9.1 POLAR COORDINATES

Transcription

1 9. Pola Coodinates Contempoay Calculus 9. POLAR COORDINATES The ectangula coodinate system is immensely useful, but it is not the only way to assign an addess to a point in the plane and sometimes it is not the most useful. In many expeimental situations, ou location is fixed and we o ou instuments, such as ada, take eadings in diffeent diections (Fig. ); this infomation can be gaphed using ectangula coodinates (e.g., with the angle on the hoizontal axis and the measuement on the vetical axis). Sometimes, howeve, it is moe useful to plot the infomation in a way simila to the way in which it was collected, as magnitudes along adial lines (Fig. ). This system is called the Pola Coodinate System. 90 shape 0 you Taking Measuements angle distance 0 0 feet Numeical Data Fig. distance (feet) angle (degees) Rectangula Coodinate Gaph of Data 90 In this section we intoduce pola coodinates and examine some of thei uses. We stat with gaphing points and functions in pola coodinates, conside how to change back and foth between the ectangula and pola coodinate systems, and see how to find the slopes of lines tangent to pola gaphs. Ou pimay easons fo consideing pola coodinates, howeve, ae that they appea in applications, and that they povide a "natual" and easy way to epesent some kinds of infomation. Example : SOS! You've just eceived a distess signal fom a ship located at A on you ada sceen (Fig. 3). Descibe its location to you captain so you vessel can speed to the escue. Solution: You could convet the elative location of the othe ship to ectangula coodinates and then tell you captain to go due east fo 7.5 miles and noth fo 3 B 90 5 A 0 5 D Pola Coodinate Gaph of Data Fig. 3 C N Fig. 3 A 7.5 E N 5 A E

2 9. Pola Coodinates Contempoay Calculus miles, but that cetainly is not the quickest way to each the othe ship. It is bette to tell the captain to sail fo 5 miles in the diection of 60. If the distessed ship was at B on the ada sceen, you vessel should sail fo 0 miles in the diection 50. (Real ada sceens have 0 at the top of the sceen, but the convention in mathematics is to put 0 in the diection of the positive x axis and to measue positive angles counteclockwise fom thee. And a eal sailo speaks of "beaing" and "ange" instead of diection and magnitude.) Pactice : Descibe the locations of the ships at C and D in Fig. 3 by giving a distance and a diection to those ships fom you cuent position at the cente of the ada sceen. Points in Pola Coodinates (, ) To constuct a pola coodinate system we need a stating point (called the oigin o pole) fo the magnitude measuements and a stating diection oigin pola axi (called the pola axis) fo the angle measuements (Fig. 4). A pola Fig. 4 coodinate pai fo a point P in the plane is an odeed pai (,) whee is the diected distance along a adial line fom O to P, and is the angle fomed by the pola axis and the segment OP (Fig. 4). The angle is positive when the angle of the adial line OP is measued counteclockwise fom the pola axis, and is negative when measued clockwise. Degee o Radian Measue fo? Eithe degee o adian measue can be used fo the angle in the pola coodinate system, but when we diffeentiate and integate tigonometic functions of we will want all of the angles to be given in adian measue. Fom now on, we will pimaily use adian measue. You should assume that all angles ae given in adian measue unless the units " " ("degees") ae shown. Example : Plot the points with the given pola coodinates: A(, 30 ), B(3, /), C(, /6), and D( 3, 70 ). Solution: To find the location of A, we look along the ay that makes an angle of 30 with the pola axis, and then take two steps in that diection (assuming step = unit). The locations of A and B ae shown in Fig. 5. To find the location of C, we look along the ay which makes an angle of /6 with 90 (/) B the pola axis, and then we take two steps backwads since = is negative. Fig. 6 shows the locations of C and D. Notice that the points B and D have diffeent addesses, (3, /) and ( 3, 70 ), but A 30 (/6) the same location. 3 Fig. 5

3 9. Pola Coodinates Contempoay Calculus 3 Pactice : Plot the points with the given pola coodinates: A(, /), B(, 0 ), C(, /3), D(, 35 ), and E(, 35 ). Which two points coincide? Each pola coodinate pai (,) gives the location of one point, but each location has lots of diffeent addesses in the pola coodinate system: the pola coodinates of a point ae not unique. This nonuniqueness of addesses comes about in two ways. Fist, the angles, ± 360, ±. 360,... all descibe the same adial line (Fig. 7), so the pola coodinates (, ), (, ± 360 ), (, ±. 360 ),... all locate the same point. Secondly, the angle ± 80 descibes the adial line pointing in exactly the opposite (, ±360 ) (, ) diection fom the adial line decibed by the angle (Fig. 8), so the pola coodinates (, ) and (, ± 80 ) locate the same point. A pola coodinate pai gives the location of exacly one point, but the location of one point is descibed by many (an infinite numbe) diffeent pola Fig. 7 coodinate pais. C D 70 (3/) Fig Fig (/6 (, ±80 ) (, ) 80 Note: In the ectangula coodinate system we use (x, y) and y = f(x): fist vaiable independent and second vaiable dependent. In the pola coodinate system we use (, ) and = f(): fist vaiable dependent and second vaiable independent, a evesal fom the ectangula coodinate usage. Pactice 3: Table contains measuements to the edge of a plateau taken by a emote senso which cashed on the plateau. Fig. 9 shows the data plotted in ectangula coodinates. Plot the data in pola coodinates and detemine the shape of the top of the plateau. angle distance angle distance 0 8 feet 50 feet Table distance (feet) angle (degees) Rectangula Coodinate Gaph of Data Fig. 9

4 9. Pola Coodinates Contempoay Calculus 4 Gaphing Functions in the Pola Coodinate System In the ectangula coodinate system, we have woked with functions given by tables of data, by gaphs, and by fomulas. Functions can be epesented in the same ways in pola coodinates. If a function is given by a table of data, we can gaph the function in pola coodinates by plotting individual points in a pola coodinate system and connecting the plotted points to see the shape of the gaph. By hand, this is a tedious pocess; by calculato o compute, it is quick and easy. If the function is given by a ectangula coodinate gaph of magnitude as a function of angle, we can ead coodinates of points on the ectangula gaph and eplot them in pola coodinates. In essence, as we go fom the ectangula coodinate gaph to the pola coodinate gaph we "wap" the ectangula gaph aound the "pole" at the oigin of the pola coodinate system. (Fig. 0) (a) ectangula coodinates (b) Fig. 0 If the function is given by a fomula, we (o ou calculato) can gaph the function to help us obtain infomation about its behavio. Typically, a gaph is ceated by evaluating the function at a lot of points and then plotting the points in the pola coodinate system. Some of the following examples illustate that functions given by simple fomulas may have athe exotic gaphs in the pola coodinate system. If a function is aleady given by a pola coodinate gaph, we can use the gaph to answe questions about the behavio of the function. It is usually easy to locate the maximum value(s) of on a pola coodinate gaph, and, by moving counteclockwise aound the gaph, we can obseve whee is inceasing, constant, o deceasing. Example 3: Gaph = and = in the pola coodinate system fo 0. Solution: = : In evey diection, we simply move units along the adial line and plot a point. The esulting pola gaph (Fig. b) is a the gaph of a constant y = k is a hoizontal line. In the pola coodinate cicle centeed at the oigin with a adius of. In the ectangula coodinate system, = = system, the gaph of a constant = k is a cicle with adius k. (a) ectangula coodinates (b) pola coodinates Fig.

5 9. Pola Coodinates Contempoay Calculus 5 = : The ectangula coodinate gaph of = is shown in Fig. a. If we ead the values of and fom the ectangula coodinate gaph and plot them in pola coodinates, the esult is the shape in Fig. b. The diffeent line thicknesses ae used in the figues to = = (a) ectangula coodinates (b) pola coodinates Fig. help you see which values fom the ectangula gaph became which pats of the cicle in the pola gaph. Pactice 4: Example 4: Gaph = and = cos() in the pola coodinate system. Gaph = and = + sin() in the pola coodinate system. Solution: = : The ectangula coodinate gaph of = is a staight line (Fig. 3a). If we ead the values of and fom the ectangula coodinate gaph and plot them in pola coodinates, the esult is the spial, called an Achimedean spial, in Fig. 3b. = + sin(): The ectangula coodinate gaph of = + sin() is shown in Fig. 4a, and it is the gaph of sine cuve shifted up unit. In pola coodinates, the esult of adding to sine is much less obvious and is shown in Fig. 4b. (a) = ectangula coodinates = + sin() (a) ectangula coodinates Fig. 3 = + sin() Fig. 4 = (b) pola coodinates / (b) pola coodinates /4 0 angle distance (adians) (metes) /6.9 /4.7 /3.6 /.0 Table Pactice 5: Plot the points in Table in the pola coodinate system and connect them with a smooth cuve. Descibe the shape of the gaph in wods.

6 9. Pola Coodinates Contempoay Calculus 6 Fig. 5 shows the effects of adding vaious constants to the ectangula and pola gaphs of = sin(). In ectangula coodinates the esult is a gaph shifted up o down by k units. In pola coodinates, the esult may be a gaph with an entiely diffeent shape (Fig. 6). = + sin() =.5 + sin() = + sin() = sin() = 0 + sin() Fig. 5: Rectangula Coodinates / 3 /4 / /4 / /4 / / /4 /4 = + sin() =.5 + sin() = + sin() Fig. 6: Pola Coodinates = sin() = 0 + sin() Fig. 7 shows the effects of adding a constant to the independent vaiable in ectangula coodinates, and the esult is a hoizontal shift of the oiginal gaph. In pola coodinates, Fig. 8, the esult is a otation of the oiginal gaph. Geneally it is difficult to find fomulas fo otated figues in ectangula coodinates, but otations ae easy in pola coodinates. = + sin() = + sin( /4) = + sin( /) = + sin( ) 3 Fig. 7: Rectangula Coodinates / / / / /4 /4 /4 /4 = + sin() = + sin( /4) = + sin( /) Fig. 8: Pola Coodinates = + sin( )

7 9. Pola Coodinates Contempoay Calculus 7 The fomulas and names of seveal functions with exotic shapes in pola coodinates ae given in the poblems. Many of them ae difficult to gaph "by hand," but by using a gaphing calculato o compute you can enjoy the shapes and easily examine the effects of changing some of the constants in thei fomulas. Conveting Between Coodinate Systems Sometimes both ectangula and pola coodinates ae needed in the same application, and it is necessay to change back and foth between the systems. In such a case we typically place the two x =. cos( ) x + y = y y =. sin( ) y oigins togethe and align the pola axis with tan( ) = x the positive x axis. Then the convesions y ae staightfowad execises using x tigonomety and ight tiangles (Fig. 9). (a) (b) x Fig. 9 Pola to Rectangula (Fig. 9a) Rectangula to Pola (Fig. 9b) x =. cos() = x + y y =. sin() tan() = y x (if x 0) Example 5: Convet (a) the pola coodinate point P(7, 0.4) to ectangula coodinates, and (b) the ectangula coodinate point R(, 5) to pola coodinates. Solution: (a) = 7 and = 0.4 (Fig. 0) so x =. cos() = 7. cos(0.4) = 7(0.9) = and y =7. sin(0.4) = 7(0.389) =.73. (b) x = and y = 5 so = x + y = = 69 and tan() = y/x = 5/ so we can take = 3 and y = actan(5/) The pola coodinate addesses =7 (3, ± n. ) and ( 3, ± (n+). ) give the location =0.4 x of the same point..73 Fig. 0 The convesion fomulas can also be used to convet function equations fom one system to the othe. y y = 3x + 5 Example 6: Convet the ectangula coodinate linea equation y = 3x + 5 (Fig. ) to a pola coodinate equation. 5 Fig. x

8 9. Pola Coodinates Contempoay Calculus 8 Solution: This simply equies that we eplace x with. cos() and y with. sin(). Then y = 3x + 5 becomes. sin() = 3. cos() + 5 so. (sin() 3cos()) = 5 and = 5/(sin() 3cos()). This final epesentation is valid only fo such that sin() 3cos() 0. Pactice 6: Convet the pola coodinate equation = 4. sin() to a ectangula coodinate equation. Example 7: Robotic Am: A obotic am has a hand at the end of a inch long foeam which is connented to an 8 inch long uppe am (Fig. ). Detemine the position of the hand, ealtive to the shoulde, when = 45 (/4) and φ = 30 (/6). shoulde foeam uppe am elbow hand Solution: The hand is. cos(/4 + /6) 3. inches to the ight of the elbow (Fig. 3) and sin((/4 + /6).6 inches above φ the elbow. Similaly, the elbow is 8. cos(/4).7 3. inches to the ight of the shoulde and 8. sin(/4).7 inches above the shoulde. Finally, the hand is appoximately = 5.8 inches to the ight of the shoulde and Fig appoximately = 4.3 inches above the shoulde. In pola coodinates, the hand is appoximately 9 inches fom the shoulde, at an angle of about 45 Fig (about adians) above the hoizontal. Pactice 7: Detemine the position of the hand, elative to the shoulde, when = 30 and φ = 45. Gaphing Functions in Pola Coodinates on a Calculato o Compute Some calculatos and computes ae pogammed to gaph pola functions simply by keying in the fomula fo, eithe as a function of o of t, but othes ae only designed to display ectangula coodinate gaphs. Howeve, we can gaph pola functions on most of them as well by using the ectangula to pola convesion fomulas, selecting the paametic mode (and the adian mode) on the calculato, and then gaphing the esulting paametic equations in the ectangula coodinate system: To gaph = () fo between 0 and 3, define x(t) = (t). cos(t) and y(t) = (t). sin(t) and gaph the paametic equations x(t), y(t) fo t taking values fom 0 to 9.43.

9 9. Pola Coodinates Contempoay Calculus 9 Which Coodinate System Should You Use? Thee ae no igid ules. Use whicheve coodinate system is easie o moe "natual" fo the poblem o data you have. Sometimes it is not clea which system to use until you have gaphed the data both ways, and some poblems ae easie if you switch back and foth between the systems. Geneally, the pola coodinate system is easie if the data consists of measuements in vaious diections (ada) you poblem involves locations in elatively featueless locations (desets, oceans, sky) otations ae involved. Typically, the ectangula coodinate system is easie if the data consists of measuements given as functions of time o location (tempeatue, height) you poblem involves locations in situations with an established gid (a city, a chess boad) tanslations ae involved. PROBLEMS. Give the locations in pola coodinates (using adian measue) of the points labeled A, B, and C in Fig. 4.. Give the locations in pola coodinates (using adian measue) of the F E C 0 A 30 points labeled D, E, and F in Fig Give the locations in pola coodinates (using adian measue) of the B D points labeled A, B, and C in Fig. 5. Fig Give the locations in pola coodinates (using adian measue) of the A points labeled D, E, and F in Fig. 5. In poblems 5 8, plot the points A D in pola coodinates, connect the dots by line segments in ode (A to B to C to D to A), and name the B E F 0 30 appoximate shape of the esulting figue. C D 5. A(3, 0 ), B(, 0 ), C(, 00 ), and D(.8, 35 ). 6. A(3, 30 ), B(, 30 ), C(3, 50 ), and D(, 80 ). 7. A(, 0.75), B(3,.69), C(,.68), and D(3, 4.887). Fig A(3, 0.54), B(,.69), C(3,.68), and D(, 4.887).

10 9. Pola Coodinates Contempoay Calculus 0 In poblems 9 4, the ectangula coodinate gaph of a function = () is shown. Sketch the pola coodinate gaph of = (). 9. The gaph in Fig The gaph in Fig. 7.. The gaph in Fig. 8. = () Fig. 6 = () Fig. 7 = () Fig. 8. The gaph in Fig The gaph in Fig The gaph in Fig. 3. = () Fig. 9 = () Fig. 30 = () Fig The ectangula coodinate gaph of = f() is shown in Fig. 3. (a) Sketch the ectangula coodinate gaphs of = + f(), = + f(), and = + f(). (b) Sketch the pola coodinate gaphs of = f(), = + f(), = + f(), and = + f(). = f() Fig The ectangula coodinate gaph of = g() is shown in Fig. 33. (a) Sketch the ectangula coodinate gaphs of = + g(), = + g(), = g() and = + g(). (b) Sketch the pola coodinate gaphs of = g(), = + g(), = + g(), and = + g(). 7. The ectangula coodinate gaph of = f() is shown in Fig. 34. Fig. 33 (a) Sketch the ectangula coodinate gaphs of = + f(), = + f(), and = + f(). (b) Sketch the pola coodinate gaphs of = f(), = + f(), = + f(), and = + f(). = f() Fig. 34

11 9. Pola Coodinates Contempoay Calculus 8. The ectangula coodinate gaph of = g() is shown in Fig. 35. (a) Sketch the ectangula coodinate gaphs of = + g(), = + g(), and = + g(). (b) Sketch the pola coodinate gaphs of = g(), = + g(), = + g(), and = + g(). = g() Fig Suppose the ectangula coodinate gaph of = f() has the hoizontal asymptote = 3 as gows abitaily lage. What does that tell us about the pola coodinate gaph of = f() fo lage values of. 0. Suppose the ectangula coodinate gaph of = f() has the vetical asymptote = /6: lim /6 What does that tell us about the pola coodinate gaph of = f() fo values of nea /6? f() = +. A compute o gaphing calculato is ecommended fo the poblems maked with a *. In poblems 40, gaph the functions in pola coodinates fo 0.. = 3. = 5 3. = /6 4. = 5/3 5. = 4. sin() 6. =. cos() 7. = + sin() 8. = + sin() 9. = + 3. sin() 30. = sin() *3. = tan() *3. = + tan() *33. = 3 cos() *34. = sin() *35. = sin() + cos() 36. = 37. =. 38. = 39. = 40. = sin(). cos(3) *4. = sin(m). cos(n) poduces lovely gaphs fo vaious small intege values of m and n. Go exploing with a gaphic calculato to find values of m and n which esult in shapes you like. *4. Gaph = *43. Gaph = cos( + a) cos( a), 0, fo a = 0, /6, /4, and /. How ae the gaphs elated?, 0, fo a = 0, /6, /4, and /. How ae the gaphs elated? *44. Gaph = sin(n), 0, fo n =,, 3, and 4 and count the numbe of "petals" on each gaph. Pedict the numbe of "petals" fo the gaphs of = sin(n) fo n = 5, 6, and 7, and then test you pediction by ceating those gaphs. *45. Repeat poblem 34 using = cos(n).

12 9. Pola Coodinates Contempoay Calculus In poblems 46 49, convet the ectangula coodinate locations to pola coodinates. 46. (0, 3), (5, 0), and (, ) 47. (, 3), (, 3), and (0, 4). 48. (0, ), (4, 4), and (3, 3) 49. (3, 4), (, 3), and ( 7, ). In poblems 50 53, convet the pola coodinate locations to ectangula coodinates. 50. (3, 0), (5, 90 ), and (, ) 5. (,3), (, 3), and (0, 4). 5. (0,3), (5,0), and (,) 53. (,3), (, 3), and (0,4). hand Poblems efe to the obotic am in Fig. 36. obotic am 0 in. φ 54. Detemine the position of the hand, ealtive to the shoulde, when = 60 and φ = in. elbow 55. Detemine the position of the hand, elative to the shoulde, when = 30 and φ = 30. shoulde Fig Detemine the position of the hand, elative to the shoulde, when = 0.6 and φ = Detemine the position of the hand, elative to the shoulde, when = 0.9 and φ = Suppose the obot's shoulde can pivot so that / /, but the elbow is boken and φ is always 0. Sketch the points the hand can each. 59. Suppose the obot's shoulde can pivot so that / /, and the elbow can pivot so that / φ /. Sketch the points the hand can each. 60. Suppose the obot's shoulde can pivot so that / /, and the elbow can pivot completely so φ. Sketch the points the hand can each. *6. Gaph = + a. fo 0 and a = 0.5, 0.8,,.5, and. What shapes do the cos() vaious values of a poduce? *6. Repeat poblem 6 with = + a. sin().

13 9. Pola Coodinates Contempoay Calculus 3 Some Exotic Cuves (and Names) Many of the following cuves wee discoveed and named even befoe pola coodinates wee invented. In most cases the path of a point moving on o aound some object is descibed. You may enjoy using you calculato to gaph some of these cuves o you can invent you own exotic shapes. (An inexpensive souce fo these shapes and names is A Catalog Of Special Plane Cuves by J. Dennis Lawence, Dove Publications, 97, and the page efeences below ae to that book ) Some Classics: Cissoid ("like ivy") of Diocles (about 00 B.C.): = a sin(). tan() p. 98 Right Stophoid ("twisting") of Baow (670): = a( sec() cos() ) p. 0 Tisectix of Maclauin (74): = a sec() 4a cos() p. 05 Lemniscate ("ibbon") of Benoulli (694): = a cos() p. Conchoid ("shell") of Nicomedes (5 B.C.): = a + b. sec() p. 37 Hippopede ("hose fette") of Poclus (about 75 B.C.): = 4b( a b sin () ) p. 44 b = 3, a =,, 3, 4 Devil's Cuve of Came (750): (sin () cos () ) = a sin () b cos () p. 5 a=, b=3 Nephoid ("kidney") of Feeth: = a. ( + sin( ) ) p. 75 a = 3 Some of ou own: (Based on thei names, what shapes do you expect fo the following cuves?) Piscatoid of Pat (99): = 3cos() fo.. Window x: (, ) and y: (, ) cos() Kemitoid of Kelcey (99) : =.5. sin(). ( 4.7). INT(/) + { 5. sin 3 () 3. sin 9 ()}. { INT(/) } fo 0 Window x: ( 3, 3) and y: (, 4) Bovine Oculoid: = + INT( /() ) fo 0 6 ( 8.85) Window x: ( 5, 5) and y: ( 4, 4) A Few Refeence Facts The pola fom of the linea equation Ax + By + C = 0 is. ( A. cos() + B. sin() ) + C = 0 The equation of the line though the pola coodinate points (, ) and (, ) is. {. sin( ) +. sin( ) } =.. sin( ) The gaph of = a. sin() + b. cos() is a cicle though the oigin with cente (b/, a/) and adius a + b. (Hint: multiply each side by, and then convet to ectangula coodinates.) The equations = oigin. ± a. cos() and = ± a. sin() ae conic sections with one focus at the If a <, the denominato is neve 0 fo 0 < and the gaph is an ellipse. If a =, the denominato is 0 fo one value of, 0 <, and the gaph is a paabola.

14 9. Pola Coodinates Contempoay Calculus 4 If a >, the denominato is 0 fo two values of, 0 <, and the gaph is a hypebola. Section 9. PRACTICE Answes Pactice : Point C is at a distance of 0 miles in the diection 30 o. D is 5 miles at 70 o. Pactice : The points ae plotted in Fig. 37. Pactice 3: See Fig. 38. The top of the plateau is oughly ectangula. 35 E 0 C B 35 /90 /3 A D Fig Pactice 4: The gaphs ae shown in Figs. 39 and 40. Note that the gaph of = cos() taces out a = = (a) ectangula coodinates Fig. 39 (b) pola coodinates 70 Fig. 38 = cos() = cos() 0 / /4 = cos() / /4 0 0 (a) ectangula coodinates (b) pola coodinates Fig. 40 cicle twice; once as goes fom 0 to, and a second time as goes fom to.

15 9. Pola Coodinates Contempoay Calculus 5 Pactice 5: The points ae plotted in Fig. 4. The points (almost) lie on a staight line. / /3 /6 0 3 Fig. 4 Pactice 6: = x + y and. sin() = y so = 4. sin() becomes x + y = 4y. Putting this last equation into the standad fom fo a cicle (by completing the squae) we have x + (y ) = 4, the equation of a cicle with cente at (0, ) and adius. Pactice 7: See Fig. 4. Fo point A, the "elbow," elative to O, the "shoulde:" x = 8. cos(30 o ) 5.6 inches and y = 8. sin(30 o ) = 9 inches. Fo point B, the "hand," elative to A: x =. cos(75 o ) 3. inches and y =. sin(75 o ).6 inches. Then the etangula coodinate location of the B elative to O is x = 8.7 inches and y = 0.6 inches. The pola coodinate location of B elative to O is = x + y 7.8 inches and 47.7 o (o 0.83 adians) 3. B O A 9 30 Fig. 4.6