Additional Topics in Trigonometry

Transcription

1 dditionl Topics in Trigonometr In hpter 9, ou studied trigonometric functions. In this chpter, ou will use our knowledge of trigonometric functions to stud dditionl topics, including soling tringles, finding res, estimting heights, representing ectors, nd writing comple numbers. In this chpter, ou should lern the following. How to use the Lw of Sines to sole oblique tringles nd how to find res of oblique tringles. (.) How to use the Lw of osines to sole oblique tringles nd how to use Heron s re Formul to find res of tringles. (.) How to write ectors, perform bsic ector opertions, nd represent ectors grphicll. (.) How to find the dot product of two ectors in the plne. (.4) How to write trigonometric forms of comple numbers nd how to multipl, diide, find powers of, nd find roots of comple numbers. (.5) Gr lkele/istockphoto.com 0,000-pound truck is prked on hill of slope d. Wht force is required to keep the truck from rolling down the hill for ring lues of d? (See Section.4, Eercise 58.) u u u u + Vectors indicte quntities tht inole both mgnitude nd direction. You cn represent ector opertions geometricll. For emple, the grphs shown boe represent ector ddition in the plne. (See Section..) 807

2 808 hpter dditionl Topics in Trigonometr. Lw of Sines Use the Lw of Sines to sole oblique tringles (S, S, or SS). Find the res of oblique tringles. Use the Lw of Sines to model nd sole rel-life problems. Introduction b c Figure. In hpter 9, ou studied techniques for soling right tringles. In this section nd the net, ou will sole oblique tringles tringles tht he no right ngles. s stndrd nottion, the ngles of tringle re lbeled,, nd, nd their opposite sides re lbeled, b, nd c, s shown in Figure.. To sole n oblique tringle, ou need to know the mesure of t lest one side nd n two other mesures of the tringle either two sides, two ngles, or one ngle nd one side. This breks down into the following four cses.. Two ngles nd n side (S or S). Two sides nd n ngle opposite one of them (SS). Three sides (SSS) 4. Two sides nd their included ngle (SS) The first two cses cn be soled using the Lw of Sines, wheres the lst two cses require the Lw of osines (see Section.). THEOREM. LW OF SINES If is tringle with sides, b, nd c, then sin b sin c sin. b h h b is cute. c is obtuse. c NOTE The Lw of Sines cn lso be written in the reciprocl form sin sin b sin. c PROOF Let h be the ltitude of either tringle. Then ou he h b sin nd h sin. Equting these two lues of h, ou he sin b sin or Note tht sin 0 nd sin 0 becuse no ngle of tringle cn he mesure of 0 or 80. In similr mnner, b constructing n ltitude from erte to side (etended), ou cn show tht sin c sin. So, the Lw of Sines is estblished. sin b sin.

3 . Lw of Sines 809 b = 8 ft Figure. 0 c 9 STUDY TIP When soling tringles, creful sketch is useful s quick test for the fesibilit of n nswer. Remember tht the longest side lies opposite the lrgest ngle, nd the shortest side lies opposite the smllest ngle. EXMPLE Gien Two ngles nd One Side S For the tringle in Figure., 0, 9, nd b 8 feet. Find the remining ngle nd sides. Solution The third ngle of the tringle is the Lw of Sines, ou he Using b 8 produces nd sin c b sin b sin sin 8 sin feet sin 9 b sin sin c sin. 8 sin feet. sin 9 EXMPLE Gien Two ngles nd One Side S pole tilts towrd the sun t n 8 ngle from the erticl, nd it csts -foot shdow. The ngle of eletion from the tip of the shdow to the top of the pole is 4. How tll is the pole? Solution From Figure., note tht 4 nd So, the third ngle is 80 8 b the Lw of Sines, ou he c = ft 4 sin c sin. ecuse c feet, the length of the pole is Figure. c sin sin sin 4.84 feet. sin 9 For prctice, tr reworking Emple for pole tht tilts w from the sun under the sme conditions.

4 80 hpter dditionl Topics in Trigonometr The mbiguous se (SS) In Emples nd, ou sw tht two ngles nd one side determine unique tringle. Howeer, if two sides nd one opposite ngle re gien, three possible situtions cn occur: () no such tringle eists, () one such tringle eists, or () two distinct tringles m stisf the conditions. THE MIGUOUS SE (SS) onsider tringle in which ou re gien, b, nd. h b sin is cute. is cute. is cute. is cute. is obtuse. is obtuse. Sketch b b b h b b h h b h Necessr condition Tringles < h None h One b One h < < b Two b None > b One possible EXMPLE Single-Solution se SS b = in. = in. For the tringle in Figure.4, inches, b inches, nd 4. Find the remining side nd ngles. 4 One solution: Figure.4 b c Solution sin b the Lw of Sines, ou he sin sin b sin Reciprocl form Multipl ech side b b. sin sin 4 Substitute for,, nd b..4. Now, ou cn determine tht Then, the remining side is c sin sin c sin sin sin 6.59 sin inches. is cute.

5 . Lw of Sines 8 b = 5 h 85 = 5 No solution: < h Figure.5 EXMPLE 4 No-Solution se SS Show tht there is no tringle for which 5, b 5, nd 85. Solution egin b mking the sketch shown in Figure.5. From this figure it ppers tht no tringle is formed. You cn erif this using the Lw of Sines. sin b sin sin b sin sin 85 sin > sin. Reciprocl form Multipl ech side b b. This contrdicts the fct tht So, no tringle cn be formed hing sides 5 nd b 5 nd n ngle of 85. EXMPLE 5 Two-Solution se SS Find two tringles for which meters, b meters, nd 0.5. Solution the Lw of Sines, ou he sin b sin sin b sin sin Reciprocl form There re two ngles, 64.8 nd , between 0 nd 80 whose sine is For 64.8, ou obtin c sin sin sin meters. sin 0.5 For 5., ou obtin c sin sin sin meters. sin 0.5 The resulting tringles re shown in Figure.6. b = m = m b = m = m Figure.6

6 8 hpter dditionl Topics in Trigonometr STUDY TIP To see how to obtin the height of the obtuse tringle in Figure.7, notice the use of the reference ngle 80 nd the difference formul for sine, s follows. h b sin80 bsin 80 cos cos 80 sin b0 cos sin b sin re of n Oblique Tringle The procedure used to proe the Lw of Sines leds to simple formul for the re of n oblique tringle. Referring to Figure.7, note tht ech tringle hs height of h b sin. onsequentl, the re of ech tringle is re bseheight cb sin bc sin. similr rguments, ou cn deelop the formuls re b sin c sin. b h h b c c is cute. Figure.7 is obtuse. RE OF N OLIQUE TRINGLE The re of n tringle is one-hlf the product of the lengths of two sides times the sine of their included ngle. Tht is, re bc sin b sin c sin. Note tht if ngle is 90, the formul gies the re for right tringle: re bc sin 90 bc bseheight. sin 90 Similr results re obtined for ngles nd equl to 90. EXMPLE 6 Finding the re of Tringulr Lot Find the re of tringulr lot hing two sides of lengths 90 meters nd 5 meters nd n included ngle of 0. Solution onsider 90 meters, b 5 meters, nd ngle 0, s shown in Figure.8. Then, the re of the tringle is b = 5 m 0 Figure.8 = 90 m re b sin 905sin 0 89 squre meters.

7 . Lw of Sines 8 ppliction EXMPLE 7 n ppliction of the Lw of Sines The course for bot rce strts t point in Figure.9 nd proceeds in the direction S 5 W to point, then in the direction S 40 E to point, nd finll bck to. Point lies 8 kilometers directl south of point. pproimte the totl distnce of the rce course. W N E S km D Figure.9 Solution ecuse lines D nd re prllel, it follows tht D. onsequentl, tringle hs the mesures shown in Figure.0. The mesure of ngle is Using the Lw of Sines, ecuse b 8, nd sin 5 8 sin sin 88 c 8 sin sin 88 The totl length of the course is pproimtel Length b sin 88 c sin kilometers. c 5 b = 8 km 40 Figure.0

8 84 hpter dditionl Topics in Trigonometr. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 4, fill in the blnks.. n tringle is tringle tht hs no right ngle.. For tringle, the Lw of Sines is gien b c sin sin.. Two nd one determine unique tringle. 4. The re of n oblique tringle is gien b bc sin b sin. In Eercises 5 4, use the Lw of Sines to sole the tringle. Round our nswers to two deciml plces b = 0 b c , 4.5, b , b 6., c , 4, b 4 00, 5, c 0 0 5, 48, b , 5, c 50 In Eercises 5 4, use the Lw of Sines to sole (if possible) the tringle. If two solutions eist, find both. Round our nswers to two deciml plces , 0, 5, 5, b 00 b , 76, 58, 58, 8, 4,.4, 4.5, b 0 b b.8 b.8. 0, b 5.. 0, 45, 5, b b , 9.5, b b 5 5 c = 0 b = c 0 0.4, 6.7,.6 4., 54.6, c , 54.6, c , 8 5, b 4.8 5, 65, c 0 0, 45, c 6 55, 4, c 4 8, 04, 5 8 6, 8, b 5 60, 9, c 0 c = WRITING OUT ONEPTS In Eercises 5 8, find lue for b such tht the tringle hs () one solution, (b) two solutions, nd (c) no solution , 60, 0, 88, Stte the Lw of Sines. 40. Write short prgrph eplining how the Lw of Sines cn be used to sole right tringle. In Eercises 4 46, find the re of the tringle hing the indicted ngle nd sides. 4. 0, 4, b , 6, c , b 57, c , b 4.5, c , 05, c , 6, b 0

9 . Lw of Sines Height ecuse of preiling winds, tree grew so tht it ws lening 4 from the erticl. t point 40 meters from the tree, the ngle of eletion to the top of the tree is 0 (see figure). Find the height h of the tree. 5. ridge Design bridge is to be built cross smll lke from gzebo to dock (see figure). The bering from the gzebo to the dock is S 4 W. From tree 00 meters from the gzebo, the berings to the gzebo nd the dock re S 74 E nd S 8 E, respectiel. Find the distnce from the gzebo to the dock. h m 0 Tree m 4 N W S Gzebo E 48. Height flgpole t right ngle to the horizontl is locted on slope tht mkes n ngle of with the horizontl. The flgpole s shdow is 6 meters long nd points directl up the slope. The ngle of eletion from the tip of the shdow to the sun is 0. () Drw tringle to represent the sitution. Show the known quntities on the tringle nd use rible to indicte the height of the flgpole. (b) Write n eqution tht cn be used to find the height of the flgpole. (c) Find the height of the flgpole. 49. ngle of Eletion 0-meter utilit pole csts 7-meter shdow directl down slope when the ngle of eletion of the sun is 4 (see figure). Find, the ngle of eletion of the ground. 50. Flight Pth plne flies 500 kilometers with bering of 6 from Nples to Elgin (see figure). The plne then flies 70 kilometers from Elgin to nton (nton is due west of Nples). Find the bering of the flight from Elgin to nton. W N S E nton 4 4 θ θ 7 m Elgin 70 km 500 km Not drwn to scle 0 m 44 Nples N 5. Rilrod Trck Design The circulr rc of rilrod cure hs chord of length 000 feet corresponding to centrl ngle of 40. () Drw digrm tht isull represents the sitution. Show the known quntities on the digrm nd use the ribles r nd s to represent the rdius of the rc nd the length of the rc, respectiel. (b) Find the rdius r of the circulr rc. (c) Find the length s of the circulr rc. 5. Glide Pth pilot hs just strted on the glide pth for lnding t n irport with runw of length 9000 feet. The ngles of depression from the plne to the ends of the runw re 7.5 nd 8.8. () Drw digrm tht isull represents the sitution. (b) Find the ir distnce the plne must trel until touching down on the ner end of the runw. (c) Find the ground distnce the plne must trel until touching down. (d) Find the ltitude of the plne when the pilot begins the descent. 54. Locting Fire The bering from the Pine Knob fire tower to the olt Sttion fire tower is N 65 E, nd the two towers re 0 kilometers prt. fire spotted b rngers in ech tower hs bering of N 80 E from Pine Knob nd S 70 E from olt Sttion (see figure). Find the distnce of the fire from ech tower. W N S E Dock 0 km olt Sttion 70 Fire Pine Knob Not drwn to scle

10 86 hpter dditionl Topics in Trigonometr 55. Distnce bot is siling due est prllel to the shoreline t speed of 0 miles per hour. t gien time, the bering to the lighthouse is S 70 E, nd 5 minutes lter the bering is S 6 E (see figure). The lighthouse is locted t the shoreline. Wht is the distnce from the bot to the shoreline? 60. Grphicl nlsis () Write the re of the shded region in the figure s function of. d 70 6 W N S E 0 cm θ 8 cm θ 0 cm 56. Distnce fmil is treling due west on rod tht psses fmous lndmrk. t gien time the bering to the lndmrk is N 6 W, nd fter the fmil trels 5 miles frther the bering is N 8 W. Wht is the closest the fmil will come to the lndmrk while on the rod? 57. ltitude The ngles of eletion to n irplne from two points nd on leel ground re 55 nd 7, respectiel. The points nd re. miles prt, nd the irplne is est of both points in the sme erticl plne. Find the ltitude of the plne. 58. Distnce The ngles of eletion nd to n irplne from the irport control tower nd from n obsertion post miles w re being continuousl monitored (see figure). Write n eqution giing the distnce d between the plne nd obsertion post in terms of nd. irport control tower θ Obsertion post 59. re You re seeding tringulr courtrd. One side of the courtrd is 5 feet long nd nother side is 46 feet long. The ngle opposite the 5-foot side is 65. () Drw digrm tht gies isul representtion of the sitution. (b) How long is the third side of the courtrd? (c) One bg of grss seed coers n re of 50 squre feet. How mn bgs of grss seed will ou need to coer the courtrd? mi Not drwn to scle φ d (b) Use grphing utilit to grph the function. (c) Determine the domin of the function. Eplin how the re of the region nd the domin of the function would chnge if the 8-centimeter line segment were decresed in length. (d) Differentite the function nd use the zero or root feture of grphing utilit to pproimte the criticl number. True or Flse? In Eercises 6 65, determine whether the sttement is true or flse. If it is flse, eplin wh or gie n emple tht shows it is flse. 6. It is not possible to crete n obtuse tringle whose longest side is one of the sides tht forms its obtuse ngle. 6. Two ngles nd one side of tringle do not necessril determine unique tringle. 6. If three sides or three ngles of n oblique tringle re known, then the tringle cn be soled. 64. The Lw of Sines is true if one of the ngles in the tringle is right ngle. 65. The re of n oblique tringle is re b sin. PSTONE 66. In the figure, tringle is to be formed b drwing line segment of length from 4, to the positie -is. For wht lue(s) of cn ou form () one tringle, (b) two tringles, nd (c) no tringles? Eplin our resoning. (0, 0) (4, ) 4 5

11 . Lw of osines 87. Lw of osines Use the Lw of osines to sole oblique tringles (SSS or SS). Use Heron s re Formul to find the re of tringle. Use the Lw of osines to model nd sole rel-life problems. Introduction Two cses remin in the list of conditions needed to sole n oblique tringle SSS nd SS. If ou re gien three sides (SSS), or two sides nd their included ngle (SS), none of the rtios in the Lw of Sines would be complete. In such cses, ou cn use the Lw of osines. See ppendi for proof of the Lw of osines. THEOREM. LW OF OSINES c b Stndrd Form b c bc cos b c c cos c b b cos lterntie Form cos b c bc cos c b c cos b c b EXMPLE Three Sides of Tringle SSS = 8 ft Figure. b = 9 ft c = 4 ft Find the three ngles of the tringle in Figure.. Solution It is good ide first to find the ngle opposite the longest side side b in this cse. Using the lterntie form of the Lw of osines, ou find tht cos c b c lterntie form ecuse cos is negtie, ou know tht is n obtuse ngle gien b t this point, it is simpler to use the Lw of Sines to determine. sin sin b sin You know tht must be cute becuse is obtuse, nd tringle cn he, t most, one obtuse ngle. So,.08 nd

12 88 hpter dditionl Topics in Trigonometr EXPLORTION Wht fmilir formul do ou obtin when ou use the third form of the Lw of osines c b b cos nd ou let 90? Wht is the reltionship between the Lw of osines nd this formul? Do ou see wh it ws wise to find the lrgest ngle first in Emple? Knowing the cosine of n ngle, ou cn determine whether the ngle is cute or obtuse. Tht is, cos > 0 cos < 0 for for 0 < 90 < < 90 < 80. cute Obtuse So, in Emple, once ou found tht ngle ws obtuse, ou knew tht ngles nd were both cute. If the lrgest ngle is cute, the remining two ngles re cute lso. EXMPLE Two Sides nd the Included ngle SS Find the remining ngles nd side of the tringle in Figure.. b = 9 m 5 Figure. c = m Solution Use the Lw of osines to find the unknown side in the figure. b c bc cos 9 9 cos Stndrd form ecuse meters, ou now know the rtio sin nd ou cn use the Lw of Sines sin b sin to sole for. sin b sin 9 sin So, rcsin nd HERON OF LEXNDRI Heron of lendri (c. 00..) ws Greek geometer nd inentor. His works describe how to find the res of tringles, qudrilterls, regulr polgons hing to sides, nd circles s well s the surfce res nd olumes of three-dimensionl objects. Heron s re Formul The Lw of osines cn be used to estblish the following formul for the re of tringle. This formul is clled Heron s re Formul fter the Greek mthemticin Heron (c. 00..). proof of this formul is gien in ppendi. THEOREM. HERON S RE FORMUL Gien n tringle with sides of lengths, b, nd c, the re of the tringle is re ss s bs c where s b c. EXMPLE Using Heron s re Formul Find the re of tringle hing sides of lengths 4 meters, b 5 meters, nd c 7 meters. Solution ecuse s b c 68 84, Heron s re Formul ields re ss s bs c squre meters.

13 . Lw of osines 89 pplictions 60 ft 60 ft P h f = 4 ft 60 ft 45 p = 60 ft H Figure. F EXMPLE 4 n ppliction of the Lw of osines The pitcher s mound on women s softbll field is 4 feet from home plte nd the distnce between the bses is 60 feet, s shown in Figure.. (The pitcher s mound is not hlfw between home plte nd second bse.) How fr is the pitcher s mound from first bse? Solution In tringle HPF, H 45 (line HP bisects the right ngle t H), f 4, nd p 60. Using the Lw of osines for this SS cse, ou he h f p fp cos H cos So, the pproimte distnce from the pitcher s mound to first bse is h feet. EXMPLE 5 n ppliction of the Lw of osines ship trels 60 miles due est, then djusts its course northwrd, s shown in Figure.4. fter treling 80 miles in tht direction, the ship is 9 miles from its point of deprture. Describe the bering from point to point. N W S E c = 60 mi b = 9 mi = 80 mi Not drwn to scle Figure.4 Solution You he 80, b 9, nd c 60. So, using the lterntie form of the Lw of osines, ou he So, cos c b c rccos nd thus the bering mesured from due north from point to point is , or N 76.5 E.

14 80 hpter dditionl Topics in Trigonometr EXMPLE 6 The Velocit of Piston In the engine shown in Figure.5, 7-inch connecting rod is fstened to crnk of rdius inches. The crnkshft rottes counterclockwise t constnt rte of 00 reolutions per minute. Find the elocit of the piston when. 7 θ θ The elocit of piston is relted to the ngle of the crnkshft. Figure.5 b θ c Lw of osines: b c c cos Figure.6 Solution Lbel the distnces s shown in Figure.5. ecuse complete reolution corresponds to rdins, it follows tht ddt rdins per minute. Gien rte: 400 (constnt rte) dt d Find: when dt You cn use the Lw of osines (Figure.6) to find n eqution tht reltes nd. Eqution: 7 cos When, d 6 cos d dt 6 sin ou cn sole for s shown. 7 cos hoose positie solution. So, when 8 nd the elocit of the piston is d 68 dt , 408 inches per minute. 0 d dt 6 sin d dt cos d dt d dt d dt 6 sin 6 cos d dt NOTE The elocit in Emple 6 is negtie becuse represents distnce tht is decresing.

15 . Lw of osines 8. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 4, fill in the blnks.. If ou re gien three sides of tringle, ou would use the Lw of to find the three ngles of the tringle.. If ou re gien two ngles nd n side of tringle, ou would use the Lw of to sole the tringle.. The stndrd form of the Lw of osines for cos c b is. c 4. The Lw of osines cn be used to estblish formul for finding the re of tringle clled Formul. In Eercises 5 0, use the Lw of osines to sole the tringle. Round our nswers to two deciml plces b 7 9 0, b, c 6 7, b, c 8 0, b 5, c 0 05, 9, b 4.5, b 5, c 55, b 5, c , b 5, c 5.4, b 0.75, c.5 0, b 6, c 7 48, b, c 4 0 5, 40, c , 6., c , 7, c 7 5 5, 7.45, b.5 4, 4 9, 0, 8, b 4 In Eercises 6, use Heron s re Formul to find the re of the tringle... 8,, b, b 6, c 7 c , 75.4, b 0., b 5, c 9 c ,.05, b 8.46, b 0.75, c 5.05 c.45 WRITING OUT ONEPTS (continued) 8. List the four cses for soling n oblique tringle. Eplin when to use the Lw of Sines nd when to use the Lw of osines. 9. Nigtion bot rce runs long tringulr course mrked b buos,, nd. The rce strts with the bots heded west for 700 meters. The other two sides of the course lie to the north of the first side, nd their lengths re 700 meters nd 000 meters. Drw figure tht gies isul representtion of the sitution, nd find the berings for the lst two legs of the rce. 0. Nigtion plne flies 80 miles from Frnklin to enterille with bering of 75. Then it flies 648 miles from enterille to Rosemount with bering of. Drw figure tht isull represents the sitution, nd find the stright-line distnce nd bering from Frnklin to Rosemount.. Sureing To pproimte the length of mrsh, sureor wlks 50 meters from point to point, then turns 75 nd wlks 0 meters to point (see figure). pproimte the length of the mrsh.. Sureing tringulr prcel of lnd hs 5 meters of frontge, nd the other boundries he lengths of 76 meters nd 9 meters. Wht ngles does the frontge mke with the two other boundries?. Sureing tringulr prcel of ground hs sides of lengths 75 feet, 650 feet, nd 575 feet. Find the mesure of the lrgest ngle. 4. Streetlight Design Determine the ngle in the design of the streetlight shown in the figure m 50 m WRITING OUT ONEPTS 7. Stte the Lw of osines. θ 4

16 8 hpter dditionl Topics in Trigonometr 5. Distnce Two ships lee port t 9.M. One trels t bering of N 5 W t miles per hour, nd the other trels t bering of S 67 W t 6 miles per hour. pproimte how fr prt the re t noon tht d. 6. Length 00-foot erticl tower is to be erected on the side of hill tht mkes 6 ngle with the horizontl (see figure). Find the length of ech of the two gu wires tht will be nchored 75 feet uphill nd downhill from the bse of the tower. 9. sebll On bsebll dimond with 90-foot sides, the pitcher s mound is 60.5 feet from home plte. How fr is it from the pitcher s mound to third bse? 40. sebll The bsebll pler in center field is pling pproimtel 0 feet from the teleision cmer tht is behind home plte. btter hits fl bll tht goes to the wll 40 feet from the cmer (see figure). () The cmer turns 8 to follow the pl. pproimtel how fr does the center fielder he to run to mke the ctch? (b) When the cmer ws turning t the rte of.4 per second. Find the speed of the center fielder., 00 ft 0 ft 8 40 ft 6 75 ft 75 ft 7. Nigtion On mp, Orlndo is 78 millimeters due south of Nigr Flls, Dener is 7 millimeters from Orlndo, nd Dener is 5 millimeters from Nigr Flls (see figure). De ner () Find the bering of Dener from Orlndo. (b) Find the bering of Dener from Nigr Flls. 8. Nigtion On mp, Minnepolis is 65 millimeters due west of lbn, Phoeni is 6 millimeters from Minnepolis, nd Phoeni is 68 millimeters from lbn (see figure). Minnepolis 6 mm Phoeni 5 mm 7 mm 65 mm 68 mm Nigr Fll s 78 mm Orlndo lbn 4. ircrft Trcking To determine the distnce between two ircrft, trcking sttion continuousl determines the distnce to ech ircrft nd the ngle between them (see figure). () Determine the distnce between the plnes when 4, b 5 miles, nd c 0 miles. (b) The plne t ngle is fling t 00 miles per hour nd the plne t ngle is fling t 75 miles per hour. Wht is the rte of seprtion of the plnes t the time of the conditions of prt ()? c b 4. ircrft Trcking Use the figure for Eercise 4 to determine the distnce between the plnes when, b 0 miles, nd c 0 miles. 4. Trusses Q is the midpoint of the line segment PR in the truss rfter shown in the figure. Wht re the lengths of the line segments PQ, QS, nd RS? R () Find the bering of Minnepolis from Phoeni. (b) Find the bering of lbn from Phoeni. P Q S

17 . Lw of osines Velocit of Piston n engine hs 7-inch connecting rod fstened to crnk (see figure). Let d be the distnce the piston is from the top of its stroke for n ngle. () Use the Lw of osines to write reltionship between nd. Use the Qudrtic Formul to write s function of. (Select the sign tht ields positie lues of. ) (b) Use the result of prt () to write d s function of. (c) omplete the tble. 46. wning Design retrctble wning boe ptio door lowers t n ngle of 50 from the eterior wll t height of 0 feet boe the ground (see figure). No direct sunlight is to enter the door when the ngle of eletion of the sun is greter thn 70. Wht is the length of the wning? 50 Sun s rs 0 ft d (d) The sprk plug fires t before top ded center. How fr is the piston from the top of its stroke? (e) Use grphing utilit to find the first nd second derities of the function d. For wht lues of is the speed of the piston 0? For wht lue in the interl 0, is it moing t the gretest speed? (f) If the engine is running t 500 reolutions per minute, find the speed of the piston when nd 0, 90, (g) Use grphing utilit to grph the second deritie. The speed of the piston is the sme when nd Is the ccelertion on the piston the sme for these two lues of? , 47. Geometr The lengths of the sides of tringulr prcel of lnd re pproimtel 00 feet, 500 feet, nd 600 feet. pproimte the re of the prcel. 48. Geometr prking lot hs the shpe of prllelogrm (see figure). The lengths of two djcent sides re 70 meters nd 00 meters. The ngle between the two sides is 70. Wht is the re of the prking lot? 70 m.5 in. 7 in. in m θ d Figure for 44 Figure for Pper Mnufcturing In certin process with continuous pper, the pper psses cross three rollers of rdii inches, 4 inches, nd 6 inches (see figure). The centers of the three-inch nd si-inch rollers re d inches prt, nd the length of the rc in contct with the pper on the four-inch roller is s inches. omplete the following tble. s 4 in. 6 in. d (inches) (degrees) s (inches) θ d True or Flse? In Eercises 49 5, determine whether the sttement is true or flse. If it is flse, eplin wh or gie n emple tht shows it is flse. 49. In Heron s re Formul, s is the erge of the lengths of the three sides of the tringle. 50. In ddition to SSS nd SS, the Lw of osines cn be used to sole tringles with SS conditions. 5. If the cosine of the lrgest ngle in tringle is negtie, then ll the ngles in tringle re cute ngles. PSTONE 5. Determine whether the Lw of Sines or the Lw of osines is needed to sole the tringle. (),, nd (b), c, nd (c) b, c, nd (d),, nd c (e) b, c, nd (f), b, nd c

18 84 hpter dditionl Topics in Trigonometr. Vectors in the Plne Represent ectors s directed line segments. Write the component forms of ectors. Perform bsic ector opertions nd represent them grphicll. Write ectors s liner combintions of unit ectors. Find the direction ngles of ectors. Use ectors to model nd sole rel-life problems. Introduction Quntities such s force nd elocit cnnot be completel chrcterized b single rel number becuse the inole both mgnitude nd direction. To represent such quntit, ou cn use directed line segment, s shown in Figure.7. The directed line segment PQ \ hs initil point P nd terminl point Q. Its mgnitude (or length) is denoted b PQ \ nd cn be found using the Distnce Formul. Terminl point Q PQ P Initil point Figure.7 Figure.8 Two directed line segments tht he the sme mgnitude nd direction re equilent. For emple, the directed line segments in Figure.8 re ll equilent. The set of ll directed line segments tht re equilent to the directed line segment PQ \ is ector in the plne, written PQ \. Vectors re denoted b lowercse, boldfce letters such s u,, nd w. EXMPLE Vector Representtion b Directed Line Segments P (0, ) R u (, ) Q (4, 5) Figure.9 S (6, ) TEHNOLOGY You cn grph ectors with grphing utilit b grphing directed line segments. onsult the user s guide for our grphing utilit for specific instructions. Let u be represented b the directed line segment from P0, to Q4, 5, nd let be represented b the directed line segment from R, to S6,, s shown in Figure.9. Show tht u nd re equilent. Solution From the Distnce Formul, it follows tht PQ \ nd RS \ he the sme mgnitude. PQ \ RS \ Moreoer, both line segments he the sme direction becuse the re both directed towrd the upper right on lines hing slope of ecuse PQ \ nd RS \ he the sme mgnitude nd direction, u nd re equilent.

19 . Vectors in the Plne 85 4 (0, 0) P (, ) Q =, 4 The stndrd position of ector Figure.0 omponent Form of Vector The directed line segment whose initil point is the origin is often the most conenient representtie of set of equilent directed line segments. This representtie of the ector is in stndrd position, s shown in Figure.0. ector whose initil point is the origin 0, 0 cn be uniquel represented b the coordintes of its terminl point,. This is the component form of ector, written s,. The coordintes nd re the components of. If both the initil point nd the terminl point lie t the origin, is the zero ector nd is denoted b 0 0, 0. OMPONENT FORM OF VETOR The component form of the ector with initil point Pp, p nd terminl point Qq is gien b PQ \, q q p, q p,. The mgnitude (or length) of is gien b q p q p. If, is unit ector. Moreoer, 0 if nd onl if is the zero ector 0. Two ectors u u, u nd, re equl if nd onl if u nd u. For instnce, in Emple, the ector u from P0, to Q4, 5 is u PQ \ 4 0, 5 4,, nd the ector from R, to S6, is RS \ 6, 4,. EXMPLE Finding the omponent Form of Vector Find the component form nd mgnitude of the ector tht hs initil point 4, 7 nd terminl point, 5. lgebric Solution Let nd P4, 7 p, p Q, 5 q, q. Then, the components of, re q p 4 5 q p 5 7. So, 5, nd the mgnitude of is Grphicl Solution Use centimeter grph pper to plot the points P4, 7 nd Q, 5. refull sketch the ector. Use the sketch to find the components of,. Then use centimeter ruler to find the mgnitude of. cm Figure. Figure. shows tht the components of re 5 nd, so 5,. Figure. lso shows tht the mgnitude of is.

20 86 hpter dditionl Topics in Trigonometr Figure. Vector Opertions The two bsic ector opertions re sclr multipliction nd ector ddition. In opertions with ectors, numbers re usull referred to s sclrs. In this tet, sclrs will lws be rel numbers. Geometricll, the product of ector nd sclr k is the ector tht is k times s long s. If k is positie, k hs the sme direction s, nd if k is negtie, k hs the direction opposite tht of, s shown in Figure.. To dd two ectors u nd geometricll, first position them (without chnging their lengths or directions) so tht the initil point of the second ector coincides with the terminl point of the first ector u. The sum u is the ector formed b joining the initil point of the first ector u with the terminl point of the second ector, s shown in Figure.. This technique is clled the prllelogrm lw for ector ddition becuse the ector u, often clled the resultnt of ector ddition, is the digonl of prllelogrm hing djcent sides u nd. u + u u Figure. DEFINITIONS OF VETOR DDITION ND SLR MULTIPLITION Let u u, u nd, be ectors nd let k be sclr ( rel number). Then the sum of u nd is the ector u u, u Sum nd the sclr multiple of k times u is the ector ku ku, u ku, ku. Sclr multiple The negtie of, is u, Negtie u nd the difference of u nd is u + ( ) u u Figure.4 u u u, u. Difference To represent u geometricll, ou cn use directed line segments with the sme initil point. The difference u is the ector from the terminl point of to the terminl point of u, which is equl to u, s shown in Figure.4.

21 . Vectors in the Plne 87 The component definitions of ector ddition nd sclr multipliction re illustrted in Emple. In this emple, notice tht ech of the ector opertions cn be interpreted geometricll. EXMPLE Vector Opertions Let, nd w 4, 4, nd find (), (b) w, nd (c) w. Solution. ecuse,, ou he,, 6,. sketch of is shown in Figure.5(). b. The difference of nd w is w, 4, 4 4, 4 7, 5. sketch of w is shown in Figure.5(b). c. The sum of nd w is w, 4, 4, 4, 4 6,, 6, 6, 0. sketch of w is shown in Figure.5(c). ( 6, 0) (, ) 5 6 (6, ) (, ) 6 7 w w (7, 5) w 4 w (6, ) 4 () (b) (c) Figure.5 Vector ddition nd sclr multipliction shre mn of the properties of ordinr rithmetic. NOTE Propert 9 cn be stted s follows: The mgnitude of the ector c is the bsolute lue of c times the mgnitude of. THEOREM.4 PROPERTIES OF VETOR DDITION ND SLR MULTIPLITION Let u,, nd w be ectors nd let c nd d be sclrs. Then the following properties re true.. u u. u w u w. u 0 u 4. u u 0 5. cdu cdu 6. c du cu du 7. cu cu c 8. u u, 9. c c 0u 0

22 88 hpter dditionl Topics in Trigonometr Unit Vectors In mn pplictions of ectors, it is useful to find unit ector tht hs the sme direction s gien nonzero ector. To do this, ou cn diide b its mgnitude. THEOREM.5 UNIT VETOR IN THE DIRETION OF If is nonzero ector in the plne, then the ector u hs length nd the sme direction s. The ector u is clled unit ector in the direction of. PROOF ecuse is positie nd u, ou cn conclude tht u hs the sme direction s. To see tht u, note tht u. So, u hs length nd the sme direction s. EXMPLE 4 Finding Unit Vector Find unit ector in the direction of 4, 5 nd erif tht the result hs mgnitude of. Solution The unit ector in the direction of is 4, , , 5 4. j = 0, This ector hs mgnitude of becuse i =, Unit ectors, 0 nd 0,, clled the stndrd unit ectors, re denoted b Figure.6 i, 0 nd j 0, Stndrd unit ectors (, ) u 4 6 (, 5) Figure.7 s shown in Figure.6. (Note tht the lowercse letters i nd j re written in boldfce to distinguish them from sclrs, ribles, or the imginr number i. ) These ectors cn be used to represent n ector, s shown.,, 0 0, i j The sclrs nd re clled the horizontl nd erticl components of, respectiel. The ector sum i j is clled liner combintion of the ectors i nd j. n ector in the plne cn be written s liner combintion of the stndrd unit ectors i nd j. For instnce, the ector in Figure.7 cn be written s u, 5, 8 i 8j.

23 . Vectors in the Plne 89 EXMPLE 5 Vector Opertions Let u i 5j nd let i j. Find 4u. Solution You could sole this problem b conerting u nd to component form. This, howeer, is not necessr. It is just s es to perform the opertions in unit ector form. 4u 4i 5j i j 8i 0j 9i j 7i j Direction ngles If u is unit ector such tht is the ngle (mesured counterclockwise) from the positie -is to u, the terminl point of u lies on the unit circle nd ou he u θ = cos θ (, ) = sin θ u, cos, sin cos i sin j s shown in Figure.8. The ngle is the direction ngle of the ector u. Suppose tht u is unit ector with direction ngle. If i bj is n ector tht mkes n ngle with the positie -is, it hs the sme direction s u nd ou cn write cos, sin cos i sin j. u Figure.8 ecuse i bj cos i sin j, it follows tht the direction ngle for is determined from sin tn sin b cos cos. EXMPLE 6 Finding Direction ngles of Vectors () u θ = 45 (, ) (, 4) (b) Figure.9 Find the direction ngle of ech ector.. u i j b. i 4j Solution. The direction ngle is tn b. So, s shown in Figure.9(). b. The direction ngle is 45, tn b 4. Moreoer, becuse i 4j lies in Qudrnt IV, lies in Qudrnt IV nd its reference ngle is rctn 4 5. So, it follows tht , s shown in Figure.9(b).

24 80 hpter dditionl Topics in Trigonometr pplictions of Vectors EXMPLE 7 Finding the omponent Form of Vector Find the component form of the ector tht represents the elocit of n irplne descending t speed of 50 miles per hour t n ngle 0 below the horizontl, s shown in Figure.0. Solution The elocit ector hs mgnitude of 50 nd direction ngle of So, cos i sin j 50cos 00i 50sin 00j i j 40.96i 5.0j 40.96, 5.0. Figure.0 You cn check tht hs mgnitude of 50, s follows , , EXMPLE 8 Using Vectors to Determine Weight force of 600 pounds is required to pull bot nd triler up rmp inclined t 5 from the horizontl. Find the combined weight of the bot nd triler. Solution sed on Figure., ou cn mke the following obsertions. \ force of grit combined weight of bot nd triler \ force ginst rmp W 5 D 5 \ force required to moe bot up rmp 600 pounds construction, tringles WD nd re similr. Therefore, ngle is 5. So, in tringle ou he sin 5 \ \ Figure. sin \ \ 600 sin 5 \ 8. onsequentl, the combined weight is pproimtel 8 pounds. NOTE In Figure., note tht \ is prllel to the rmp.

25 . Vectors in the Plne 8 STUDY TIP Recll from Section 9.8 tht in ir nigtion, berings cn be mesured in degrees clockwise from north. EXMPLE 9 Using Vectors to Find Speed nd Direction n irplne is treling t speed of 500 miles per hour with bering of 0 t fied ltitude with negligible wind elocit s shown in Figure.(). When the irplne reches certin point, it encounters wind with elocit of 70 miles per hour in the direction N 45 E, s shown in Figure.(b). Wht re the resultnt speed nd direction of the irplne? Wind 0 θ () Figure. (b) Solution Using Figure., the elocit of the irplne (lone) is 500cos 0, sin 0 50, 50 nd the elocit of the wind is 70cos 45, sin 45 5, 5. So, the elocit of the irplne (in the wind) is 50 5, , 48.5 nd the resultnt speed of the irplne is Finll, if 5.5 miles per hour. which implies tht.6. is the direction ngle of the flight pth, ou he tn rctn So, the true direction of the irplne is pproimtel

26 8 hpter dditionl Topics in Trigonometr. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 0, fill in the blnks.. cn be used to represent quntit tht inoles both mgnitude nd direction.. The directed line segment PQ \ hs point P nd point Q.. The of the directed line segment PQ \ is denoted b PQ \. 4. The set of ll directed line segments tht re equilent to gien directed line segment PQ \ is in the plne. 5. In order to show tht two ectors re equilent, ou must show tht the he the sme nd the sme. 6. The directed line segment whose initil point is the origin is sid to be in. 7. ector tht hs length of is clled. 8. The two bsic ector opertions re sclr nd ector. 9. The ector u is clled the of ector ddition. 0. The ector sum i j is clled of the ectors i nd j, nd the sclrs nd re clled the nd components of, respectiel. In Eercises, find the component form nd the mgnitude of the ector (, ) (, ) (, 5) 4 4 ( 4, ) (, 4) 5 (, ) Initil Point 5., 6. 4, 7., 5 8., 7 9., 0.,., 5., In Eercises 8, use the figure to sketch grph of the specified ector. To print n enlrged cop of the grph, go to the website u 6. u 7. u 8. u In Eercises 9 6, find () u, (b) u, nd (c) u. Then sketch ech resultnt ector. 9. u,,, 0. u,, 4, 0. u 5,, 0, 0. u 0, 0,,. u i j, i j 4. u i j, j 5. u i, j 6. u j, i In Eercises 7 46, find unit ector in the direction of the gien ector. Verif tht the result hs mgnitude of. 7. u, 0 8. u 0, 9., 40. 5, 4. i j 4. 6i j 4. w 4j 44. w 6i 45. w i j 46. w 7j i In Eercises 47 50, find the ector with the gien mgnitude nd the sme direction s u. Mgnitude Terminl Point,, 5, 5, 7 8, 9 9, 5, 9, 40 Direction u, u, u, u, u

27 . Vectors in the Plne 8 In Eercises 5 56, find the component form of nd sketch the specified ector opertions geometricll, where u i j, nd w i j. 5. u 5. 4 w 5. u w 54. u w 55. u w 56. u w In Eercises 57 60, find the mgnitude nd direction ngle of the ector i 6j 58. 5i 4j cos 60i sin 60j 8cos 5i sin 5j In Eercises 6 68, find the component form of gien its mgnitude nd the ngle it mkes with the positie -is. Sketch in the direction i 4j 68. in the direction i j In Eercises 69 7, find the component form of the sum of u nd with the gien mgnitudes nd direction ngles nd u Mgnitude Mgnitude u 5 5 u 4 4 u 0 50 u 50 0 In Eercises 7 76, use the Lw of osines to find the ngle between the ectors. (ssume ) i j, i j, w i j w i j i j, i j, w i j w i j. ngle ngle u 0 90 u u u 0 0 WRITING OUT ONEPTS 77. Wht conditions must be met in order for two ectors to be equilent? Which ectors in the figure pper to be equilent? 78. The ectors u nd he the sme mgnitudes in the two figures. In which figure will the mgnitude of the sum be greter? Gie reson for our nswer. () (b) Resultnt Force In Eercises 79 nd 80, find the ngle between the forces gien the mgnitude of their resultnt. (Hint: Write force s ector in the direction of the positie -is nd force s ector t n ngle with the positie -is.) Force D u u Force Resultnt Force pounds 60 pounds 90 pounds pounds 000 pounds 750 pounds E 8. Velocit gun with muzzle elocit of 00 feet per second is fired t n ngle of 6 boe the horizontl. Find the erticl nd horizontl components of the elocit. 8. Velocit Detroit Tigers pitcher Joel Zum ws recorded throwing pitch t elocit of 04 miles per hour. If he threw the pitch t n ngle of 5 below the horizontl, find the erticl nd horizontl components of the elocit. (Source: Dmon Lichtenwlner, sebll Info Solutions)

28 84 hpter dditionl Topics in Trigonometr 8. Resultnt Force Forces with mgnitudes of 5 newtons nd 00 newtons ct on hook (see figure). The ngle between the two forces is 45. Find the direction nd mgnitude of the resultnt of these forces. 5 newtons Figure for 8 Figure for Resultnt Force Forces with mgnitudes of 000 newtons nd 900 newtons ct on mchine prt t ngles of 0 nd 45, respectiel, with the -is (see figure). Find the direction nd mgnitude of the resultnt of these forces. 85. Resultnt Force Three forces with mgnitudes of 75 pounds, 00 pounds, nd 5 pounds ct on n object t ngles of 0, 45, nd 0, respectiel, with the positie -is. Find the direction nd mgnitude of the resultnt of these forces. 86. Resultnt Force Three forces with mgnitudes of 70 pounds, 40 pounds, nd 60 pounds ct on n object t ngles of 0, 45, nd 5, respectiel, with the positie -is. Find the direction nd mgnitude of the resultnt of these forces. ble Tension In Eercises 87 nd 88, use the figure to determine the tension in ech cble supporting the lod in. 0 in lb 00 newtons 89. Tow Line Tension loded brge is being towed b two tugbots, nd the mgnitude of the resultnt is 6000 pounds directed long the is of the brge (see figure). Find the tension in the tow lines if the ech mke n 8 ngle with the is of the brge in lb 000 newtons newtons 90. Rope Tension To crr 00-pound clindricl weight, two people lift on the ends of short ropes tht re tied to n eelet on the top center of the clinder. Ech rope mkes 0 ngle with the erticl. Drw figure tht gies isul representtion of the sitution, nd find the tension in the ropes. 9. Work he object is pulled 0 feet cross floor, using force of 00 pounds. The force is eerted t n ngle of 50 boe the horizontl (see figure). Find the work done. (Use the formul for work, W FD, where F is the component of the force in the direction of motion nd D is the distnce.) Rope Tension tetherbll weighing pound is pulled outwrd from the pole b horizontl force u until the rope mkes 45 ngle with the pole (see figure). Determine the resulting tension in the rope nd the mgnitude of u. Tension Nigtion n irplne is fling in the direction of 48, with n irspeed of 875 kilometers per hour. ecuse of the wind, its groundspeed nd direction re 800 kilometers per hour nd 40, respectiel (see figure). Find the direction nd speed of the wind. 00 lb 0 ft lb u Wind 800 kilometers per hour 875 kilometers per hour 94. Nigtion n irplne s elocit with respect to the ir is 580 miles per hour, nd its bering is. The wind, t the ltitude of the plne, is from the southwest nd hs elocit of 60 miles per hour. () Drw figure tht gies isul representtion of the problem. (b) Wht is the true direction of the plne, nd wht is its speed with respect to the ground? W N S E

29 . Vectors in the Plne 85 True or Flse? In Eercises 95 nd 96, decide whether the sttement is true or flse. If it is flse, eplin wh or gie n emple tht shows it is flse. 95. If u nd he the sme mgnitude nd direction, then u. 96. If u i bj is unit ector, then b. 97. Proof Proe tht cos i sin j is unit ector for n lue of. PSTONE 98. The initil nd terminl points of ector re, 4 nd 9,, respectiel. () Write in component form. (b) Write s the liner combintion of the stndrd unit ectors i nd j. (c) Sketch with its initil point t the origin. (d) Find the mgnitude of. 99. Writing In our own words, stte the difference between sclr nd ector. Gie emples of ech. 00. Writing Gie geometric descriptions of the opertions of ddition of ectors nd multipliction of ector b sclr. 0. Writing Identif the quntit s sclr or s ector. Eplin our resoning. () The muzzle elocit of bullet (b) The price of compn s stock (c) The ir temperture in room (d) The weight of n utomobile 0. Technolog Write progrm for our grphing utilit tht grphs two ectors nd their difference gien the ectors in component form. In Eercises 0 nd 04, use the progrm in Eercise 0 to find the difference of the ectors shown in the figure (, 6) (4, 5) (5, ) (9, 4) ( 0, 70) (80, 80) (0, 60) ( 00, 0) SETION PROJET dding Vectors Grphicll The pseudo code below cn be trnslted into progrm for grphing utilit. Progrm Input. Input b. Input c. Input d. Drw line from 0, 0 to, b. Drw line from 0, 0 to c, d. dd c nd store in e. dd b d nd store in f. Drw line from 0, 0 to e, f. Drw line from, b to c, d. Drw line from c, d to e, f. Puse to iew grph. End progrm. The progrm sketches two ectors u i bj nd ci dj in stndrd position. Then, using the prllelogrm lw for ector ddition, the progrm lso sketches the ector sum u. efore running the progrm, ou should set lues tht produce n pproprite iewing window. () n irplne is fling t heding of 00 nd speed of 400 miles per hour. The irplne encounters wind of elocit 75 miles per hour in the direction 40. Use the progrm to find the resultnt speed nd direction of the irplne. (b) fter encountering the wind, is the irplne in prt () treling t higher speed or lower speed? Eplin. (c) onsider the irplne described in prt (), t heding of 00 nd speed of 400 miles per hour. Use the progrm to find the wind elocit in the direction of 40 tht will produce resultnt direction of 0. (d) onsider the irplne described in prt (), t heding of 00 nd speed of 400 miles per hour. Use the progrm to find the wind direction t speed of 75 miles per hour tht will produce resultnt direction of 0.

30 86 hpter dditionl Topics in Trigonometr.4 Vectors nd Dot Products Find the dot product of two ectors nd use the properties of the dot product. Find the ngle between two ectors nd determine whether two ectors re orthogonl. Write ector s the sum of two ector components. Use ectors to find the work done b force. The Dot Product of Two Vectors So fr ou he studied two ector opertions ector ddition nd multipliction b sclr ech of which ields nother ector. In this section, ou will stud third ector opertion, the dot product. This product ields sclr, rther thn ector. DEFINITION OF THE DOT PRODUT The dot product of u u, u nd, is u u u. THEOREM.6 PROPERTIES OF THE DOT PRODUT Let u,, nd w be ectors in the plne or in spce nd let c be sclr. u u u w u u w 5. cu cu u c Proofs of Properties nd 4 re gien in ppendi. EXMPLE Finding Dot Products Find ech dot product.. 4, 5, b.,, c. Solution. b. c. 4, 5, ,, 0 0, 4, , 4, NOTE In Emple, be sure ou see tht the dot product of two ectors is sclr ( rel number), not ector. Moreoer, notice tht the dot product cn be positie, zero, or negtie.

31 .4 Vectors nd Dot Products 87 EXMPLE Using Properties of Dot Products Let u,,, 4, nd w,. Find ech dot product.. u w b. u c. u w Solution egin b finding the dot product of u nd.. b. c. u,, u w 4, 4, 8 u u 4 8 u w 4,, 4 6 Notice tht the first product is ector, wheres the second nd third re sclrs. EXMPLE Dot Product nd Mgnitude The dot product of u with itself is 5. Wht is the mgnitude of u? Solution ecuse u u u nd u u 5, it follows tht u u u 5. The ngle etween Two Vectors The ngle between two nonzero ectors is the ngle, 0, between their respectie stndrd position ectors, s shown in Figure.. This ngle cn be found using the dot product. THEOREM.7 NGLE ETWEEN TWO VETORS u θ u If is the ngle between two nonzero ectors u nd, then cos u u. Figure. Origin PROOF onsider the tringle determined b ectors u,, nd u, s shown in Figure.. the Lw of osines, ou cn write u u u cos u u u u cos u u u u u cos u u u u u u cos u u u u cos u u cos cos u u.

32 88 hpter dditionl Topics in Trigonometr =, 5 u = 4, θ Figure.4 EXMPLE 4 Finding the ngle etween Two Vectors Find the ngle between u 4, nd, 5. Solution The two ectors nd re shown in Figure.4. cos u u 4,, 5 4,, This implies tht the ngle between the two ectors is rccos Rewriting the epression for the ngle between two ectors in the form u u cos lterntie form of dot product produces n lterntie w to clculte the dot product. From this form, ou cn see tht becuse u nd re lws positie, u nd cos will lws he the sme sign. Figure.5 shows the fie possible orienttions of two ectors. u cos θ u θ < < < cos < 0 Opposite Direction Obtuse ngle 90 ngle cute ngle Sme Direction Figure.5 u θ cos 0 u θ 0 < < 0 < cos < 0 u cos DEFINITION OF ORTHOGONL VETORS The ectors u nd re orthogonl if u 0. The terms orthogonl nd perpendiculr men essentill the sme thing meeting t right ngles. Note tht the zero ector is orthogonl to eer ector u, becuse 0 u 0.

33 .4 Vectors nd Dot Products 89 4 = 6, u =, Figure.6 w F Figure.7 w EXMPLE 5 Determining Orthogonl Vectors re the ectors u, nd 6, 4 orthogonl? Solution Find the dot product of the two ectors. u, 6, ecuse the dot product is 0, the two ectors re b definition orthogonl (see Figure.6). Finding Vector omponents You he lred seen pplictions in which two ectors re dded to produce resultnt ector. Mn pplictions in phsics nd engineering pose the reerse problem decomposing gien ector into the sum of two ector components. onsider bot on n inclined rmp, s shown in Figure.7. The force F due to grit pulls the bot down the rmp nd ginst the rmp. These two orthogonl forces, nd w, re ector components of F. Tht is, w 0 F w w. Vector components of F The negtie of component w represents the force needed to keep the bot from rolling down the rmp, wheres w represents the force tht the tires must withstnd ginst the rmp. DEFINITION OF VETOR OMPONENTS Let u nd be nonzero ectors such tht w u u w w θ w where w nd w re orthogonl nd w is prllel to (or sclr multiple of), s shown in Figure.8. The ectors w nd w re clled ector components of u. The ector is the projection of u onto nd is denoted b w w proj u. is cute. The ector w is clled the ector component of u orthogonl to nd is gien b w u w. u w From the definition of ector components, ou cn see tht it is es to find the component w once ou he found the projection of u onto. To find the projection, ou cn use the dot product. θ THEOREM.8 PROJETION OF u ONTO w Let u nd be nonzero ectors. The projection of u onto is is obtuse. Figure.8 proj u u.

34 840 hpter dditionl Topics in Trigonometr = 6, w u =, 5 w 4 5 Figure.9 EXMPLE 6 Decomposing Vector into Orthogonl omponents Find the projection of u, 5 onto 6,. Then write u s the sum of two orthogonl ectors, one of which is proj u. Solution The projection of u onto is s shown in Figure.9. The other component, w, is w u w So, w proj u u, 5 6 5, 5 9 7, 5 5. u w w 6 5, 5 9 7, 5 5, , 6 5, 5 w 0 F Figure.40 EXMPLE 7 Finding Force 00-pound crt sits on rmp inclined t 0, s shown in Figure.40. Wht force is required to keep the crt from rolling down the rmp? Solution ecuse the force due to grit is erticl nd downwrd, ou cn represent the grittionl force b the ector F 00j. Force due to grit To find the force required to keep the crt from rolling down the rmp, project F onto unit ector in the direction of the rmp, s follows. cos 0i sin 0j i j Therefore, the projection of F onto is w proj F F F i j. Unit ector long rmp F 0i 00j The mgnitude of this force is 00, nd so force of 00 pounds is required to keep the crt from rolling down the rmp.

35 .4 Vectors nd Dot Products 84 Work The work W done b constnt force F cting long the line of motion of n object is gien b W mgnitude of forcedistnce F PQ \ s shown in Figure.4(). If the constnt force F is not directed long the line of motion, s shown in Figure.4(b), the work W done b the force is gien b \ W proj PQ F PQ \ cos F PQ \ F PQ \. F F θ proj PQ F P Q P Q Work = F PQ Work = proj PQ F PQ () Force cts long the line of motion. (b) Force cts t n ngle with the line of motion. Figure.4 This notion of work is summrized in the following definition. DEFINITION OF WORK The work W done b constnt force F s its point of ppliction moes long the ector PQ \ is gien b either of the following.. W proj PQ \F PQ \ Projection form. W F PQ \ Dot product form EXMPLE 8 Finding Work To close sliding brn door, person pulls on rope with constnt force of 50 pounds t constnt ngle of 60, s shown in Figure.4. Find the work done in moing the brn door feet to its closed position. P F ft proj PQ F 60 Q Solution Using projection, ou cn clculte the work s follows. W proj PQ \F PQ \ Projection form for work cos 60FPQ \ 50 Figure.4 ft 00 foot-pounds So, the work done is 00 foot-pounds. You cn erif this result b finding the ectors F nd PQ \ nd clculting their dot product.

36 84 hpter dditionl Topics in Trigonometr.4 Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 6, fill in the blnks.. The of two ectors ields sclr, rther thn ector.. The dot product of u u, u nd, is u.. If is the ngle between two nonzero ectors u nd, then cos. 4. The ectors u nd re if u The projection of u onto is gien b proj u. 6. The work W done b constnt force F s its point of ppliction moes long the ector PQ \ is gien b W or W. In Eercises 7 0, find the dot product of u nd. 7. u 7, 8. u 6, 0,, 9. u 4i j 0. u i 4j i j 7i j In Eercises 4, use the ectors u, nd 4, to find the indicted quntit. Stte whether the result is ector or sclr.. u u. u. u 4. u In Eercises 5 0, use the dot product to find the mgnitude of u. 5. u 8, 5 6. u 4, 6 7. u 0i 5j 8. u i 6j 9. u 6j 0. u i In Eercises 0, find the ngle between the ectors.. u, 0. u, 0, 4, 0. u i 4j 4. u i j j i j 5. u i j 6. u 6i j 6i 4j 8i 4j 7. u 5i 5j 8. u i j 6i 6j 4i j 9. u cos i sin j cos 4 i sin 4 j 0. In Eercises 4, use ectors to find the interior ngles of the tringle with the gien ertices..,,, 4,, 5., 4,, 7, 8,., 0,,, 0, 6 4., 5,, 9, 7, 9 In Eercises 5 8, find u, where is the ngle between u nd u cos cos 4 i sin 4 j i sin j u 4, 0, u 00, 50, u 9, 6, u 4,, In Eercises 9 44, determine whether u nd re orthogonl, prllel, or neither. 9. u, u, 5, 5 4, 5 4. u 4i j 4. u i 5i 6j i j 4. u i j 44. u cos, sin i j sin, cos In Eercises 45 48, find the projection of u onto. Then write u s the sum of two orthogonl ectors, one of which is proj u. 45. u, 46. u 4, 6,, 47. u 0, 48. u,, 5 4, In Eercises 49 5, find two ectors in opposite directions tht re orthogonl to the ector u. (There re mn correct nswers.) 49. u, u 8, 5. u i j 5. u 5 i j 4 6

37 .4 Vectors nd Dot Products 84 Work In Eercises 5 nd 54, find the work done in moing prticle from P to Q if the mgnitude nd direction of the force re gien b. 5. P0, 0, Q4, 7,, P,, Q, 5, i j WRITING OUT ONEPTS 55. Under wht conditions is the dot product of two ectors equl to the product of the lengths of the ectors? 56. Two forces of the sme mgnitude F nd F ct t ngles nd respectiel. ompre the work done b F with the work done b F in moing long the ector if (). \ PQ (b) 60 nd Reenue The ector u 4600, 550 gies the numbers of units of two models of cellulr phones produced b telecommunictions compn. The ector 79.99, gies the prices (in dollrs) of the two models of cellulr phones, respectiel. () Find the dot product u nd interpret the result in the contet of the problem. (b) Identif the ector opertion used to increse the prices b 5%. 58. rking Lod truck with gross weight of 0,000 pounds is prked on slope of d (see figure). ssume tht the onl force to oercome is the force of grit. d Weight = 0,000 lb () Find the force required to keep the truck from rolling down the hill in terms of the slope d. (b) Use grphing utilit to complete the tble. d Force d Force (c) Find the force perpendiculr to the hill when d Work Determine the work done b person lifting 45-newton bg of sugr meters. 60. Work Determine the work done b crne lifting 400-pound cr 5 feet. 6. Work force of 45 pounds eerted t n ngle of 0 boe the horizontl is required to slide tble cross floor (see figure). The tble is drgged 0 feet. Determine the work done in sliding the tble. 0 0 ft 45 lb 6. Work trctor pulls log 800 meters, nd the tension in the cble connecting the trctor nd log is pproimtel 5,69 newtons. The direction of the force is 5 boe the horizontl. pproimte the work done in pulling the log. True or Flse? In Eercises 6 65, determine whether the sttement is true or flse. If it is flse, eplin wh or gie n emple tht shows it is flse. 6. The work W done b constnt force F cting long the line of motion of n object is represented b ector. 64. sliding door moes long the line of ector PQ \. If force is pplied to the door long ector tht is orthogonl to PQ \, then no work is done. 65. The dot product of two ectors is sclr tht is lws nonnegtie. PSTONE 66. Wht is known bout, the ngle between two nonzero ectors u nd, under ech condition (see figure)? u θ Origin () u 0 (b) u > 0 (c) u < Proe the following Properties of the Dot Product. () 0 0 (b) (c) cu u c 68. Proe tht 4u u u. u w u u w

38 844 hpter dditionl Topics in Trigonometr.5 Trigonometric Form of omple Number Plot comple numbers in the comple plne nd find bsolute lues of comple numbers. Write the trigonometric forms of comple numbers. Multipl nd diide comple numbers written in trigonometric form. Use DeMoire s Theorem to find powers of comple numbers. Find nth roots of comple numbers. The omple Plne Just s rel numbers cn be represented b points on the rel number line, ou cn represent comple number z bi s the point, b in coordinte plne (the comple plne). The horizontl is is clled the rel is nd the erticl is is clled the imginr is, s shown in Figure.4. Imginr is (, ) or + i (, ) or i Rel is Figure.4 The bsolute lue of the comple number bi is defined s the distnce between the origin 0, 0 nd the point, b. DEFINITION OF THE SOLUTE VLUE OF OMPLEX NUMER The bsolute lue of the comple number z bi is bi b. (, 5) Figure.44 Imginr is 5 4 Rel is NOTE If the comple number bi is rel number (tht is, if b 0), then this definition grees with tht gien for the bsolute lue of rel number EXMPLE Finding the bsolute Vlue of omple Number Plot z 5i nd find its bsolute lue. Solution 0i 0. The number is plotted in Figure.44. It hs n bsolute lue of z 5 9.

39 .5 Trigonometric Form of omple Number 845 (, b) b r Imginr is θ Rel is Trigonometric Form of omple Number In Section.4, ou lerned how to dd, subtrct, multipl, nd diide comple numbers. To work effectiel with powers nd roots of comple numbers, it is helpful to write comple numbers in trigonometric form. In Figure.45, consider the nonzero comple number bi. letting be the ngle from the positie rel is (mesured counterclockwise) to the line segment connecting the origin nd the point, b, ou cn write nd r cos Figure.45 b r sin where r b. onsequentl, ou he bi r cos r sin i from which ou cn obtin the trigonometric form of comple number. TRIGONOMETRI FORM OF OMPLEX NUMER The trigonometric form of the comple number z bi is z rcos i sin where r cos, b r sin, r b, nd tn b. The number r is the modulus of z, nd is clled n rgument of z. The trigonometric form of comple number is lso clled the polr form. ecuse there re infinitel mn choices for, the trigonometric form of comple number is not unique. Normll, is restricted to the interl 0 <, lthough on occsion it is conenient to use < 0. EXMPLE Writing omple Number in Trigonometric Form Write the comple number z i in trigonometric form. Solution The bsolute lue of z is r i Imginr is 4π Rel is 6 4 nd the reference ngle tn b is gien b. z = 4 z = i 4 ecuse tn nd becuse z i lies in Qudrnt III, ou choose to be So, the trigonometric form is 4. z rcos i sin 4 cos 4 4 i sin. Figure.46 See Figure.46.

40 846 hpter dditionl Topics in Trigonometr EXMPLE Writing omple Number in Stndrd Form Write the comple number in stndrd form bi. Solution ou cn write ecuse cos 6i. z 8 cos i sin nd z 8 cos i sin i sin TEHNOLOGY You cn use grphing utilit to conert comple number in trigonometric (or polr) form to stndrd form. For specific kestrokes, see the user s mnul for our grphing utilit. Multipliction nd Diision of omple Numbers The trigonometric form dpts nicel to multipliction nd diision of comple numbers. Suppose ou re gien two comple numbers z nd z r cos i sin r cos i sin. The product of z nd is gien b z z z r r cos i sin cos i sin r r cos cos sin sin isin cos cos sin. Using the sum nd difference formuls for cosine nd sine, ou cn rewrite this eqution s z z r r cos i sin. This estblishes the first prt of the following rule. The second prt is left for ou to erif (see Eercise 5). PRODUT ND QUOTIENT OF TWO OMPLEX NUMERS Let z nd z r cos i sin r cos i sin be comple numbers. z z r r cos i sin Product z r cos z r i sin, z 0 Quotient Note tht this rule ss tht to multipl two comple numbers ou multipl moduli nd dd rguments, wheres to diide two comple numbers ou diide moduli nd subtrct rguments.

41 .5 Trigonometric Form of omple Number 847 EXMPLE 4 Multipling omple Numbers TEHNOLOGY Some grphing utilities cn multipl nd diide comple numbers in trigonometric form. If ou he ccess to such grphing utilit, use it to find z z nd z z in Emples 4 nd 5. Find the product z z of the comple numbers. Solution z cos i sin z 8 cos i sin 6 6 Use the formul for multipling comple numbers. z z r r cos i sin 8 cos 6 cos 5 6 cos i sin 60 i 6i You cn check this result b first conerting the comple numbers to the stndrd forms z i nd nd then multipling, s in Section.4. z z i4 4i 4 4i i 4 6i i sin 5 6 i sin 5 nd re coterminl. z 4 4i 6 EXMPLE 5 Diiding omple Numbers Find the quotient z z of the comple numbers. z 4cos 00 i sin 00 z 8cos 75 i sin 75 Solution Use the formul for diiding comple numbers. z r cos z r i sin 4 cos00 75 i sin cos 5 i sin 5 i i

42 848 hpter dditionl Topics in Trigonometr Powers of omple Numbers The trigonometric form of comple number is used to rise comple number to power. To ccomplish this, consider repeted use of the multipliction rule. z rcos i sin z rcos i sin rcos i sin r cos i sin z r cos i sin rcos i sin r cos i sin z 4 r 4 cos 4 i sin 4 z 5 r 5 cos 5 i sin 5. This pttern leds to DeMoire s Theorem, which is nmed fter the French mthemticin brhm DeMoire ( ). The Grnger ollection, New York RHM DEMOIVRE ( ) DeMoire is remembered for his work in probbilit theor nd DeMoire s Theorem. His book The Doctrine of hnces (published in 78) includes the theor of recurring series nd the theor of prtil frctions. THEOREM.9 DEMOIVRE S THEOREM If z rcos i sin is comple number nd n is positie integer, then z n rcos i sin n r n cos n i sin n. EXMPLE 6 Finding Powers of omple Number i 6 cos i sin 6 cos i sin EXMPLE 7 Finding Powers of omple Number Use DeMoire s Theorem to find i. Solution First conert the comple number to trigonometric form using r So, the trigonometric form is nd z i cos i sin. Then, b DeMoire s Theorem, ou he i cos i sin cos i sin 4096cos 8 i sin r, n 6 cos, sin 0 rctn.

43 .5 Trigonometric Form of omple Number 849 Roots of omple Numbers Recll tht consequence of the Fundmentl Theorem of lgebr is tht polnomil eqution of degree n hs n solutions in the comple number sstem. So, the eqution 6 hs si solutions, nd in this prticulr cse ou cn find the si solutions b fctoring nd using the Qudrtic Formul. 6 onsequentl, the solutions re ±, 0 ± i, nd ± i. Ech of these numbers is sith root of. In generl, n nth root of comple number is defined s follows. DEFINITION OF N nth ROOT OF OMPLEX NUMER The comple number u bi is n nth root of the comple number z if z u n bi n. EXPLORTION The nth roots of comple number re useful for soling some polnomil equtions. For instnce, eplin how ou cn use DeMoire s Theorem to sole the polnomil eqution [ Hint: Write 6 s 6cos i sin. ] To find formul for n nth root of comple number, let u be n nth root of z, where u scos i sin nd z rcos i sin. DeMoire s Theorem nd the fct tht u n z, ou he s n cos n i sin n rcos i sin. Tking the bsolute lue of ech side of this eqution, it follows tht s n r. Substituting bck into the preious eqution nd diiding b r, ou get cos n i sin n cos i sin. So, it follows tht cos n cos nd sin n sin. ecuse both sine nd cosine he period of, these lst two equtions he solutions if nd onl if the ngles differ b multiple of. onsequentl, there must eist n integer k such tht n k k n. substituting this lue of into the trigonometric form of u, ou get the result stted in Theorem.0 on the following pge.

44 850 hpter dditionl Topics in Trigonometr THEOREM.0 nth ROOTS OF OMPLEX NUMER For positie integer n, the comple number z rcos i sin hs ectl n distinct nth roots gien b nr cos k n i sin where k 0,,,..., n. k n Imginr is NOTE When k eceeds n, the roots begin to repet. For instnce, if k n, the ngle n n n n r Figure.47 π n π n Rel is is coterminl with n, which is lso obtined when k 0. The formul for the nth roots of comple number z hs nice geometricl interprettion, s shown in Figure.47. Note tht becuse the nth roots of z ll he n n the sme mgnitude r, the ll lie on circle of rdius r with center t the origin. Furthermore, becuse successie nth roots he rguments tht differ b n, the n roots re equll spced round the circle. You he lred found the sith roots of b fctoring nd b using the Qudrtic Formul. Emple 8 shows how ou cn sole the sme problem with the formul for nth roots. EXMPLE 8 Find the nth Roots of Rel Number + i + 0i i i Figure.48 Imginr is + i + 0i Rel is Find ll sith roots of. Solution First write in the trigonometric form cos 0 i sin 0. Then, b Theorem.0, with n 6 nd r, the roots he the form 6 cos So, for k 0,,,, 4, nd 5, the sith roots re s follows. (See Figure.48.) cos 0 i sin 0 cos i sin cos i sin i cos i sin cos 4 i sin i cos 5 i sin i k 6 i sin 0 k k k 6 cos i sin. i Increment b In Figure.48, notice tht the roots obtined in Emple 8 ll he mgnitude of nd re equll spced round the unit circle. lso notice tht the comple roots occur in conjugte pirs, s discussed in Section.5. The n distinct nth roots of re clled the nth roots of unit. n 6

45 .5 Trigonometric Form of omple Number 85 EXMPLE 9 Finding the nth Roots of omple Number Find the three cube roots of z i. Solution ecuse z lies in Qudrnt II, the trigonometric form of z is i Figure.49 Imginr is + i i Rel is z i 8 cos 5 i sin 5. Theorem.0, the cube roots he the form 68 cos Finll, for k 0,, nd, ou obtin the roots 68 cos i 68 cos 560 i sin 560 cos 65i sin cos See Figure k i sin 5º 60k. rctn i sin cos 45 i sin i 5 60 i sin cos 85 i sin i..5 Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 4, fill in the blnks.. The of comple number bi is the distnce between the origin 0, 0 nd the point, b.. The of comple number z bi is gien b z rcos i sin, where r is the of z nd is the of z.. Theorem sttes tht if z rcos i sin is comple number nd n is positie integer, then z n r n cos n i sin n. 4. The comple number u bi is n of the comple number z if z u n bi n. In Eercises 5 0, plot the comple number nd find its bsolute lue i 6. 5 i 7. 7i i 0. 8 i In Eercises 4, write the comple number in trigonometric form.. Imginr.. Imginr is z = i z = i 4 is Rel is Rel is 4 z = 6 4 z = + i Rel is In Eercises 5 4, represent the comple number grphicll, nd find the trigonometric form of the number. 5. i i 7. i i Imginr is 4 Imginr is Rel is

46 85 hpter dditionl Topics in Trigonometr 9. i 0. 5 i. 5i. i. 7 4i 4. i i 8. i 9. i 0. i. 5 i. 8 i. 8 5i i In Eercises 5 44, find the stndrd form of the comple number. Then represent the comple number grphicll. 5. cos 60 i sin cos 5 i sin cos0 i sin0 8. 8cos 5 i sin cos i sin cos 5 5 i sin 4. 7cos 0 i sin cos i sin 4. 5cos98 45 i sin cos80º 0 i sin80º 0 In Eercises 45 48, use grphing utilit to represent the comple number in stndrd form cos i sin cos i sin cos 55 i sin cos 58º i sin 58º In Eercises 49 60, perform the opertion nd lee the result in trigonometric form. cos 49. i sin cos i sin cos i sin 4 cos i sin cos 0i sin 0 cos 0i sin 0 5. cos 00i sin cos 00 i sin cos 80 i sin 80cos 0 i sin cos 5 i sin 5cos 0 i sin cos 50 i sin 50 9cos 0 i sin 0 cos 0 i sin 0 cos 40 i sin 40 cos i sin cos i sin 5cos 4. i sin 4. 4cos. i sin. cos 9 i sin 9 cos i sin 6cos 40 i sin 40 7cos 00 i sin 00 In Eercises 6 68, () write the trigonometric forms of the comple numbers, (b) perform the indicted opertion using the trigonometric forms, nd (c) perform the indicted opertion using the stndrd forms, nd check our result with tht of prt (b). 6. i i 6. i i 6. i i 64. i i 65. 4i i 66. i 6 i i 68. i 4 i In Eercises 69 7, sketch the grphs of ll comple numbers z stisfing the gien condition z In Eercises 7 nd 74, represent the powers z, z, z, nd z 4 grphicll. Describe the pttern. 7. z i 74. z i z 5 In Eercises 75 9, use DeMoire s Theorem to find the indicted power of the comple number. Write the result in stndrd form. 75. i i i i i 0 4 i 5cos 0 i sin 0 8. cos 60 i sin

47 .5 Trigonometric Form of omple Number cos i sin cos i sin cos. i sin cos 0 i sin i i 89. cos 5 i sin cos 0i sin cos i sin cos i sin 8 In Eercises 9 08, () use Theorem.0 to find the indicted roots of the comple number, (b) represent ech of the roots grphicll, nd (c) write ech of the roots in stndrd form. 9. Squre roots of 5cos 0 i sin Squre roots of 6cos 60 i sin ube roots of 8 cos i sin 96. Fifth roots of cos 5 5 i sin ube roots of 5 i 98. ube roots of 4 i 99. Squre roots of 5i 00. Fourth roots of 65i 0. Fourth roots of 6 0. Fourth roots of i 0. Fifth roots of 04. ube roots of ube roots of Fourth roots of Fifth roots of 4 i 08. Sith roots of 64i In Eercises 8, use Theorem.0 to find ll the solutions of the eqution nd represent the solutions grphicll.. 4 i i i 0 7. i i WRITING OUT ONEPTS In Eercises 09 nd 0, use the figure. One of the fourth roots of comple number z is shown. 09. How mn roots re not shown? 0. Describe the other roots. Imginr is z 0 Rel is True or Flse? In Eercises 9, determine whether the sttement is true or flse. If it is flse, eplin wh or gie n emple tht shows it is flse. 9. lthough the squre of the comple number bi is gien b bi b, the bsolute lue of the comple number z bi is defined s bi b. 0. Geometricll, the nth roots of n comple number z re ll equll spced round the unit circle centered t the origin.. The product of the two comple numbers z nd z r cos i sin r cos i sin is zero onl when r 0 nd/or r 0.. DeMoire s Theorem, 4 6i 8 cos i sin86.. DeMoire s Theorem, i 64cos i sin. PSTONE 4. Use the grph of the roots of comple number. () Write ech of the roots in trigonometric form. (b) Identif the comple number whose roots re gien. Use grphing utilit to erif our results. (i) Imginr is 0 0 Rel is 5. Gien two comple numbers z r cos i sin nd z r cos i sin, z 0, show tht z r cos z r i sin. 6. Show tht z rcos i sin is the comple conjugte of z rcos i sin. 7. Use the trigonometric forms of z nd z in Eercise 6 to find () zz nd (b) zz, z Show tht the negtie of z rcos i sin is z r cos i sin. 9. Show tht i is ninth root of. 0. Show tht 4 i is fourth root of. (ii) Imginr is Rel is

48 854 hpter dditionl Topics in Trigonometr HPTER SUMMRY Section. Reiew Eercises Use the Lw of Sines to sole oblique tringles (S, S, or SS) (p. 808). Find the res of oblique tringles (p. 8). 6 Use the Lw of Sines to model nd sole rel-life problems (p. 8). 7 0 Section. Use the Lw of osines to sole oblique tringles (SSS or SS) (p. 87). 0 Use Heron s re Formul to find the re of tringle (p. 88). 5 8 Use the Lw of osines to model nd sole rel-life problems (p. 89). 9, 40 Section. Represent ectors s directed line segments (p. 84) Write the component forms of ectors (p. 85) Perform bsic ector opertions nd represent them grphicll (p. 86). 5 6 Write ectors s liner combintions of unit ectors (p. 88) Find the direction ngles of ectors (p. 89) Use ectors to model nd sole rel-life problems (p. 80) Section.4 Find the dot product of two ectors nd use the properties of the dot product (p. 86) Find the ngle between two ectors nd determine whether two ectors re orthogonl (p. 87) Write ector s the sum of two ector components (p. 89) Use ectors to find the work done b force (p. 84). 99, 00 Section.5 Plot comple numbers in the comple plne nd find bsolute lues of comple numbers (p. 844) Write the trigonometric forms of comple numbers (p. 845) Multipl nd diide comple numbers written in trigonometric form (p. 846)., Use DeMoire s Theorem to find powers of comple numbers (p. 848). 6 Find nth roots of comple numbers (p. 849). 7 8

49 Reiew Eercises 855 REVIEW EXERISES See for worked-out solutions to odd-numbered eercises. In Eercises, use the Lw of Sines to sole (if possible) the tringle. If two solutions eist, find both. Round our nswers to two deciml plces... In Eercises 6, find the re of the tringle hing the indicted ngle nd sides.., b 7, c , 4, c , 8, b 6 6., b, c 7. Height From certin distnce, the ngle of eletion to the top of building is 7. t point 50 meters closer to the building, the ngle of eletion is. pproimte the height of the building. 8. Geometr Find the length of the side w of the prllelogrm. w c 8 c b b. 7, 8, b , 0, c 5. 6, 98, c , 45, c , 48, b , 6, , b 0, c , 0, b. 75, 5., b.7. 5, 6., b = 8 = Height tree stnds on hillside of slope 8 from the horizontl. From point 75 feet down the hill, the ngle of eletion to the top of the tree is 45 (see figure). Find the height of the tree. Figure for 9 0. Rier Width sureor finds tht tree on the opposite bnk of rier flowing due est hs bering of N 0 E from certin point nd bering of N 5 W from point 400 feet downstrem. Find the width of the rier. In Eercises 0, use the Lw of osines to sole the tringle. Round our nswers to two deciml plces... b = 4 = 8 b = 4 = 7 00 c = ft 45. 6, b 9, c , b 50, c , b 5.0, c , b 8.8, c ,, c 8. 50, 0, c ,.5, b , b.4, c 9.5 In Eercises 4, determine whether the Lw of Sines or the Lw of osines is needed to sole the tringle. Then sole the tringle.. b 9, c, 64. 4, c 5, 5., b 5, c ,, c.8 In Eercises 5 8, use Heron s re Formul to find the re of the tringle. 5., b 6, c , b 8, c 0 7.., b 5.8, c , b 4, c 5 8 c

50 856 hpter dditionl Topics in Trigonometr 9. Sureing To pproimte the length of mrsh, sureor wlks 45 meters from point to point. Then the sureor turns 65 nd wlks 00 meters to point (see figure). pproimte the length of the mrsh u i j, 5i j u 7i j, 4i j u 4i, i 6j u 6j, i j 00 m m In Eercises 59 6, find the component form of w nd sketch the specified ector opertions geometricll, where u 6i 5j nd 0i j. 59. w u 60. w 6. w 6. 4u 5 w 40. Nigtion Two plnes lee n irport t pproimtel the sme time. One is fling 45 miles per hour t bering of 55, nd the other is fling 50 miles per hour t bering of 67. Drw figure tht gies isul representtion of the sitution nd determine the distnce between the plnes fter the he flown for hours. In Eercises 4 44, grph the ector with the gien initil point nd terminl point. Initil Point 4. 0, 0 4., 4 4., , 8 In Eercises 45 50, find the component form of the ector stisfing the conditions ( 5, 4) (, ) 47. Initil point: 0, 0; terminl point: 7, 48. Initil point:, 5; terminl point: 5, , 50., 0 Terminl Point 8, 7 5, 7 8, 4 8, 5 In Eercises 5 58, find () u, (b) u, (c) 4u, nd (d) 5u. 5. u,,, 6 5. u 4, 5, 0, 5. u 5,, 4, u, 8,, 6 4 ( (0, ) ) 7 6, 4 6 In Eercises 6 66, write ector u s liner combintion of the stndrd unit ectors i nd j. 6. u, u 6, u hs initil point, 4 nd terminl point 9, u hs initil point, 7 nd terminl point 5, 9. In Eercises 67 nd 68, write the ector in the form cos i sin j i 0j 68. 4i j In Eercises 69 74, find the mgnitude nd the direction ngle of the ector cos 60i sin 60j cos 50i sin 50j 7. 5i 4j 7. 4i 7j 7. i j 74. 8i j 75. Resultnt Force Forces with mgnitudes of 85 pounds nd 50 pounds ct on single point. The ngle between the forces is 5. Describe the resultnt force. 76. Rope Tension 80-pound weight is supported b two ropes, s shown in the figure. Find the tension in ech rope lb 77. Nigtion n irplne hs n irspeed of 40 miles per hour t bering of 5. The wind elocit is 5 miles per hour in the direction of N 0E. Find the resultnt speed nd direction of the irplne. 78. Nigtion n irplne hs n irspeed of 74 kilometers per hour t bering of 0. The wind elocit is kilometers per hour from the west. Find the resultnt speed nd direction of the irplne.

51 Reiew Eercises 857 In Eercises 79 8, find the dot product of u nd. 79. u 6, u 7,, 9 4, 4 8. u i 7j 8. u 7i j i 5j 6i j In Eercises 8 86, use the ectors u 4, nd 5, to find the indicted quntit. Stte whether the result is ector or sclr. 8. u u 84. u 85. uu 86. u In Eercises 87 90, find the ngle u cos 7 7 i sin 4 4 j cos 5 5 i sin 6 6 j u cos 45i sin 45j cos 00i sin 00j u, 4,, u,, 4, between the ectors. In Eercises 9 94, determine whether u nd re orthogonl, prllel, or neither. 9. u, 8 9. u 4, 8,, 4 9. u i 94. u i j i j i 6j In Eercises 95 98, find the projection of u onto. Then write u s the sum of two orthogonl ectors, one of which is proj u. 95. u 4,, 8, 96. u 5, 6, 0, u, 7,, 98. u, 5, 5, Work In Eercises 99 nd 00, find the work done in moing prticle from P to Q if the mgnitude nd direction of the force re gien b. 99. P5,, Q8, 9,, P, 9, Q, 8, i 6j In Eercises 0 04, plot the comple number nd find its bsolute lue. 0. 7i 0. 6i 0. 5 i 04. i In Eercises 05 0, write the comple number in trigonometric form i i i i 0. i In Eercises nd, () write the two comple numbers in trigonometric form, nd (b) use the trigonometric forms to find z z nd z /z, where z 0... z i, z i, z 0i In Eercises 6, use DeMoire s Theorem to find the indicted power of the comple number. Write the result in stndrd form. 5 cos. i sin cos i sin i 6 6. i 8 Grphicl Resoning In Eercises 7 nd 8, use the grph of the roots of comple number. () Write ech of the roots in trigonometric form. (b) Identif the comple number whose roots re gien. Use grphing utilit to erif our results. 7. Imginr 8. 4 is z i Rel is Imginr is Rel is In Eercises 9, () use Theorem.0 to find the indicted roots of the comple number, (b) represent ech of the roots grphicll, nd (c) write ech of the roots in stndrd form. 9. Sith roots of 79i 0. Fourth roots of 56i. ube roots of 8. Fifth roots of 04 In Eercises 8, use Theorem.0 to find ll solutions of the eqution nd represent the solutions grphicll i i

52 858 hpter dditionl Topics in Trigonometr HPTER TEST Tke this test s ou would tke test in clss. When ou re finished, check our work ginst the nswers gien in the bck of the book. In Eercises 6, use the informtion to sole (if possible) the tringle. If two solutions eist, find both solutions. Round our nswers to two deciml plces.. 4, 68,.. 0, 8, ,., b , b 7., c , 5, b 6., 4, b tringulr prcel of lnd hs borders of lengths 60 meters, 70 meters, nd 8 meters. Find the re of the prcel of lnd. 40 mi 8. n irplne flies 70 miles from point to point with bering of 4. It then flies 40 miles from point to point with bering of 7(see figure). Find the distnce nd bering from point to point. 70 mi 7 In Eercises 9 nd 0, find the component form of the ector stisfing the gien conditions. 9. Initil point of :, 7; terminl point of :, 6 0. Mgnitude of : ; direction of : u, 5 In Eercises 4, nd < > 6, 5. Find the resultnt ector nd sketch its grph. u <, 7 > 4. u. u. 5u 4. 4u Figure for 8 5. Find unit ector in the direction of u 4, Forces with mgnitudes of 50 pounds nd 0 pounds ct on n object t ngles of 45 nd 60, respectiel, with the -is. Find the direction nd mgnitude of the resultnt of these forces. 7. Find the ngle between the ectors u, 5 nd,. 8. re the ectors u 6, 0 nd 5, orthogonl? 9. Find the projection of u 6, 7 onto 5,. Then write u s the sum of two orthogonl ectors pound motorccle is heded up hill inclined t. Wht force is required to keep the motorccle from rolling down the hill when stopped t red light?. Write the comple number z 4 4i in trigonometric form.. Write the comple number z 6cos 0 i sin 0 in stndrd form. In Eercises nd 4, use DeMoire s Theorem to find the indicted power of the comple number. Write the result in stndrd form cos 6 i 6 i sin Find the fourth roots of 56 i. 6. Find ll solutions of the eqution 7i 0 nd represent the solutions grphicll.

53 P.S. Problem Soling 859 P.S. PROLEM SOLVING. In the figure, nd re positie ngles nd the sides re mesured in centimeters. () Write s function of nd determine its domin. (b) Differentite the function nd use the deritie to find the mimum of the function. Wht is the rnge of the function? (c) Use grphing utilit to grph the function. (d) If ddt 0. rdin per second, find ddt when (e) Write c s function of nd determine its domin. (f) Use grphing utilit to grph the function in prt (e). Wht is the rnge of the function? (g) If ddt 0. rdin per second, find dcdt when (h) Use grphing utilit to complete the tble. c α (i) Eplin the lue for c in the tble when. onsider two forces () Find F F s function of. (b) Use grphing utilit to grph the function in prt () for 0 <. (c) Use the grph in prt (b) to determine the rnge of the function. Wht is its mimum, nd for wht lue of does it occur? Wht is its minimum, nd for wht lue of does it occur? (d) Eplin wh the mgnitude of the resultnt is neer 0.. Write the ector w in terms of u nd, gien tht the terminl point of w bisects the line segment. () (b) 8 c w γ u β 9 F 0, 0 nd F 5cos, sin u w Use the Lw of osines to proe tht b c bc cos 5. Use the Lw of osines to proe tht b c bc cos 6. Let R nd r be the rdii of the circumscribed nd inscribed circles of tringle, respectiel (see figure), nd let s b c. R () Proe tht R (b) Proe tht r s s bs c. s (c) Gien tringle with 5, b 55, nd c 7, find the res of (i) the tringle, (ii) the circumscribed circle, nd (iii) the inscribed circle. (d) Find the length of the lrgest circulr trck tht cn be built on tringulr piece of propert with sides of lengths 00 feet, 50 feet, nd 5 feet. 7. () Use n re formul for oblique tringles to find the re of the tringle in the figure. (b) Find the equtions of the two nonerticl lines nd use integrtion to find the re of the tringle. b 8. Proe tht if u is orthogonl to nd w, then u is orthogonl to c dw 0 5 for n sclrs c nd d. r c sin 5 b c. b c. b sin c sin.

54 860 hpter dditionl Topics in Trigonometr 9. Gien two ectors u nd () proe tht u u u. (b) The eqution in prt () is clled the Prllelogrm Lw. Use the figure to write geometric interprettion of the Prllelogrm Lw. 0. In the figure, bem of light is directed t the blue mirror, reflected to the red mirror, nd then reflected bck to the blue mirror. Find the distnce PT tht the light trels from the red mirror bck to the blue mirror. 5. For ech pir of ectors, find the following. (i) u (ii) (iii) u (i) u u () (i) u u () u,, (c) u,,. The fmous formul (b) (d) is clled Euler s Formul, fter the Swiss mthemticin Leonhrd Euler (707 78). This formul gies rise to one of the most wonderful equtions in mthemtics. ei ft e bi e cos b i sin b u u u + O T Q 6 ft θ P u 0,, u, 4 5, 5 This elegnt eqution reltes the fie most fmous numbers in mthemtics 0,,, e, nd i in single eqution. Show how Euler s Formul cn be used to derie this eqution. θ α Red mirror α lue mirror. hiking prt is lost in ntionl prk. Two rnger sttions he receied n emergenc SOS signl from the prt. Sttion is 75 miles due est of Sttion. The bering from Sttion to the signl is S 60 E nd the bering from Sttion to the signl is S 75 W. () Find the distnce from ech sttion to the SOS signl. (b) rescue prt is in the prk 0 miles from Sttion t bering of S 80 E. Find the distnce nd the bering the rescue prt must trel to rech the lost hiking prt. 4. The figure shows z nd z. Describe z z nd z z. z θ Imginr is 5. trithlete sets course to swim S 5 E from point on shore to buo 4 mile w. fter swimming 00 rds through strong current, the trithlete is off course t bering of S 5 E. Find the bering nd distnce the trithlete needs to swim to correct her course. 6. Find the olume of the right tringulr prism in terms of, where V h. is the re of the bse nd h is the height of the prism. 60 θ z Rel is 00 d 5 4 mi uo W N S E