Clear the screen and grah r = - sin u. Do ou see a attern? = 5 = 7 = The curve in Figure 8 is an eamle of a limaçon (a French word for snail) without an inner loo.

8 7 CHAPTER 0 Polar Coordinates; Vectors DEFINITIN Limaçons without an inner loo are characterized b equations of the form r = a + b cos u r = a + b sin u r = a - b cos u r = a - b sin u where a 7 0, b 7 0, and a 7 b. The grah of a limaçon without an inner loo does not ass through the ole. Now Work PRBLEM EXAMPLE 9 Grahing a Polar Equation (Limaçon with an Inner Loo) Grah the equation: r = + cos u First, check for smmetr. Polar Ais: Relace u b -u. The result is r = + cos-u = + cos u The test is satisfied, so the grah is smmetric with resect to the olar ais. Table U r cos U + () = + a b L.7 + a b = + (0) = + a- b = 0 + a- b L (-) = - The Line U P Relace u b - u. The result is : The test fails, so the grah ma or ma not be smmetric with resect to the line u =. r = + cos - u = + cos cos u + sin sin u = - cos u The Pole: Relace r b -r. The test fails, so the grah ma or ma not be smmetric with resect to the ole. Relace u b u +. The test fails, so the grah ma or ma not be smmetric with resect to the ole. Net, identif oints on the grah of r = + cos u b assigning values to the angle u and calculating the corresonding values of r. Due to the eriodicit of the cosine function and the smmetr with resect to the olar ais, we onl need to assign values to u from 0 to, as given in Table. Now lot the oints r, u from Table, beginning at, 0 and ending at -,. See Figure 9(a). Finall, reflect this ortion of the grah about the olar ais (the -ais) to obtain the comlete grah. See Figure 9(b). Figure 9 = Eloration Grah r = - cos u. Clear the screen and grah r = + sin u. Clear the screen and grah r = - sin u. Do ou see a attern? 5,,.7, ( 6 ) 0, (, 0) 0 0.7, 5 6 (a) (, ) 7 5, (, ) (.7, 6 ) (, 0) 0, 0 0.7, 5 6 (, ) (b) r cos 7 The curve in Figure 9(b) is an eamle of a limaçon with an inner loo.

9 SECTIN 0. Polar Equations and Grahs 75 DEFINITIN Limaçons with an inner loo are characterized b equations of the form r = a + b cos u r = a + b sin u r = a - b cos u r = a - b sin u where a 7 0, b 7 0, and a 6 b. The grah of a limaçon with an inner loo will ass through the ole twice. Now Work PRBLEM 5 EXAMPLE 0 Grahing a Polar Equation (Rose) Grah the equation: r = cosu Check for smmetr. Polar Ais: If we relace u b -u, the result is r = cos-u = cosu The test is satisfied, so the grah is smmetric with resect to the olar ais. The Line U P If we relace u b - u, we obtain : Table U 0 6 r cos(u) () = a b = (0) = 0 a- b = - (-) = - The test is satisfied, so the grah is smmetric with resect to the line u =. The Pole: r = cos - u = cos - u = cosu Since the grah is smmetric with resect to both the olar ais and the line u = it must be smmetric with resect to the ole., Net, construct Table. Due to the eriodicit of the cosine function and the smmetr with resect to the olar ais, the line u = and the ole, we consider, onl values of u from 0 to. Plot and connect these oints in Figure 0(a). Finall, because of smmetr, reflect this ortion of the grah first about the olar ais (the -ais) and then about the line u = (the -ais) to obtain the comlete grah. See Figure 0(b). Figure 0 Eloration Grah r = cosu; clear the screen and grah r = cos6u. How man etals did each of these grahs have? Clear the screen and grah, in order, each on a clear screen, r = cosu, r = cos5u, and r = cos7u. What do ou notice about the number of etals? 5, 6 0, (, 0) (, ), 7 (a) (, ), 7 (b) r cos () The curve in Figure 0(b) is called a rose with four etals., 6 (, 0) 5 0

10 76 CHAPTER 0 Polar Coordinates; Vectors DEFINITIN Rose curves are characterized b equations of the form r = a cosnu, r = a sinnu, a Z 0 and have grahs that are rose shaed. If n Z 0 is even, the rose has n etals; if n Z ; is odd, the rose has n etals. Now Work PRBLEM 9 Table 5 U EXAMPLE 0 (0) = r sin(u) a b = () = a b = (0) = 0 r ;.9 ; ;.9 0 Grahing a Polar Equation (Lemniscate) Grah the equation: r = sinu We leave it to ou to verif that the grah is smmetric with resect to the ole. Because of the smmetr with resect to the ole, we onl need to consider values of u between u = 0 and u =. Note that there are no oints on the grah for (quadrant II), since r 6 0 for such values. Table 5 lists oints on the 6 u 6 grah for values of u = 0 through u = The oints from Table 5 where r Ú 0. are lotted in Figure (a). The remaining oints on the grah ma be obtained b using smmetr. Figure (b) shows the final grah drawn. Figure = = = ( (0, 0) ).9, = (, ) (.9, ).9, 6 = 0 = = = (.9, ) (0, 0) =, 6 = 0 = 5 = = 7 = 5 = = 7 (a) (b) r = sin () The curve in Figure (b) is an eamle of a lemniscate (from the Greek word ribbon). DEFINITIN Lemniscates are characterized b equations of the form r = a sinu r = a cosu where a Z 0, and have grahs that are roeller shaed. Now Work PRBLEM 5 EXAMPLE Grahing a Polar Equation (Siral) Grah the equation: r = e u>5 The tests for smmetr with resect to the ole, the olar ais, and the line u = fail. Furthermore, there is no number u for which r = 0, so the grah does not ass through the ole. bserve that r is ositive for all u, r increases as u increases, r : 0

11 SECTIN 0. Polar Equations and Grahs 77 Table U Table 7 r e U> as u : - q, and r : q as u : q. With the hel of a calculator, we obtain the values in Table 6. See Figure. Figure r = e u/5 5 (.7, (.87, ) ).7, (, 0) (.5, ) 0.57, 7 The curve in Figure is called a logarithmic siral, since its equation ma be written as u = 5 ln r and it sirals infinitel both toward the ole and awa from it. Classification of Polar Equations The equations of some lines and circles in olar coordinates and their corresonding equations in rectangular coordinates are given in Table 7. Also included are the names and grahs of a few of the more frequentl encountered olar equations. Lines Descrition Line assing through the ole making an angle a with the olar ais Vertical line Horizontal line Rectangular equation = (tan a) = a = b Polar equation u = a r cos u = a r sin u = b Tical grah Circles Descrition Center at the ole, radius a Passing through the ole, tangent to the line u =, center on the olar ais, radius a Rectangular equation Polar equation Tical grah + = a, a 7 0 r = a, a = ;a, a 7 0 r = ;a cos u, a 7 0 Passing through the ole, tangent to the olar ais, center on the line u =, radius a + = ;a, a 7 0 r = ;a sin u, a 7 0 a a a (continued)

12 78 CHAPTER 0 Polar Coordinates; Vectors Table 7 (Continued) ther Equations Name Cardioid Limaçon without inner loo Limaçon with inner loo Polar equations r = a ; a cos u, a 7 0 r = a ; b cos u, 0 6 b 6 a r = a ; b cos u, 0 6 a 6 b r = a ; a sin u, a 7 0 r = a ; b sin u, 0 6 b 6 a r = a ; b sin u, 0 6 a 6 b Tical grah Name Lemniscate Rose with three etals Rose with four etals Polar equations r = a cos(u), a 7 0 r = a sin(u), a 7 0 r = a sin(u), a 7 0 r = a sin(u), a 7 0 r = a cos(u), a 7 0 r = a cos(u), a 7 0 Tical grah Sketching Quickl If a olar equation involves onl a sine (or cosine) function, ou can quickl obtain a sketch of its grah b making use of Table 7, eriodicit, and a short table. EXAMPLE Sketching the Grah of a Polar Equation Quickl Grah the equation: r = + sin u You should recognize the olar equation: Its grah is a cardioid. The eriod of sin u is, so form a table using 0 u, comute r, lot the oints r, u, and sketch the grah of a cardioid as u varies from 0 to. See Table 8 and Figure. Table 8 U r sin U Figure r = + sin u 0 + (0) = + () = (, ) + (0) = + (-) = 0 + (0) = 5 (, ) (, 0) 5 ( 0, ) 7 0

13 SECTIN 0. Polar Equations and Grahs 79 Calculus Comment For those of ou who are lanning to stud calculus, a comment about one imortant role of olar equations is in order. In rectangular coordinates, the equation + =, whose grah is the unit circle, is not the grah of a function. In fact, it requires two functions to obtain the grah of the unit circle: = - Uer semicircle = - - Lower semicircle In olar coordinates, the equation r =, whose grah is also the unit circle, does define a function. For each choice of u, there is onl one corresonding value of r, that is, r =. Since man roblems in calculus require the use of functions, the oortunit to eress nonfunctions in rectangular coordinates as functions in olar coordinates becomes etremel useful. Note also that the vertical-line test for functions is valid onl for equations in rectangular coordinates. Historical Feature Jakob Bernoulli (65 705) Polar coordinates seem to have been invented b Jakob Bernoulli (65 705) in about 69, although, as with most such ideas, earlier traces of the notion eist. Earl users of calculus remained committed to rectangular coordinates, and olar coordinates did not become widel used until the earl 800s. Even then, it was mostl geometers who used them for describing odd curves. Finall, about the mid-800s, alied mathematicians realized the tremendous simlification that olar coordinates make ossible in the descrition of objects with circular or clindrical smmetr. From then on their use became widesread. 0. Assess Your Understanding Are You Preared? Answers are given at the end of these eercises. If ou get a wrong answer, read the ages listed in red.. If the rectangular coordinates of a oint are, -6, the. Is the sine function even, odd, or neither? (. 558) oint smmetric to it with resect to the origin is. (. 60 6). The difference formula for cosine is cosa - B =. 5. sin 5 =. ( ) (. 60). The standard equation of a circle with center at -, 5 and radius is. (. 8 85) 6. cos =. ( ) Concets and Vocabular 7. An equation whose variables are olar coordinates is called a(n). 8. True or False The tests for smmetr in olar coordinates are necessar, but not sufficient. 9. To test if the grah of a olar equation ma be smmetric with resect to the olar ais, relace u b. 0. To test if the grah of a olar equation ma be smmetric with resect to the line u =, relace u b.. True or False A cardiod asses through the ole.. Rose curves are characterized b equations of the form r = a cos (n u) or r = a sin (n u), a Z 0.If n Z 0 is even, the rose has etals; if n Z ; is odd, the rose has etals. Skill Building In Problems 8, transform each olar equation to an equation in rectangular coordinates. Then identif and grah the equation.. r =. r = 5. u = 6. u =- 7. r sin u = 8. r cos u = 9. r cos u = - 0. r sin u = -

14 70 CHAPTER 0 Polar Coordinates; Vectors. r = cos u. r = sin u. r = - sin u. r = - cos u 5. r sec u = 6. r csc u = 8 7. r csc u = - 8. r sec u = - In Problems 9 6, match each of the grahs (A) through (H) to one of the following olar equations. 9. r = 0. u =. r = cos u. r cos u =. r = + cos u. r = sin u 5. u = 6. r sin u = (A) (B) 5 (C) (D) (E) 5 7 (F) In Problems 7 60, identif and grah each olar equation. 7. r = + cos u 8. r = + sin u 9. r = - sin u 0. r = - cos u. r = + sin u. r = - cos u. r = - cos u. r = + sin u 5. r = + sin u 6. r = - sin u 7. r = - cos u 8. r = + cos u 9. r = cosu 50. r = sinu 5. r = sin5u 5. r = cosu 5. r = 9 cosu 5. r = sinu 55. r = u 56. r = u 57. r = - cos u 58. r = + cos u 59. r = - cos u 60. r = cosu Mied Practice In Problems 6 66, grah each air of olar equations on the same olar grid. Find the olar coordinates of the oint(s) of intersection and label the oint(s) on the grah. 6. r = 8 cos u; r = sec u 6. r = 8 sin u; r = csc u 6. r = sin u; r = + cos u 6. r = ; r = + cos u 65. r = + sin u; r = + cos u 66. r = + cos u; r = cos u Alications and Etensions In Problems 67 70, the olar equation for each grah is either r = a + b cos u or r = a + b sin u, a 7 0. Select the correct equation and find the values of a and b , 5 (G) 7, 5 (H) 7 (6, 0) (6, )

15 SECTIN 0. Polar Equations and Grahs ( 5, ) ( 5, ) (, 0) (, 0) In Problems 7 80, grah each olar equation. 7. r = (arabola) - cos u 7. r = (ellise) - cos u 7. r = - cos u (herbola) 7. r = - cos u (arabola) 75. r = u, u Ú 0 (siral of Archimedes) 76. r = (recirocal siral) u 77. r = csc u -, 0 6 u 6 (conchoid) 78. r = sin u tan u (cissoid) 79. r = tan u, - (kaa curve) 6 u r = cos u 8. Show that the grah of the equation r sin u = a is a 8. Show that the grah of the equation r cos u = a is a vertical horizontal line a units above the ole if a Ú 0 and ƒaƒ units below the ole if a 6 0. line a units to the right of the ole if a Ú 0 and ƒaƒ units to the left of the ole if a Show that the grah of the equation r = a sin u, a 7 0, is a 8. Show that the grah of the equation r = -a sin u, a 7 0, is circle of radius a with center at 0, a in rectangular a circle of radius a with center at 0, -a in rectangular coordinates. coordinates. 85. Show that the grah of the equation r = a cos u, a 7 0, is 86. Show that the grah of the equation r = -a cos u, a 7 0, is a circle of radius a with center at a, 0 in rectangular a circle of radius a with center at -a, 0 in rectangular coordinates. coordinates. Elaining Concets: Discussion and Writing 87. Elain wh the following test for smmetr is valid: Relace r b -r and u b -u in a olar equation. If an equivalent equation results, the grah is smmetric with resect to the line u = (-ais). (a) Show that the test on age 7 fails for r = cos u, et this new test works. (b) Show that the test on age 7 works for r = sin u, et this new test fails. 88. Write down two different tests for smmetr with resect to the olar ais. Find eamles in which one test works and the other fails.which test do ou refer to use? Justif our answer. 89. The tests for smmetr given on age 7 are sufficient, but not necessar. Elain what this means. 90. Elain wh the vertical-line test used to identif functions in rectangular coordinates does not work for equations eressed in olar coordinates. Are You Preared? Answers. -, 6. cos A cos B + sin A sin B = 9. dd