Polar Coordinates and Polar Graphs. Polar Coordinates

Transcription

1 _1.qxd 11// :5 PM Pge 79 SECTION 1. Polr Coordintes nd Polr Grphs 79 O Polr coordintes Figure 1. Section 1. r = directed distnce θ = directed ngle P = (r, θ) Polr xis Polr Coordintes nd Polr Grphs Understnd the polr coordinte system. Rewrite rectngulr coordintes nd equtions in polr form nd vice vers. Sketch the grph of n eqution given in polr form. Find the slope of tngent line to polr grph. Identify severl types of specil polr grphs. Polr Coordintes So fr, you hve been representing grphs s collections of points x, y on the rectngulr coordinte system. The corresponding equtions for these grphs hve been in either rectngulr or prmetric form. In this section you will stu coordinte system clled the polr coordinte system. To form the polr coordinte system in the plne, fix point O, clled the pole (or origin), nd construct from O n initil ry clled the polr xis, s shown in Figure 1.. Then ech point P in the plne cn be ssigned polr coordintes r,, s follows. r directed distnce from O to P directed ngle, counterclockwise from polr xis to segment OP Figure 1.7 shows three points on the polr coordinte system. Notice tht in this system, it is convenient to locte points with respect to grid of concentric circles intersected by rdil lines through the pole. θ = (, ) 1 θ = (, ) θ = (, ) () Figure 1.7 (b) (c) POLAR COORDINATES The mthemticin credited with first using polr coordintes ws Jmes Bernoulli, who introduced them in 191. However, there is some evidence tht it my hve been Isc Newton who first used them. With rectngulr coordintes, ech point x, y hs unique representtion. This is not true with polr coordintes. For instnce, the coordintes r, nd r, represent the sme point [see prts (b) nd (c) in Figure 1.7]. Also, becuse r is directed distnce, the coordintes r, nd r, represent the sme point. In generl, the point r, cn be written s r, r, n or r, r, n 1 where n is ny integer. Moreover, the pole is represented by,, where is ny ngle.

2 _1.qxd 11// :5 PM Pge 7 7 CHAPTER 1 Conics, Prmetric Equtions, nd Polr Coordintes Pole y θ (Origin) r x (r, θ) (x, y) y x Polr xis (x-xis) Relting polr nd rectngulr coordintes Figure 1.8 Coordinte Conversion To estblish the reltionship between polr nd rectngulr coordintes, let the polr xis coincide with the positive x-xis nd the pole with the origin, s shown in Figure 1.8. Becuse x, y lies on circle of rdius r, it follows tht r x y. Moreover, for r >, the definition of the trigonometric functions implies tht tn y cos x nd sin y x, r, r. If r <, you cn show tht the sme reltionships hold. THEOREM 1.1 Coordinte Conversion The polr coordintes r, of point re relted to the rectngulr coordintes x, y of the point s follows. 1. x r cos. tn y x y y r sin r x y (r, θ) = (, ) 1 1 (x, y) = (, ) 1 θ (, ) (r, ) = 1 (x, y) = (, ) To convert from polr to rectngulr coordintes, let x r cos nd y r sin. Figure 1.9 x EXAMPLE 1 Polr-to-Rectngulr Conversion. For the point r,,, x r cos cos nd y r sin sin. So, the rectngulr coordintes re x, y,. b. For the point r,,, x cos nd So, the rectngulr coordintes re See Figure 1.9. y sin x, y,.. EXAMPLE Rectngulr-to-Polr Conversion θ (, ) (r, ) = (x, y) = ( 1, 1) 1 1 y (r, θ ) = (, ) (x, y) = (, ) 1 To convert from rectngulr to polr coordintes, let tn nd r x y. Figure 1. yx x. For the second qudrnt point x, y 1, 1, tn y x 1. Becuse ws chosen to be in the sme qudrnt s x, y, you should use positive vlue of r. r x y 1 1 This implies tht one set of polr coordintes is r,,. b. Becuse the point x, y, lies on the positive y-xis, choose nd r, nd one set of polr coordintes is r,,. See Figure 1..

3 _1.qxd 11// :5 PM Pge 71 SECTION 1. Polr Coordintes nd Polr Grphs 71 Polr Grphs One wy to sketch the grph of polr eqution is to convert to rectngulr coordintes nd then sketch the grph of the rectngulr eqution. 1 EXAMPLE Grphing Polr Equtions () Circle: r (b) Rdil line: 1 1 (c) Verticl line: r sec Figure Describe the grph of ech polr eqution. Confirm ech description by converting to rectngulr eqution.. r b. c. r sec Solution. The grph of the polr eqution r consists of ll points tht re two units from the pole. In other words, this grph is circle centered t the origin with rdius of. [See Figure 1.1().] You cn confirm this by using the reltionship r x y to obtin the rectngulr eqution x y. Rectngulr eqution b. The grph of the polr eqution consists of ll points on the line tht mkes n ngle of with the positive x-xis. [See Figure 1.1(b).] You cn confirm this by using the reltionship tn yx to obtin the rectngulr eqution y x. Rectngulr eqution c. The grph of the polr eqution r sec is not evident by simple inspection, so you cn begin by converting to rectngulr form using the reltionship r cos x. r sec r cos 1 x 1 Polr eqution Rectngulr eqution From the rectngulr eqution, you cn see tht the grph is verticl line. [See Figure 1.1(c.)] TECHNOLOGY Sketching the grphs of complicted polr equtions by hnd cn be tedious. With technology, however, the tsk is not difficult. If your grphing utility hs polr mode, use it to grph the equtions in the exercise set. If your grphing utility doesn t hve polr mode, but does hve prmetric mode, you cn grph r f by writing the eqution s x f cos y f sin. For instnce, the grph of r 1 shown in Figure 1. ws produced with grphing clcultor in prmetric mode. This eqution ws grphed using the prmetric equtions x 1 cos Spirl of Archimedes Figure 1. y 1 sin with the vlues of vrying from to. This curve is of the form r nd is clled spirl of Archimedes.

4 _1.qxd 11// : PM Pge 7 7 CHAPTER 1 Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE Sketching Polr Grph NOTE One wy to sketch the grph of r cos by hnd is to mke tble of vlues. By extending the tble nd plotting the points, you will obtin the curve shown in Exmple. r Sketch the grph of r cos. Solution Begin by writing the polr eqution in prmetric form. x cos cos nd y cos sin After some experimenttion, you will find tht the entire curve, which is clled rose curve, cn be sketched by letting vry from to, s shown in Figure 1.. If you try duplicting this grph with grphing utility, you will find tht by letting vry from to, you will ctully trce the entire curve twice Figure 1. 5 Use grphing utility to experiment with other rose curves they re of the form r cos n or r sin n. For instnce, Figure 1. shows the grphs of two other rose curves. r =.5 cos θ r = sin 5θ... Rose curves Figure 1. Generted by Derive

5 _1.qxd 11// : PM Pge 7 SECTION 1. Polr Coordintes nd Polr Grphs 7 r = f( θ) Tngent line (r, θ) Slope nd Tngent Lines To find the slope of tngent line to polr grph, consider differentible function given by r f. To find the slope in polr form, use the prmetric equtions x r cos f cos nd y r sin f sin. Using the prmetric form of given in Theorem 1.7, you hve d d f cos f sin f sin f cos which estblishes the following theorem. Tngent line to polr curve Figure 1.5 THEOREM 1.11 Slope in Polr Form If f is differentible function of, then the slope of the tngent line to the grph of r f t the point r, is d d f cos f sin f sin f cos provided tht t r,. See Figure 1.5. d From Theorem 1.11, you cn mke the following observtions. 1. Solutions to yield horizontl tngents, provided tht.. Solutions to yield verticl tngents, provided tht. If nd re simultneously, no conclusion cn be drwn bout tngent lines. d d d d d d EXAMPLE 5 Finding Horizontl nd Verticl Tngent Lines Find the horizontl nd verticl tngent lines of r sin,. ( 1, ) (, ) 1,, Horizontl nd verticl tngent lines of r sin Figure 1. Solution nd Begin by writing the eqution in prmetric form. x r cos sin cos y r sin sin sin sin Next, differentite x nd y with respect to nd set ech derivtive equl to. cos sin cos d sin cos sin d,, So, the grph hs verticl tngent lines t, nd,, nd it hs horizontl tngent lines t, nd 1,, s shown in Figure 1..

6 _1.qxd 11// : PM Pge 7 7 CHAPTER 1 Conics, Prmetric Equtions, nd Polr Coordintes EXAMPLE Finding Horizontl nd Verticl Tngent Lines (, ) (, ) (, ) ( 1, ) ( 1, 5 ) Horizontl nd verticl tngent lines of r 1 cos Figure 1.7 Find the horizontl nd verticl tngents to the grph of r 1 cos. Solution Using y r sin, differentite nd set equl to. 1 cos cos sin sin d So, cos 1 nd cos 1, nd you cn conclude tht when, nd. Similrly, using x r cos, you hve, sin cos sin sin cos 1. d y r sin 1 cos sin cos 1cos 1 x r cos cos cos So, sin or cos 1, nd you cn conclude tht when,, nd 5. From these results, nd from the grph shown in Figure 1.7, you cn conclude tht the grph hs horizontl tngents t, nd,, nd hs verticl tngents t 1,, 1, 5, nd,. This grph is clled crdioid. Note tht both derivtives nd d re when Using this informtion lone, you don t know whether the grph hs horizontl or verticl tngent line t the pole. From Figure 1.7, however, you cn see tht the grph hs cusp t the pole. d d d d., f( θ) = cos θ Theorem 1.11 hs n importnt consequence. Suppose the grph of r f psses through the pole when nd f. Then the formul for simplifies s follows. f sin f cos f sin f cos f sin f cos sin tn cos So, the line is tngent to the grph t the pole,,. This rose curve hs three tngent lines,, nd t the pole. Figure THEOREM 1.1 Tngent Lines t the Pole If f nd f, then the line is tngent t the pole to the grph of r f. Theorem 1.1 is useful becuse it sttes tht the zeros of r f cn be used to find the tngent lines t the pole. Note tht becuse polr curve cn cross the pole more thn once, it cn hve more thn one tngent line t the pole. For exmple, the rose curve f cos hs three tngent lines t the pole, s shown in Figure 1.8. For this curve, f cos is when is,, nd 5. Moreover, the derivtive f sin is not for these vlues of.

7 _1.qxd 11// : PM Pge 75 SECTION 1. Polr Coordintes nd Polr Grphs 75 Specil Polr Grphs Severl importnt types of grphs hve equtions tht re simpler in polr form thn in rectngulr form. For exmple, the polr eqution of circle hving rdius of nd centered t the origin is simply r. Lter in the text you will come to pprecite this benefit. For now, severl other types of grphs tht hve simpler equtions in polr form re shown below. (Conics re considered in Section 1..) Limçons r ± b cos r ± b sin >, b > b < 1 Limçon with inner loop b 1 Crdioid (hert-shped) 1 < b < Dimpled limçon b Convex limçon Rose Curves n petls if n is odd n petls if n is even n n = n = n = 5 n = r cos Rose curve n r cos Rose curve n r sin Rose curve n r sin Rose curve n Circles nd Lemnisctes r cos Circle r sin Circle r sin Lemniscte r cos Lemniscte TECHNOLOGY The rose curves described bove re of the form r cos n or r sin n, where n is positive integer tht is greter thn or equl to. Use grphing utility to grph r cos n or r sin n for some noninteger vlues of n. Are these grphs lso rose curves? For exmple, try sketching the grph of r cos,. Generted by Mple FOR FURTHER INFORMATION For more informtion on rose curves nd relted curves, see the rticle A Rose is Rose... by Peter M. Murer in The Americn Mthemticl Monthly. The computer-generted grph t the left is the result of n lgorithm tht Murer clls The Rose. To view this rticle, go to the website

8 _1.qxd 11// : PM Pge 7 7 CHAPTER 1 Conics, Prmetric Equtions, nd Polr Coordintes Exercises for Section 1. See for worked-out solutions to odd-numbered exercises. In Exercises 1, plot the point in polr coordintes nd find the corresponding rectngulr coordintes for the point. (c) (d) 1.,., 7.,., 7 5.,.., 1.57 In Exercises 7 1, use the ngle feture of grphing utility to find the rectngulr coordintes for the point given in polr coordintes. Plot the point. 7. 5, 8., , , 1. In Exercises 11 1, the rectngulr coordintes of point re given. Plot the point nd find two sets of polr coordintes for the point for < , 1 1., 5 1., 1., 15., 1 1. In Exercises 17, use the ngle feture of grphing utility to find one set of polr coordintes for the point given in rectngulr coordintes. 17., 18., 19.., 5 1. Plot the point,.5 if the point is given in () rectngulr coordintes nd (b) polr coordintes.. Grphicl Resoning () Set the window formt of grphing utility to rectngulr coordintes nd locte the cursor t ny position off the xes. Move the cursor horizontlly nd verticlly. Describe ny chnges in the displyed coordintes of the points. (b) Set the window formt of grphing utility to polr coordintes nd locte the cursor t ny position off the xes. Move the cursor horizontlly nd verticlly. Describe ny chnges in the displyed coordintes of the points. (c) Why re the results in prts () nd (b) different? In Exercises, mtch the grph with its polr eqution. [The grphs re lbeled (), (b), (c), nd (d).] () 5, (b),. r sin. r cos 5. r 1 cos. r sec In Exercises 7, convert the rectngulr eqution to polr form nd sketch its grph. 7. x y 8. x y x 9. y. x x y xy y 9x. x y 9x y In Exercises 5, convert the polr eqution to rectngulr form nd sketch its grph. 5. r. r 7. r sin 8. r 5 cos 9. r. 1. r sec. r csc In Exercises 5, use grphing utility to grph the polr eqution. Find n intervl for over which the grph is trced only once.. r cos. r 51 sin 5. r sin. r cos 7. r 8. r 1 cos sin r sin 5 r cos 51. r sin 5. r 1 5. Convert the eqution r h cos k sin to rectngulr form nd verify tht it is the eqution of circle. Find the rdius nd the rectngulr coordintes of the center of the circle.

9 _1.qxd 11// : PM Pge 77 SECTION 1. Polr Coordintes nd Polr Grphs Distnce Formul () Verify tht the Distnce Formul for the distnce between the two points r 1, 1 nd r, in polr coordintes is (b) Describe the positions of the points reltive to ech other if 1. Simplify the Distnce Formul for this cse. Is the simplifiction wht you expected? Explin. (c) Simplify the Distnce Formul if 1 9. Is the simplifiction wht you expected? Explin. (d) Choose two points on the polr coordinte system nd find the distnce between them. Then choose different polr representtions of the sme two points nd pply the Distnce Formul gin. Discuss the result. In Exercises 55 58, use the result of Exercise 5 to pproximte the distnce between the two points in polr coordintes. In Exercises 59 nd, find / nd the slopes of the tngent lines shown on the grph of the polr eqution. 59. r sin. r 1 sin (, ) d r 1 r r 1 r cos1. 55.,,, ,.5, 7, ,.5, 5, 1, In Exercises 1, use grphing utility to () grph the polr eqution, (b) drw the tngent line t the given vlue of, nd (c) find / t the given vlue of. Hint: Let the increment between the vlues of equl /. 1. r 1 cos,. r cos,. r sin,. r, In Exercises 5 nd, find the points of horizontl nd verticl tngency (if ny) to the polr curve. 5. r 1 sin. r sin, 7, In Exercises 7 nd 8, find the points of horizontl tngency (if ny) to the polr curve. 7. r csc 8. r sin cos 7 1,,, 1, 1 (, ) 1 In Exercises 9 7, use grphing utility to grph the polr eqution nd find ll points of horizontl tngency. 9. r sin cos 7. r cos sec 71. r csc 5 7. r cos In Exercises 7 8, sketch grph of the polr eqution nd find the tngents t the pole. 7. r sin 7. r cos 75. r 1 sin 7. r 1 cos 77. r cos 78. r sin r sin 8. r cos In Exercises 81 9, sketch grph of the polr eqution. 81. r 5 8. r 8. r 1 cos 8. r 1 sin 85. r cos 8. r 5 sin 87. r csc 88. r sin cos 89. r 9. r r cos 9. r sin In Exercises 9 9, use grphing utility to grph the eqution nd show tht the given line is n symptote of the grph. Nme of Grph 9. Conchoid 9. Conchoid 95. Hyperbolic spirl 9. Strophoid 11. Sketch the grph of r sin over ech intervl. () (b) (c) 1. Think About It Use grphing utility to grph the polr eqution r 1 cos for () (b) nd (c) Use the grphs to describe the effect of the ngle. Write the eqution s function of sin for prt (c).. Polr Eqution r sec r csc r r cos sec Writing About Concepts, Asymptote x 1 y 1 y x 97. Describe the differences between the rectngulr coordinte system nd the polr coordinte system. 98. Give the equtions for the coordinte conversion from rectngulr to polr coordintes nd vice vers. 99. For constnts nd b, describe the grphs of the equtions r nd in polr coordintes. 1. How re the slopes of tngent lines determined in polr coordintes? Wht re tngent lines t the pole nd how re they determined? b,

10 _1.qxd 78 11// : PM CHAPTER 1 Pge 78 Conics, Prmetric Equtions, nd Polr Coordintes 1. Verify tht if the curve whose polr eqution is r f is rotted bout the pole through n ngle, then n eqution for the rotted curve is r f. True or Flse? In Exercises , determine whether the sttement is true or flse. If it is flse, explin why or give n exmple tht shows it is flse. 1. The polr form of n eqution for curve is r f sin. Show tht the form becomes 115. If r1, 1 nd r, represent the sme point on the polr coordinte system, then r1 r. () r f cos if the curve is rotted counterclockwise rdins bout the pole. (b) r f sin if the curve is rotted counterclockwise rdins bout the pole. (c) r f cos if the curve is rotted counterclockwise rdins bout the pole. 11. If r, 1 nd r, represent the sme point on the polr coordinte system, then 1 n for some integer n If x >, then the point x, y on the rectngulr coordinte system cn be represented by r, on the polr coordinte system, where r x y nd rctn y x The polr equtions r sin nd r sin hve the sme grph. In Exercises 15 18, use the results of Exercises 1 nd Write n eqution for the limçon r sin fter it hs been rotted by the given mount. Use grphing utility to grph the rotted limçon. () (b) (c) (d) (b) (c) (d) 17. Sketch the grph of ech eqution. (b) r 1 sin () r 1 sin Anmorphic Art Use the nmorphic trnsformtions 1. Write n eqution for the rose curve r sin fter it hs been rotted by the given mount. Verify the results by using grphing utility to grph the rotted rose curve. () Section Project: r y 1 nd x, 8 to sketch the trnsformed polr imge of the rectngulr grph. When the reflection (in cylindricl mirror centered t the pole) of ech polr imge is viewed from the polr xis, the viewer will see the originl rectngulr imge. () y (b) x (c) y x 5 (d) x y Prove tht the tngent of the ngle between the rdil line nd the tngent line t the point r, on the grph of r f (see figure) is given by tn r dr d. Polr curve: r = f (θ ) Tngent line ψ Rdil line P = (r, θ ) θ O A Polr xis In Exercises 19 11, use the result of Exercise 18 to find the ngle between the rdil nd tngent lines to the grph for the indicted vlue of. Use grphing utility to grph the polr eqution, the rdil line, nd the tngent line for the indicted vlue of. Identify the ngle. Polr Eqution 19. r 1 cos Vlue of 11. r 1 cos 111. r cos 11. r sin 11. r 1 cos 11. r 5 Museum of Science nd Industry in Mnchester, Englnd This exmple of nmorphic rt is from the Museum of Science nd Industry in Mnchester, Englnd. When the reflection of the trnsformed polr pinting is viewed in the mirror, the viewer sees fces. FOR FURTHER INFORMATION For more informtion on nmor- phic rt, see the rticle Anmorphisms by Philip Hickin in the Mthemticl Gzette.