Additional Topics in Trigonometry

Transcription

1 dditionl Topis in Trigonometr 6 6. Lw of Sines 6. Lw of osines 6. Vetors in the Plne 6. Vetors nd Dot Produts 6.5 Trigonometri Form of omple Numer In Mthemtis Trigonometr is used to sole tringles, represent etors, nd to write trigonometri forms of omple numers. In Rel Life Trigonometr is used to find res, estimte heights, nd represent etors inoling fore, eloit, nd other quntities. For instne, trigonometr nd etors n e used to find the tension in the tow lines s loded rge is eing towed two tugots. (See Eerise 9, pge 56.) Lu Tettoni/Terr/oris IN REERS There re mn reers tht use trigonometr. Seerl re listed elow. Pilot Eerise 5, pge 5 iil Engineer Eerise 55, pge wning Designer Eerise 58, pge Lndsper Eerise, pge 9 7

2 8 hpter 6 dditionl Topis in Trigonometr 6. LW OF SINES Owen Frnken/oris Wht ou should lern Use the Lw of Sines to sole olique tringles (S or S). Use the Lw of Sines to sole olique tringles (SS). Find the res of olique tringles. Use the Lw of Sines to model nd sole rel-life prolems. Wh ou should lern it You n use the Lw of Sines to sole rel-life prolems inoling olique tringles. For instne, in Eerise 5 on pge 6, ou n use the Lw of Sines to determine the distne from ot to the shoreline. Introdution In hpter, ou studied tehniques for soling right tringles. In this setion nd the net, ou will sole olique tringles tringles tht he no right ngles. s stndrd nottion, the ngles of tringle re leled,, nd, nd their opposite sides re leled,, nd, s shown in Figure 6.. FIGURE 6. To sole n olique 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 reks down into the following four ses.. Two ngles nd n side (S or S). Two sides nd n ngle opposite one of them (SS). Three sides (SSS). Two sides nd their inluded ngle (SS) The first two ses n e soled using the Lw of Sines, wheres the lst two ses require the Lw of osines (see Setion 6.). Lw of Sines If is tringle with sides,, nd, then sin sin sin. h h is ute. is otuse. The Lw of Sines n lso e written in the reiprol form sin sin sin. For proof of the Lw of Sines, see Proofs in Mthemtis on pge 87.

3 Setion 6. Lw of Sines 9 Emple Gien Two ngles nd One Side S = 8 ft FIGURE When soling tringles, reful sketh is useful s quik test for the fesiilit of n nswer. Rememer tht the longest side lies opposite the lrgest ngle, nd the shortest side lies opposite the smllest ngle. For the tringle in Figure 6., 0, 9, nd 8 feet. Find the remining ngle nd sides. The third ngle of the tringle is the Lw of Sines, ou he Using 8 produes nd sin sin sin. 8 sin sin 9.59 feet sin sin 9 8 sin sin feet. sin sin 9 Now tr Eerise 5. Emple Gien Two ngles nd One Side S pole tilts towrd the sun t n 8 ngle from the ertil, nd it sts -foot shdow. The ngle of eletion from the tip of the shdow to the top of the pole is. How tll is the pole? From Figure 6., note tht nd So, the third ngle is the Lw of Sines, ou he = ft sin sin. euse feet, the length of the pole is FIGURE 6. sin sin.8 feet. sin sin 9 Now tr Eerise 5. For prtie, tr reworking Emple for pole tht tilts w from the sun under the sme onditions.

4 0 hpter 6 dditionl Topis in Trigonometr The miguous se (SS) onsider tringle in whih ou re gien,, nd. Sketh Neessr ondition The miguous 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 possile situtions n our: () no suh tringle eists, () one suh tringle eists, or () two distint tringles m stisf the onditions. is ute. is ute. is ute. is ute. is otuse. is otuse. h < h h h h h sin h < < Tringles None One One Two None One possile h > Emple Single- se SS = in. = in. For the tringle in Figure 6., inhes, inhes, nd. Find the remining side nd ngles. One solution: FIGURE 6. the Lw of Sines, ou he sin sin Reiprol form sin sin Multipl eh side. sin sin Sustitute for,, nd... is ute. Now, ou n determine tht Then, the remining side is sin sin sin sin inhes. sin sin Now tr Eerise 5.

5 Setion 6. Lw of Sines = 5 = 5 h 85 No solution: < h FIGURE 6.5 Emple No- se SS Show tht there is no tringle for whih 5, 5, nd 85. egin mking the sketh shown in Figure 6.5. From this figure it ppers tht no tringle is formed. You n erif this using the Lw of Sines. sin sin sin sin sin 85 sin > sin. Reiprol form Multipl eh side. This ontrdits the ft tht So, no tringle n e formed hing sides 5 nd 5 nd n ngle of 85. Now tr Eerise 7. Emple 5 Two- se SS Find two tringles for whih meters, meters, nd 0.5. the Lw of Sines, ou he sin sin sin sin Reiprol form There re two ngles, 6.8 nd , etween 0 nd 80 whose sine is For 6.8, ou otin For 5., ou otin sin sin sin meters. sin sin 0.5 sin sin..9 meters. sin sin 0.5 The resulting tringles re shown in Figure 6.6. = m = m = m = m FIGURE 6.6 Now tr Eerise 9.

6 hpter 6 dditionl Topis in Trigonometr To see how to otin the height of the otuse tringle in Figure 6.7, notie the use of the referene ngle 80 nd the differene formul for sine, s follows. h sin80 sin 80 os os 80 sin 0 os sin sin re of n Olique Tringle The proedure used to proe the Lw of Sines leds to simple formul for the re of n olique tringle. Referring to Figure 6.7, note tht eh tringle hs height of h sin. onsequentl, the re of eh tringle is re seheight sin sin. similr rguments, ou n deelop the formuls re sin sin. h h is ute. FIGURE 6.7 is otuse. re of n Olique Tringle The re of n tringle is one-hlf the produt of the lengths of two sides times the sine of their inluded ngle. Tht is, re sin sin sin. Note tht if ngle is 90, the formul gies the re for right tringle: re sin 90 seheight. sin 90 Similr results re otined for ngles nd equl to 90. Emple 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 inluded ngle of 0. = 5 m FIGURE = 90 m onsider 90 meters, 5 meters, nd ngle 0, s shown in Figure 6.8. Then, the re of the tringle is re sin 905sin 0 89 squre meters. Now tr Eerise 9.

7 Setion 6. Lw of Sines pplition Emple 7 N W E S 5 n pplition of the Lw of Sines The ourse for ot re strts t point in Figure 6.9 nd proeeds in the diretion S 5 W to point, then in the diretion S 0 E to point, nd finll k to. Point lies 8 kilometers diretl south of point. pproimte the totl distne of the re ourse. 8 km 0 euse lines D nd re prllel, it follows tht! D. onsequentl, tringle hs the mesures shown in Figure 6.0. The mesure of ngle is Using the Lw of Sines,. sin 5 sin 88 sin 0 D FIGURE euse 8, = 8 km 0 8 sin 5 sin sin 0 sin nd The totl length of the ourse is pproimtel Length 9.5 kilometers. FIGURE Now tr Eerise 9. LSSROOM DISUSSION Using the Lw of Sines In this setion, ou he een using the Lw of Sines to sole olique tringles. n the Lw of Sines lso e used to sole right tringle? If so, write short prgrph eplining how to use the Lw of Sines to sole eh tringle. Is there n esier w to sole these tringles?. $S#. $S# = 0 = 0

8 hpter 6 dditionl Topis in Trigonometr 6. EXERISES VOULRY: Fill in the lnks.. n tringle is tringle tht hs no right ngle.. For tringle, the Lw of Sines is gien. Two nd one determine unique tringle.. The re of n olique tringle is gien sin sin. SKILLS ND PPLITIONS See for worked-out solutions to odd-numered eerises. sin sin. In Eerises 5, use the Lw of Sines to sole the tringle. Round our nswers to two deiml ples , 55, 8, 6, 60, = 0 0, ,., 8 0, 5 0, 5 0, 9, = 0 65, 5, ,.5, = , 8 5, 0 = 5 5.6, , 0, 5 8 8, In Eerises 5, use the Lw of Sines to sole (if possile) the tringle. If two solutions eist, find oth. Round our nswers to two deiml ples In Eerises 5 8, find lues for suh tht the tringle hs () one solution, () two solutions, nd () no solution In Eerises 9, find the re of the tringle hing the indited ngle nd sides , 5, 00, 0 5, 95.0, 0, 0, 76, 76, 58, 58, 0, 0, 5, 5, 6, 60, 0, 88, 0, 0, 5, 5 5, 7 0, 6.,, 5, 5, 5, 8,,.,.5, 5 5, 9.5, 5 5, , 6, 8, 57,.5, 05, , 6, 0

9 Setion 6. Lw of Sines 5 5. HEIGHT euse of preiling winds, tree grew so tht it ws lening from the ertil. t point 0 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. 9. RIDGE DESIGN ridge is to e uilt ross smll lke from gzeo to dok (see figure). The ering from the gzeo to the dok is S W. From tree 00 meters from the gzeo, the erings to the gzeo nd the dok re S 7 E nd S 8 E, respetiel. Find the distne from the gzeo to the dok. h 9 0 m 0 Tree m N W S Gzeo E 6. HEIGHT flgpole t right ngle to the horizontl is loted on slope tht mkes n ngle of with the horizontl. The flgpole s shdow is 6 meters long nd points diretl 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 rile to indite the height of the flgpole. () Write n eqution tht n e used to find the height of the flgpole. () Find the height of the flgpole. 7. NGLE OF ELEVTION 0-meter utilit pole sts 7-meter shdow diretl down slope when the ngle of eletion of the sun is (see figure). Find, the ngle of eletion of the ground. 8. FLIGHT PTH plne flies 500 kilometers with ering 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 ering of the flight from Elgin to nton. W N S E nton θ θ 7 m 0 m Elgin 70 km 500 km Not drwn to sle Nples N 50. RILROD TRK DESIGN The irulr r of rilrod ure hs hord of length 000 feet orresponding to entrl ngle of 0. () Drw digrm tht isull represents the sitution. Show the known quntities on the digrm nd use the riles r nd s to represent the rdius of the r nd the length of the r, respetiel. () Find the rdius r of the irulr r. () Find the length s of the irulr r. 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. () Find the ir distne the plne must trel until touhing down on the ner end of the runw. () Find the ground distne the plne must trel until touhing down. (d) Find the ltitude of the plne when the pilot egins the desent. 5. LOTING FIRE The ering from the Pine Kno fire tower to the olt Sttion fire tower is N 65 E, nd the two towers re 0 kilometers prt. fire spotted rngers in eh tower hs ering of N 80 E from Pine Kno nd S 70 E from olt Sttion (see figure). Find the distne of the fire from eh tower. W N S Dok E 0 km olt Sttion 70 Fire Pine Kno Not drwn to sle

10 6 hpter 6 dditionl Topis in Trigonometr 5. DISTNE ot is siling due est prllel to the shoreline t speed of 0 miles per hour. t gien time, the ering to the lighthouse is S 70 E, nd 5 minutes lter the ering is S 6 E (see figure). The lighthouse is loted t the shoreline. Wht is the distne from the ot to the shoreline? d 5. DISTNE fmil is treling due west on rod tht psses fmous lndmrk. t gien time the ering to the lndmrk is N 6 W, nd fter the fmil trels 5 miles frther the ering is N 8 W. Wht is the losest the fmil will ome to the lndmrk while on the rod? 55. LTITUDE The ngles of eletion to n irplne from two points nd on leel ground re 55 nd 7, respetiel. The points nd re. miles prt, nd the irplne is est of oth points in the sme ertil plne. Find the ltitude of the plne. 56. DISTNE The ngles of eletion nd to n irplne from the irport ontrol tower nd from n osertion post miles w re eing ontinuousl monitored (see figure). Write n eqution giing the distne d etween the plne nd osertion post in terms of nd. irport ontrol tower EXPLORTION 70 θ 6 Osertion post TRUE OR FLSE? In Eerises 57 59, determine whether the sttement is true or flse. Justif our nswer. 57. If tringle ontins n otuse ngle, then it must e olique. 58. Two ngles nd one side of tringle do not neessril determine unique tringle. 59. If three sides or three ngles of n olique tringle re known, then the tringle n e soled. mi Not drwn to sle φ W d N S E 60. GRPHIL ND NUMERIL NLYSIS In the figure, nd re positie ngles. () Write s funtion of. () Use grphing utilit to grph the funtion in prt (). Determine its domin nd rnge. () Use the result of prt () to write s funtion of. (d) Use grphing utilit to grph the funtion in prt (). Determine its domin nd rnge. (e) omplete the tle. Wht n ou infer? α FIGURE FOR 60 FIGURE FOR 6 6. GRPHIL NLYSIS m 8 m 0 m () Write the re of the shded region in the figure s funtion of. () Use grphing utilit to grph the funtion. () Determine the domin of the funtion. Eplin how the re of the region nd the domin of the funtion would hnge if the eight-entimeter line segment were deresed in length. 6. PSTONE In the figure, tringle is to e formed drwing line segment of length from, to the positie -is. For wht lue(s) of n ou form () one tringle, () two tringles, nd () no tringles? Eplin our resoning. (0, 0) γ β 9 θ θ (, ) 5

11 Setion 6. Lw of osines 7 6. LW OF OSINES Wht ou should lern Use the Lw of osines to sole olique tringles (SSS or SS). Use the Lw of osines to model nd sole rel-life prolems. Use Heron s re Formul to find the re of tringle. Wh ou should lern it You n use the Lw of osines to sole rel-life prolems inoling olique tringles. For instne, in Eerise 5 on pge, ou n use the Lw of osines to pproimte how fr sell pler hs to run to mke th. Introdution Two ses remin in the list of onditions needed to sole n olique tringle SSS nd SS. If ou re gien three sides (SSS), or two sides nd their inluded ngle (SS), none of the rtios in the Lw of Sines would e omplete. In suh ses, ou n use the Lw of osines. Lw of osines Stndrd Form os os os lterntie Form os os os For proof of the Lw of osines, see Proofs in Mthemtis on pge 88. Emple Three Sides of Tringle SSS Find the three ngles of the tringle in Figure 6.. Dniel endj/istokphoto.om = 8 ft FIGURE 6. = 9 ft = ft It is good ide first to find the ngle opposite the longest side side in this se. Using the lterntie form of the Lw of osines, ou find tht os euse os is negtie, ou know tht is n otuse ngle gien t this point, it is simpler to use the Lw of Sines to determine. sin sin 8 sin You know tht must e ute euse is otuse, nd tringle n he, t most, one otuse ngle. So,.08 nd Now tr Eerise

12 8 hpter 6 dditionl Topis in Trigonometr Do ou see wh it ws wise to find the lrgest ngle first in Emple? Knowing the osine of n ngle, ou n determine whether the ngle is ute or otuse. Tht is, os os > 0 < 0 for for 0 < 90 < < 90 < 80. ute Otuse So, in Emple, one ou found tht ngle ws otuse, ou knew tht ngles nd were oth ute. If the lrgest ngle is ute, the remining two ngles re ute lso. Emple Two Sides nd the Inluded ngle SS Find the remining ngles nd side of the tringle in Figure 6.. = 9 m FIGURE 6. 5 = m When soling n olique tringle gien three sides, ou use the lterntie form of the Lw of osines to sole for n ngle. When soling n olique tringle gien two sides nd their inluded ngle, ou use the stndrd form of the Lw of osines to sole for n unknown. Use the Lw of osines to find the unknown side in the figure. os 9 9 os euse 5.07 meters, ou now know the rtio sin nd ou n use the reiprol form of the Lw of Sines to sole for. sin sin sin sin 0.70 There re two ngles etween 0 nd 80 whose sine is 0.70,.7 nd For.7, 9 sin For 5., euse side is the longest side of the tringle, must e the lrgest ngle of the tringle. So,.7 nd 0.. Now tr Eerise 7.

13 Setion 6. Lw of osines 9 pplitions Emple n pplition of the Lw of osines 60 ft 60 ft P h F f = ft 60 ft 5 p = 60 ft H FIGURE 6. The pither s mound on women s softll field is feet from home plte nd the distne etween the ses is 60 feet, s shown in Figure 6.. (The pither s mound is not hlfw etween home plte nd seond se.) How fr is the pither s mound from first se? In tringle HPF, H 5 (line HP isets the right ngle t H), f, nd p 60. Using the Lw of osines for this SS se, ou he h f p fp os H os So, the pproimte distne from the pither s mound to first se is h feet. Now tr Eerise. Emple n pplition of the Lw of osines ship trels 60 miles due est, then djusts its ourse northwrd, s shown in Figure 6.. fter treling 80 miles in tht diretion, the ship is 9 miles from its point of deprture. Desrie the ering from point to point. N W S E = 60 mi = 9 mi = 80 mi FIGURE 6. You he 80, 9, nd 60. So, using the lterntie form of the Lw of osines, ou he os So, ros , nd thus the ering mesured from due north from point to point is , or N 76.5 E. Now tr Eerise 9.

14 0 hpter 6 dditionl Topis in Trigonometr Heron s re Formul HISTORIL NOTE Heron of lendri (. 00..) ws Greek geometer nd inentor. His works desrie how to find the res of tringles, qudrilterls, regulr polgons hing to sides, nd irles s well s the surfe res nd olumes of three-dimensionl ojets. The Lw of osines n e used to estlish the following formul for the re of tringle. This formul is lled Heron s re Formul fter the Greek mthemtiin Heron (. 00..). Heron s re Formul Gien n tringle with sides of lengths,, nd, the re of the tringle is re s s s s where s. For proof of Heron s re Formul, see Proofs in Mthemtis on pge 89. Emple 5 Using Heron s re Formul Find the re of tringle hing sides of lengths meters, 5 meters, nd 7 meters. euse s 68 8, Heron s re Formul ields re s s s s 8.89 squre meters. Now tr Eerise 59. You he now studied three different formuls for the re of tringle. Stndrd Formul: re h Olique Tringle: re sin sin sin Heron s re Formul: re s s s s LSSROOM DISUSSION The re of Tringle Use the most pproprite formul to find the re of eh tringle elow. Show our work nd gie our resons for hoosing eh formul... ft ft ft 50 ft. ft d. ft ft ft ft 5 ft

15 Setion 6. Lw of osines 6. EXERISES VOULRY: Fill in the lnks. See for worked-out solutions to odd-numered eerises.. 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 os is.. The Lw of osines n e used to estlish formul for finding the re of tringle lled Formul. SKILLS ND PPLITIONS In Eerises 5 0, use the Lw of osines to sole the tringle. Round our nswers to two deiml ples = = 0 = = 7 = 8 = = 5 0 = , 55, 75.,., 0, 8, 0 5, 75 0, 5 0, 5, 5, In Eerises 6, omplete the tle soling the prllelogrm shown in the figure. (The lengths of the digonls re gien nd d. ) 5, 0.75, 6,, 0, 6., φ θ 7 7, 5 7 =.5 = , 7.5,, 9, 7 9 0, 8, d In Eerises 7, determine whether the Lw of Sines or the Lw of osines is needed to sole the tringle. Then sole the tringle In Eerises 0, use Heron s re Formul to find the re of the tringle , 5, 0 0,, 70,, 8,, 7, 5,. 60,, 7 8,,.5, 75.,.,.05,, 6, 0., 5, 8.6, 0.75, 7,, 0. 5, 5 8, d

16 hpter 6 dditionl Topis in Trigonometr. NVIGTION ot re runs long tringulr ourse mrked uos,, nd. The re strts with the ots heded west for 700 meters. The other two sides of the ourse 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 erings for the lst two legs of the re.. NVIGTION plne flies 80 miles from Frnklin to enterille with ering of 75. Then it flies 68 miles from enterille to Rosemount with ering of. Drw figure tht isull represents the sitution, nd find the stright-line distne nd ering from Frnklin to Rosemount.. SURVEYING 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. 75. SURVEYING tringulr prel of lnd hs 5 meters of frontge, nd the other oundries he lengths of 76 meters nd 9 meters. Wht ngles does the frontge mke with the two other oundries? 5. SURVEYING tringulr prel of ground hs sides of lengths 75 feet, 650 feet, nd 575 feet. Find the mesure of the lrgest ngle. 6. STREETLIGHT DESIGN Determine the ngle in the design of the streetlight shown in the figure. 0 m 50 m 8. LENGTH 00-foot ertil tower is to e ereted on the side of hill tht mkes 6 ngle with the horizontl (see figure). Find the length of eh of the two gu wires tht will e nhored 75 feet uphill nd downhill from the se of the tower ft 75 ft 9. NVIGTION 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). Dener 5 mm 7 mm 75 ft Nigr Flls 78 mm Orlndo () Find the ering of Dener from Orlndo. () Find the ering of Dener from Nigr Flls. 50. NVIGTION On mp, Minnepolis is 65 millimeters due west of ln, Phoeni is 6 millimeters from Minnepolis, nd Phoeni is 68 millimeters from ln (see figure). θ Phoeni Minnepolis 6 mm 65 mm 68 mm ln 7. DISTNE Two ships lee port t 9.M. One trels t ering of N 5 W t miles per hour, nd the other trels t ering of S 67 W t 6 miles per hour. pproimte how fr prt the re t noon tht d. () Find the ering of Minnepolis from Phoeni. () Find the ering of ln from Phoeni. 5. SELL On sell dimond with 90-foot sides, the pither s mound is 60.5 feet from home plte. How fr is it from the pither s mound to third se?

17 Setion 6. Lw of osines 5. SELL The sell pler in enter field is pling pproimtel 0 feet from the teleision mer tht is ehind home plte. tter hits fl ll tht goes to the wll 0 feet from the mer (see figure). The mer turns 8 to follow the pl. pproimtel how fr does the enter fielder he to run to mke the th? 5. IRRFT TRKING To determine the distne etween two irrft, trking sttion ontinuousl determines the distne to eh irrft nd the ngle etween them (see figure). Determine the distne etween the plnes when, 5 miles, nd 0 miles. 5. IRRFT TRKING Use the figure for Eerise 5 to determine the distne etween the plnes when, 0 miles, nd 0 miles. 55. 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? P Q S 0 ft R ft ENGINE DESIGN n engine hs seen-inh onneting rod fstened to rnk (see figure). () Use the Lw of osines to write n eqution giing the reltionship etween nd. () Write s funtion of. (Selet the sign tht ields positie lues of. ) () Use grphing utilit to grph the funtion in prt (). (d) Use the grph in prt () to determine the mimum distne the piston moes in one le..5 in. θ FIGURE FOR 56 FIGURE FOR PPER MNUFTURING In proess with ontinuous pper, the pper psses ross three rollers of rdii inhes, inhes, nd 6 inhes (see figure). The enters of the three-inh nd si-inh rollers re d inhes prt, nd the length of the r in ontt with the pper on the four-inh roller is s inhes. omplete the tle. 58. WNING DESIGN retrtle wning oe ptio door lowers t n ngle of 50 from the eterior wll t height of 0 feet oe the ground (see figure). No diret 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? 0 ft 7 in s in. in. 6 in. d (inhes) (degrees) s (inhes) Suns rs θ d 59. GEOMETRY The lengths of the sides of tringulr prel of lnd re pproimtel 00 feet, 500 feet, nd 600 feet. pproimte the re of the prel.

18 hpter 6 dditionl Topis in Trigonometr 60. GEOMETRY prking lot hs the shpe of prllelogrm (see figure). The lengths of two djent sides re 70 meters nd 00 meters. The ngle etween the two sides is 70. Wht is the re of the prking lot? 70 m GEOMETRY You wnt to u tringulr lot mesuring 50 rds 80 rds 0 rds. The prie of the lnd is $000 per re. How muh does the lnd ost? (Hint: re 80 squre rds) 6. GEOMETRY You wnt to u tringulr lot mesuring 50 feet 860 feet 90 feet. The prie of the lnd is $00 per re. How muh does the lnd ost? (Hint: re,560 squre feet) EXPLORTION 00 m TRUE OR FLSE? In Eerises 6 nd 6, determine whether the sttement is true or flse. Justif our nswer. 6. In Heron s re Formul, s is the erge of the lengths of the three sides of the tringle. 6. In ddition to SSS nd SS, the Lw of osines n e used to sole tringles with SS onditions. 65. WRITING tringle hs side lengths of 0 entimeters, 6 entimeters, nd 5 entimeters. n the Lw of osines e used to sole the tringle? Eplin. 66. WRITING Gien tringle with 7 meters, 87, nd 0, n the Lw of osines e used to sole the tringle? Eplin. 67. IRUMSRIED ND INSRIED IRLES Let R nd r e the rdii of the irumsried nd insried irles of tringle, respetiel (see figure), nd let s. () Proe tht R () Proe tht sin sin sin. r s s s. s IRUMSRIED ND INSRIED IRLES 68 nd 69, use the results of Eerise 67. In Eerises 68. Gien tringle with 5, 55, nd 7, find the res of () the tringle, () the irumsried irle, nd () the insried irle. 69. Find the length of the lrgest irulr running trk tht n e uilt on tringulr piee of propert with sides of lengths 00 feet, 50 feet, nd 5 feet. 70. THINK OUT IT Wht fmilir formul do ou otin when ou use the third form of the Lw of osines os, nd ou let 90? Wht is the reltionship etween the Lw of osines nd this formul? 7. THINK OUT IT In Emple, suppose 5. fter soling for, whih ngle would ou sole for net, or? re there two possile solutions for tht ngle? If so, how n ou determine whih ngle is the orret solution? 7. WRITING Desrie how the Lw of osines n e used to sole the miguous se of the olique tringle, where feet, 0 feet, nd 0. Is the result the sme s when the Lw of Sines is used to sole the tringle? Desrie the dntges nd the disdntges of eh method. 7. WRITING In Eerise 7, the Lw of osines ws used to sole tringle in the two-solution se of SS. n the Lw of osines e used to sole the no-solution nd single-solution ses of SS? Eplin. 7. PSTONE Determine whether the Lw of Sines or the Lw of osines is needed to sole the tringle. (),, nd (),, nd (),, nd (d),, nd (e),, nd (f),, nd R r 75. PROOF Use the Lw of osines to proe tht os. 76. PROOF Use the Lw of osines to proe tht os.

19 Setion 6. Vetors in the Plne 5 6. VETORS IN THE PLNE Wht ou should lern Represent etors s direted line segments. Write the omponent forms of etors. Perform si etor opertions nd represent them grphill. Write etors s liner omintions of unit etors. Find the diretion ngles of etors. Use etors to model nd sole rel-life prolems. Wh ou should lern it You n use etors to model nd sole rel-life prolems inoling mgnitude nd diretion. For instne, in Eerise 0 on pge 57, ou n use etors to determine the true diretion of ommeril jet. Introdution Quntities suh s fore nd eloit inole oth mgnitude nd diretion nd nnot e ompletel hrterized single rel numer. To represent suh quntit, ou n use direted line segment, s shown in Figure 6.5. The direted line segment PQ \ hs initil point P nd terminl point Q. Its mgnitude (or length) is denoted PQ \ nd n e found using the Distne Formul. Terminl point Q PQ P Initil point FIGURE 6.5 FIGURE 6.6 Two direted line segments tht he the sme mgnitude nd diretion re equilent. For emple, the direted line segments in Figure 6.6 re ll equilent. The set of ll direted line segments tht re equilent to the direted line segment PQ \ is etor in the plne, written PQ \. Vetors re denoted lowerse, oldfe letters suh s u,, nd w. ill hmn/photo Reserhers, In. Emple Vetor Representtion Direted Line Segments Let u e represented the direted line segment from P0, 0 to Q,, nd let e represented the direted line segment from R, to S,, s shown in Figure 6.7. Show tht u nd re equilent. 5 (, ) S (, ) (, ) R u Q P (0, 0) FIGURE 6.7 From the Distne Formul, it follows tht PQ \ nd RS \ he the sme mgnitude. PQ \ 0 0 Moreoer, oth line segments he the sme diretion euse the re oth direted towrd the upper right on lines hing slope of 0 0. euse PQ \ nd RS \ he the sme mgnitude nd diretion, u nd re equilent. Now tr Eerise. RS \

20 6 hpter 6 dditionl Topis in Trigonometr omponent Form of Vetor The direted line segment whose initil point is the origin is often the most onenient representtie of set of equilent direted line segments. This representtie of the etor is in stndrd position. etor whose initil point is the origin 0, 0 n e uniquel represented the oordintes of its terminl point,. This is the omponent form of etor, written s,. The oordintes nd re the omponents of. If oth the initil point nd the terminl point lie t the origin, is the zero etor nd is denoted 0 0, 0. TEHNOLOGY You n grph etors with grphing utilit grphing direted line segments. onsult the user s guide for our grphing utilit for speifi instrutions. omponent Form of Vetor The omponent form of the etor with initil point Pp, p nd terminl point Qq, q is gien PQ \ q p, q p,. The mgnitude (or length) of is gien q p q p. If, is unit etor. Moreoer, 0 if nd onl if is the zero etor 0. Two etors u u, u nd, re equl if nd onl if u nd u. For instne, in Emple, the etor u from P0, 0 to Q, is u PQ \ 0, 0,, nd the etor from R, to S, is RS \,,. Emple Finding the omponent Form of Vetor Find the omponent form nd mgnitude of the etor tht hs initil point, 7 nd terminl point, 5. lgeri Let P, 7 p, p nd Q, 5 q, q. Then, the omponents of, re q p 5 q p 5 7. So, 5, nd the mgnitude of is Grphil Use entimeter grph pper to plot the points P, 7 nd Q, 5. refull sketh the etor. Use the sketh to find the omponents of,. Then use entimeter ruler to find the mgnitude of. m FIGURE 6.8 Now tr Eerise 9. Figure 6.8 shows tht the omponents of re 5 nd, so 5,. Figure 6.8 lso shows tht the mgnitude of is.

21 Setion 6. Vetors in the Plne 7 FIGURE 6.9 Vetor Opertions The two si etor opertions re slr multiplition nd etor ddition. In opertions with etors, numers re usull referred to s slrs. In this tet, slrs will lws e rel numers. Geometrill, the produt of etor nd slr k is the etor tht is k times s long s. If k is positie, k hs the sme diretion s, nd if k is negtie, k hs the diretion opposite tht of, s shown in Figure 6.9. To dd two etors u nd geometrill, first position them (without hnging their lengths or diretions) so tht the initil point of the seond etor oinides with the terminl point of the first etor u. The sum u is the etor formed joining the initil point of the first etor u with the terminl point of the seond etor, s shown in Figure 6.0. This tehnique is lled the prllelogrm lw for etor ddition euse the etor u, often lled the resultnt of etor ddition, is the digonl of prllelogrm hing djent sides u nd. u + u u FIGURE 6.0 Definitions of Vetor ddition nd Slr Multiplition Let u u, u nd, e etors nd let k e slr ( rel numer). Then the sum of u nd is the etor u u, u Sum nd the slr multiple of k times u is the etor ku ku, u ku, ku. Slr multiple The negtie of, is u, Negtie u nd the differene of u nd is u + ( ) u u FIGURE 6. u u dd. See Figure 6.. u, u. Differene To represent u geometrill, ou n use direted line segments with the sme initil point. The differene u is the etor from the terminl point of to the terminl point of u, whih is equl to u, s shown in Figure 6..

22 8 hpter 6 dditionl Topis in Trigonometr The omponent definitions of etor ddition nd slr multiplition re illustrted in Emple. In this emple, notie tht eh of the etor opertions n e interpreted geometrill. Emple Vetor Opertions Let, 5 nd w,, nd find eh of the following etors... w. w. euse, 5, ou he, 5, 5, 0. sketh of is shown in Figure 6... The differene of w nd is w,, 5, 5 5,. sketh of w is shown in Figure 6.. Note tht the figure shows the etor differene w s the sum w.. The sum of nd w is w, 5,, 5,, 5 6, 8 6, 5 8,. sketh of w is shown in Figure (, 0) 6 (, 5) w w (, ) 5 (5, ) (, ) 0 w 8 (, 5) + w FIGURE 6. FIGURE 6. FIGURE 6. Now tr Eerise.

23 Setion 6. Vetors in the Plne 9 Vetor ddition nd slr multiplition shre mn of the properties of ordinr rithmeti. Properties of Vetor ddition nd Slr Multiplition Let u,, nd w e etors nd let nd d e slrs. Then the following properties re true.. u u. u w u w. u 0 u. u u 0 5. du du 6. du u du u u, 7. u u 8. 0u 0 9. Propert 9 n e stted s follows: the mgnitude of the etor is the solute lue of times the mgnitude of. HISTORIL NOTE The Grnger olletion Willim Rown Hmilton ( ), n Irish mthemtiin, did some of the erliest work with etors. Hmilton spent mn ers deeloping sstem of etor-like quntities lled quternions. lthough Hmilton ws onined of the enefits of quternions, the opertions he defined did not produe good models for phsil phenomen. It ws not until the ltter hlf of the nineteenth entur tht the Sottish phsiist Jmes Mwell (8 879) restrutured Hmilton s quternions in form useful for representing phsil quntities suh s fore, eloit, nd elertion. Unit Vetors In mn pplitions of etors, it is useful to find unit etor tht hs the sme diretion s gien nonzero etor. To do this, ou n diide its mgnitude to otin u unit etor Unit etor in diretion of Note tht u is slr multiple of. The etor u hs mgnitude of nd the sme diretion s. The etor u is lled unit etor in the diretion of. Emple Finding Unit Vetor Find unit etor in the diretion of, 5 nd erif tht the result hs mgnitude of. The unit etor in the diretion of is, 5 5, 5 9 9, This etor hs mgnitude of euse Now tr Eerise.

24 50 hpter 6 dditionl Topis in Trigonometr j = 0, i =, 0 FIGURE 6.5 The unit etors, 0 nd 0, re lled the stndrd unit etors nd re denoted i, 0 nd j 0, s shown in Figure 6.5. (Note tht the lowerse letter i is written in oldfe to distinguish it from the imginr numer i. ) These etors n e used to represent n etor,, s follows.,, 0 0, i j The slrs nd re lled the horizontl nd ertil omponents of, respetiel. The etor sum i j is lled liner omintion of the etors i nd j. n etor in the plne n e written s liner omintion of the stndrd unit etors i nd j. 8 6 (, ) u 6 (, 5) FIGURE 6.6 Emple 5 Writing Liner omintion of Unit Vetors Let u e the etor with initil point, 5 nd terminl point,. Write u s liner omintion of the stndrd unit etors i nd j. egin writing the omponent form of the etor u. u, 5, 8 i 8j This result is shown grphill in Figure 6.6. Now tr Eerise 5. Emple 6 Vetor Opertions Let u i 8j nd let i j. Find u. You ould sole this prolem onerting u nd to omponent form. This, howeer, is not neessr. It is just s es to perform the opertions in unit etor form. u i 8j i j 6i 6j 6i j i 9j Now tr Eerise 59.

25 Setion 6. Vetors in the Plne 5 (, ) Diretion ngles If u is unit etor suh tht is the ngle (mesured ounterlokwise) from the positie -is to u, the terminl point of u lies on the unit irle nd ou he u θ = os θ = sin θ u, os, sin os i sin j s shown in Figure 6.7. The ngle is the diretion ngle of the etor u. Suppose tht u is unit etor with diretion ngle. If i j is n etor tht mkes n ngle with the positie -is, it hs the sme diretion s u nd ou n write os, sin u FIGURE 6.7 os i sin j. euse i j os i sin j, it follows tht the diretion ngle for is determined from tn sin os sin os Quotient identit Multipl numertor nd denomintor.. Simplif. Emple 7 Finding Diretion ngles of Vetors (, ) Find the diretion ngle of eh etor.. u i j. i j u θ = 5 FIGURE 6.8 (, ) FIGURE The diretion ngle is tn. 5, So, s shown in Figure The diretion ngle is tn. Moreoer, euse i j lies in Qudrnt IV, lies in Qudrnt IV nd its referene ngle is rtn , So, it follows tht s shown in Figure 6.9. Now tr Eerise

26 5 hpter 6 dditionl Topis in Trigonometr pplitions of Vetors Emple 8 Finding the omponent Form of Vetor Find the omponent form of the etor tht represents the eloit of n irplne desending t speed of 50 miles per hour t n ngle 0 elow the horizontl, s shown in Figure 6.0. The eloit etor hs mgnitude of 50 nd diretion ngle of os i sin j os 00i 50sin 00j i 500.0j 0.96i 5.0j FIGURE , 5.0 You n hek tht hs mgnitude of 50, s follows , , Now tr Eerise 8. Emple 9 Using Vetors to Determine Weight W 5 D 5 FIGURE 6. fore of 600 pounds is required to pull ot nd triler up rmp inlined t 5 from the horizontl. Find the omined weight of the ot nd triler. sed on Figure 6., ou n mke the following osertions. \ fore of grit omined weight of ot nd triler \ fore ginst rmp \ fore required to moe ot up rmp 600 pounds onstrution, tringles WD nd re similr. Therefore, ngle is 5. So, in tringle ou he sin 5 \ \ sin \ \ 600 sin 5 \ 8. onsequentl, the omined weight is pproimtel 8 pounds. (In Figure 6., note tht \ is prllel to the rmp.) Now tr Eerise 95.

27 Setion 6. Vetors in the Plne 5 Emple 0 Using Vetors to Find Speed nd Diretion Rell from Setion.8 tht in ir nigtion, erings n e mesured in degrees lokwise from north. n irplne is treling t speed of 500 miles per hour with ering of 0 t fied ltitude with negligile wind eloit s shown in Figure 6.(). When the irplne rehes ertin point, it enounters wind with eloit of 70 miles per hour in the diretion N 5 E, s shown in Figure 6.(). Wht re the resultnt speed nd diretion of the irplne? 0 Wind θ () FIGURE 6. () Using Figure 6., the eloit of the irplne (lone) is 500os 0, sin 0 50, 50 nd the eloit of the wind is 70os 5, sin 5 So, the eloit of the irplne (in the wind) is nd the resultnt speed of the irplne is Finll, if tn 5, , , miles per hour. whih implies tht is the diretion ngle of the flight pth, ou he rtn.065 So, the true diretion of the irplne is pproimtel Now tr Eerise

28 5 hpter 6 dditionl Topis in Trigonometr 6. EXERISES VOULRY: Fill in the lnks. See for worked-out solutions to odd-numered eerises.. n e used to represent quntit tht inoles oth mgnitude nd diretion.. The direted line segment PQ \ hs point P nd point Q.. The of the direted line segment PQ \ is denoted PQ \.. The set of ll direted line segments tht re equilent to gien direted line segment PQ \ is in the plne. 5. In order to show tht two etors re equilent, ou must show tht the he the sme nd the sme. 6. The direted line segment whose initil point is the origin is sid to e in. 7. etor tht hs mgnitude of is lled. 8. The two si etor opertions re slr nd etor. 9. The etor u is lled the of etor ddition. 0. The etor sum i j is lled of the etors i nd j, nd the slrs nd re lled the nd omponents of, respetiel. SKILLS ND PPLITIONS In Eerises nd, show tht u nd re equilent... 6 (0, 0) In Eerises, find the omponent form nd the mgnitude of the etor (, 5) (, ) (, ) u (, ) (6, 5) (, ) 6 (, ) u (, ) 5 (, ) (0, ) (, ) (0, 5) (, ) Initil Point 9., 5 0., 7.,.,., 5., (, ) 5 (, ) Terminl Point 5, 5, 7 8, 9 9, 5, 9, 0 In Eerises 5 0, use the figure to sketh grph of the speified etor. To print n enlrged op of the grph, go to the wesite u 8. u 9. u 0. u u 5 (, ) 5 (, )

29 Setion 6. Vetors in the Plne 55 In Eerises 8, find () u, () u, nd () u, Then sketh eh resultnt etor.. u,,,. u,,. u 5,, 0, 0. u 0, 0, In Eerises 9 8, find unit etor in the diretion of the gien etor. Verif tht the result hs mgnitude of. In Eerises 9 5, find the etor with the gien mgnitude nd the sme diretion s u u i j, u i j, Mgnitude Diretion In Eerises 5 56, the initil nd terminl points of etor re gien. Write liner omintion of the stndrd unit etors i nd j. Initil Point 5., 5. 0, 55. 6, 56., 5 i j j 7. u i, j 8. u j, 5. w j 6. w 6i u, u, 5 u, 5 u, Terminl Point,, 6 0,, i 9. u, 0 0. u 0,.,. 5,. i j. 6i j 7. w i j 8. w 7j i In Eerises 57 6, find the omponent form of nd sketh the speified etor opertions geometrill, where u i j, nd w i j u w 59. u w 60. u w 6. u w 6. u w In Eerises 6 66, find the mgnitude nd diretion ngle of the etor. 6. 6i 6j 6. 5i j 65. os 60i sin 60j 66. 8os 5i sin 5j, 0, In Eerises 67 7, find the omponent form of gien its mgnitude nd the ngle it mkes with the positie -is. Sketh Mgnitude ngle 7. in the diretion i j 7. in the diretion i j In Eerises 75 78, find the omponent form of the sum of u nd with diretion ngles nd Mgnitude ngle In Eerises 79 nd 80, use the Lw of osines to find the ngle etween the etors. ( ssume ) u 5 5 u u 0 50 u 50 0 i j, i j, u u u u w i j w i j 80 0 RESULTNT FORE In Eerises 8 nd 8, find the ngle etween the fores gien the mgnitude of their resultnt. (Hint: Write fore s etor in the diretion of the positie -is nd fore s etor t n ngle with the positie -is.) Fore Fore Resultnt Fore 8. 5 pounds 60 pounds 90 pounds pounds 000 pounds 750 pounds u 8. VELOITY gun with muzzle eloit of 00 feet per seond is fired t n ngle of 6 oe the horizontl. Find the ertil nd horizontl omponents of the eloit. 8. Detroit Tigers pither Joel Zum ws reorded throwing pith t eloit of 0 miles per hour. If he threw the pith t n ngle of 5 elow the horizontl, find the ertil nd horizontl omponents of the eloit. (Soure: Dmon Lihtenwlner, sell Info s)

30 56 hpter 6 dditionl Topis in Trigonometr 85. RESULTNT FORE Fores with mgnitudes of 5 newtons nd 00 newtons t on hook (see figure). The ngle etween the two fores is 5. Find the diretion nd mgnitude of the resultnt of these fores. 000 newtons 9. TOW LINE TENSION loded rge is eing towed two tugots, nd the mgnitude of the resultnt is 6000 pounds direted long the is of the rge (see figure). Find the tension in the tow lines if the eh mke n 8 ngle with the is of the rge. 5 newtons 5 00 newtons newtons 8 8 FIGURE FOR 85 FIGURE FOR RESULTNT FORE Fores with mgnitudes of 000 newtons nd 900 newtons t on mhine prt t ngles of 0 nd 5, respetiel, with the -is (see figure). Find the diretion nd mgnitude of the resultnt of these fores. 87. RESULTNT FORE Three fores with mgnitudes of 75 pounds, 00 pounds, nd 5 pounds t on n ojet t ngles of 0, 5, nd 0, respetiel, with the positie -is. Find the diretion nd mgnitude of the resultnt of these fores. 88. RESULTNT FORE Three fores with mgnitudes of 70 pounds, 0 pounds, nd 60 pounds t on n ojet t ngles of 0, 5, nd 5, respetiel, with the positie -is. Find the diretion nd mgnitude of the resultnt of these fores. 89. trffi light weighing pounds is suspended two les (see figure). Find the tension in eh le. 90. Repet Eerise 89 if 0 nd LE TENSION In Eerises 9 nd 9, use the figure to determine the tension in eh le supporting the lod l θ = 0 θ = 0 in in. 0 in. 9. ROPE TENSION To rr 00-pound lindril weight, two people lift on the ends of short ropes tht re tied to n eelet on the top enter of the linder. Eh rope mkes 0 ngle with the ertil. Drw figure tht gies isul representtion of the sitution, nd find the tension in the ropes. In Eerises 95 98, fore of F pounds is required to pull n ojet weighing W pounds up rmp inlined t degrees from the horizontl. 95. Find F if W 00 pounds nd 96. Find W if F 600 pounds nd 97. Find if F 5000 pounds nd W 5,000 pounds. 98. Find F if W 5000 pounds nd 99. WORK he ojet is pulled 0 feet ross floor, using fore of 00 pounds. The fore is eerted t n ngle of 50 oe the horizontl (see figure). Find the work done. (Use the formul for work, W FD, where F is the omponent of the fore in the diretion of motion nd D is the distne.) l 0 ft.. 6. Tension 5 l FIGURE FOR 99 FIGURE FOR ROPE TENSION tetherll weighing pound is pulled outwrd from the pole horizontl fore u until the rope mkes 5 ngle with the pole (see figure). Determine the resulting tension in the rope nd the mgnitude of u. u 5000 l

31 Setion 6. Vetors in the Plne NVIGTION n irplne is fling in the diretion of 8, with n irspeed of 875 kilometers per hour. euse of the wind, its groundspeed nd diretion re 800 kilometers per hour nd 0, respetiel (see figure). Find the diretion nd speed of the wind. 0. NVIGTION ommeril jet is fling from Mimi to Settle. The jet s eloit with respet to the ir is 580 miles per hour, nd its ering is. The wind, t the ltitude of the plne, is lowing from the southwest with eloit of 60 miles per hour. () Drw figure tht gies isul representtion of the sitution. () Write the eloit of the wind s etor in omponent form. () Write the eloit of the jet reltie to the ir in omponent form. (d) Wht is the speed of the jet with respet to the ground? (e) Wht is the true diretion of the jet? EXPLORTION 0 8 TRUE OR FLSE? In Eerises 0 0, use the figure to determine whether the sttement is true or flse. Justif our nswer. 0. d 0. s u Wind 875 kilometers per hour 05. u 06. w s 07. w d 08. d u t 0. t w d t s w W 800 kilometers per hour N S E. PROOF Proe tht os i sin j is unit etor for n lue of.. PSTONE The initil nd terminl points of etor re, nd 9,, respetiel. () Write in omponent form. () Write s the liner omintion of the stndrd unit etors i nd j. () Sketh with its initil point t the origin. (d) Find the mgnitude of.. GRPHIL RESONING onsider two fores F 0, 0 nd F 5os, sin. () Find F F s funtion of. () Use grphing utilit to grph the funtion in prt () for 0 <. () Use the grph in prt () to determine the rnge of the funtion. Wht is its mimum, nd for wht lue of does it our? Wht is its minimum, nd for wht lue of does it our? (d) Eplin wh the mgnitude of the resultnt is neer 0.. TEHNOLOGY Write progrm for our grphing utilit tht grphs two etors nd their differene gien the etors in omponent form. In Eerises 5 nd 6, use the progrm in Eerise to find the differene of the etors shown in the figure (, 6) (, 5) (5, ) (9, ) WRITING In our own words, stte the differene etween slr nd etor. Gie emples of eh. 8. WRITING Gie geometri desriptions of the opertions of ddition of etors nd multiplition of etor slr. 9. WRITING Identif the quntit s slr or s etor. Eplin our resoning. () The muzzle eloit of ullet () The prie of ompn s stok () The ir temperture in room (d) The weight of n utomoile 5 ( 0, 70) (80, 80) (0, 60) ( 00, 0) 50 50

32 58 hpter 6 dditionl Topis in Trigonometr 6. VETORS ND DOT PRODUTS Wht ou should lern Find the dot produt of two etors nd use the properties of the dot produt. Find the ngle etween two etors nd determine whether two etors re orthogonl. Write etor s the sum of two etor omponents. Use etors to find the work done fore. Wh ou should lern it You n use the dot produt of two etors to sole rel-life prolems inoling two etor quntities. For instne, in Eerise 76 on pge 66, ou n use the dot produt to find the fore neessr to keep sport utilit ehile from rolling down hill. The Dot Produt of Two Vetors So fr ou he studied two etor opertions etor ddition nd multiplition slr eh of whih ields nother etor. In this setion, ou will stud third etor opertion, the dot produt. This produt ields slr, rther thn etor. Definition of the Dot Produt The dot produt of u u, u nd, is u u u. Properties of the Dot Produt Let u,, nd w e etors in the plne or in spe nd let e slr.... u u 0 0 u w u u w. 5. u u u For proofs of the properties of the dot produt, see Proofs in Mthemtis on pge 90. Emple Finding Dot Produts Joe Redle/Gett Imges Find eh dot produt.., 5,.,,.., 5, ,,.,, 0. 0,, Now tr Eerise 7. In Emple, e sure ou see tht the dot produt of two etors is slr ( rel numer), not etor. Moreoer, notie tht the dot produt n e positie, zero, or negtie.

33 Setion 6. Vetors nd Dot Produts 59 Emple Using Properties of Dot Produts Let u,,,, nd w,. Find eh dot produt... egin finding the dot produt of u nd... u w u u,, u w,, 8 u u 8 Notie tht the produt in prt () is etor, wheres the produt in prt () is slr. n ou see wh? Now tr Eerise 7. Emple Dot Produt nd Mgnitude u u θ Origin FIGURE 6. The dot produt of u with itself is 5. Wht is the mgnitude of u? euse u u u nd u u 5, it follows tht u u u 5. Now tr Eerise 5. The ngle etween Two Vetors The ngle etween two nonzero etors is the ngle, 0, etween their respetie stndrd position etors, s shown in Figure 6.. This ngle n e found using the dot produt. ngle etween Two Vetors If is the ngle etween two nonzero etors u nd, then os u u. For proof of the ngle etween two etors, see Proofs in Mthemtis on pge 90.

34 60 hpter 6 dditionl Topis in Trigonometr Emple Finding the ngle etween Two Vetors 6 =, 5 5 u =, θ 5 6 FIGURE 6. Find the ngle etween u, nd, 5. The two etors nd re shown in Figure 6.. os u 7 5 This implies tht the ngle etween the two etors is ros u,, 5,, Now tr Eerise 5. Rewriting the epression for the ngle etween two etors in the form u u os lterntie form of dot produt produes n lterntie w to lulte the dot produt. From this form, ou n see tht euse u nd re lws positie, u nd os will lws he the sme sign. Figure 6.5 shows the fie possile orienttions of two etors. u θ os Opposite Diretion FIGURE 6.5 u θ < < < os < 0 Otuse ngle u θ os 0 90 ngle u θ 0 < < 0 < os < ute ngle u 0 os Sme Diretion Definition of Orthogonl Vetors The etors u nd re orthogonl if u 0. The terms orthogonl nd perpendiulr men essentill the sme thing meeting t right ngles. Note tht the zero etor is orthogonl to eer etor u, euse 0 u 0.

35 Setion 6. Vetors nd Dot Produts 6 TEHNOLOGY The grphing utilit progrm, Finding the ngle etween Two Vetors, found on the wesite for this tet t demi.engge.om, grphs two etors u, nd, d in stndrd position nd finds the mesure of the ngle etween them. Use the progrm to erif the solutions for Emples nd 5. Emple 5 Determining Orthogonl Vetors re the etors u, nd 6, orthogonl? Find the dot produt of the two etors. u, 6, 6 0 euse the dot produt is 0, the two etors re orthogonl (see Figure 6.6). = 6, u =, FIGURE 6.6 Now tr Eerise 5. Finding Vetor omponents You he lred seen pplitions in whih two etors re dded to produe resultnt etor. Mn pplitions in phsis nd engineering pose the reerse prolem deomposing gien etor into the sum of two etor omponents. onsider ot on n inlined rmp, s shown in Figure 6.7. The fore F due to grit pulls the ot down the rmp nd ginst the rmp. These two orthogonl fores, nd w, re etor omponents of F. Tht is, w F w w. Vetor omponents of F The negtie of omponent w represents the fore needed to keep the ot from rolling down the rmp, wheres w represents the fore tht the tires must withstnd ginst the rmp. proedure for finding nd is shown on the following pge. w w w F FIGURE 6.7 w

36 6 hpter 6 dditionl Topis in Trigonometr Definition of Vetor omponents Let u nd e nonzero etors suh tht u w w where w nd w re orthogonl nd w is prllel to (or slr multiple of), s shown in Figure 6.8. The etors w nd w re lled etor omponents of u. The etor is the projetion of u onto nd is denoted w w proj u. The etor w is gien w u w. w u u w θ θ is ute. FIGURE 6.8 w is otuse. w From the definition of etor omponents, ou n see tht it is es to find the omponent w one ou he found the projetion of u onto. To find the projetion, ou n use the dot produt, s follows. So, nd u w w w u w u w 0 w proj u u. w is slr multiple of. Tke dot produt of eh side with. w nd re orthogonl. Projetion of u onto Let u nd e nonzero etors. The projetion of u onto is proj u u.

37 Setion 6. Vetors nd Dot Produts 6 Emple 6 Deomposing Vetor into omponents = 6, w 5 6 w 5 u =, 5 FIGURE 6.9 Find the projetion of u, 5 onto 6,. Then write u s the sum of two orthogonl etors, one of whih is proj u. The projetion of u onto is s shown in Figure 6.9. The other omponent, w, is So, w proj u u w u w, 5 6 5, 5 9 7, 5 5. u w w 6 5, 5 9 7,, Now tr Eerise , 6 5, 5 Emple 7 Finding Fore w 0 FIGURE 6.0 F 00-pound rt sits on rmp inlined t 0, s shown in Figure 6.0. Wht fore is required to keep the rt from rolling down the rmp? euse the fore due to grit is ertil nd downwrd, ou n represent the grittionl fore the etor F 00j. Fore due to grit To find the fore required to keep the rt from rolling down the rmp, projet F onto unit etor in the diretion of the rmp, s follows. os 0i sin 0j i j Therefore, the projetion of F onto is w proj F F F i j. Unit etor long rmp The mgnitude of this fore is 00, nd so fore of 00 pounds is required to keep the rt from rolling down the rmp. Now tr Eerise 75.

38 6 hpter 6 dditionl Topis in Trigonometr Work The work W done onstnt fore F ting long the line of motion of n ojet is gien W mgnitude of foredistne F PQ \ s shown in Figure 6.. If the onstnt fore F is not direted long the line of motion, s shown in Figure 6., the work W done the fore is gien \ W proj PQ F PQ \ os F PQ \ F PQ \. Projetion form for work \ proj PQ F os F lterntie form of dot produt F F θ proj PQ F P Q P Q Fore ts long the line of motion. Fore ts t ngle with the line of motion. FIGURE 6. FIGURE 6. This notion of work is summrized in the following definition. Definition of Work The work W done onstnt fore F s its point of pplition moes long the etor PQ \ is gien either of the following.. W proj PQ \F PQ \ Projetion form. W F PQ \ Dot produt form Emple 8 Finding Work To lose sliding rn door, person pulls on rope with onstnt fore of 50 pounds t onstnt ngle of 60, s shown in Figure 6.. Find the work done in moing the rn door feet to its losed position. P ft proj PQ F Q Using projetion, ou n lulte the work s follows. W proj PQ \F PQ \ Projetion form for work 60 os 60F PQ \ F 00 foot-pounds 50 FIGURE 6. ft So, the work done is 00 foot-pounds. You n erif this result finding the etors F nd PQ \ nd lulting their dot produt. Now tr Eerise 79.

39 Setion 6. Vetors nd Dot Produts EXERISES VOULRY: Fill in the lnks.. The of two etors ields slr, rther thn etor.. The dot produt of u u, u nd, is u.. If is the ngle etween two nonzero etors u nd, then os. See for worked-out solutions to odd-numered eerises.. The etors u nd re if u The projetion of u onto is gien proj u. 6. The work W done onstnt fore F s its point of pplition moes long the etor PQ \ is gien W or W. SKILLS ND PPLITIONS In Eerises 7, find the dot produt of u nd. 7. u 7, 8. u 6, 0,, i j i j In Eerises 5, use the etors u,,,, nd w, to find the indited quntit. Stte whether the result is etor or slr. u u w.. In Eerises 5 0, use the dot produt to find the mgnitude of u. In Eerises 0, find the ngle. u, 0. u, 0, etween the etors.. u i j. u i j j 5. u i j 6. u 6i j 6i j, 9. u, 0. u, 5 5. u u 6. u, 8. u i j. u i j 7i j. u i j. u i j i j 7. u 8. uw 9. w u 0. u w. w. u u w 5. u 8, 5 6. u, 6 7. u 0i 5j 8. u i 6j 9. u 6j 0. u i, 0 i j 8i j 7. u 5i 5j 8. u i j In Eerises, grph the etors nd find the degree mesure of the ngle etween the etors.. u i j. u 6i j. u 5i 5j. u i j In Eerises 5 8, use etors to find the interior ngles of the tringle with the gien erties. 5.,,,,, , 0,,, 0, 6 8. In Eerises 9 5, find u, where is the ngle etween u nd i 6j u os os i sin j u os os 7i 5j 8i 8j i sin j i sin j i sin j u, 0, u 00, 50, u 9, 6, 5. u,, i j i j 8i j,,, 7, 8,, 5,, 9, 7, 9 6

40 66 hpter 6 dditionl Topis in Trigonometr In Eerises 5 58, determine whether u nd re orthogonl, prllel, or neither. 5. u, 0 5. u, 5, u i j 56. u i In Eerises 59 6, find the projetion of u onto. Then write u s the sum of two orthogonl etors, one of whih is proj u. In Eerises 6 66, use the grph to determine mentll the projetion of u onto. (The oordintes of the terminl points of the etors in stndrd position re gien.) Use the formul for the projetion of u onto to erif our result In Eerises 67 70, find two etors in opposite diretions tht re orthogonl to the etor u. (There re mn orret nswers.) 67. u, u 8, 69. u i 70. u i 5 j j WORK In Eerises 7 nd 7, find the work done in moing prtile from P to Q if the mgnitude nd diretion of the fore re gien , 5 5i 6j i j 6,, (, ) u (, ) u (6, ) 5 6 (6, ) 6 P0, 0, Q, 7,, P,, Q, 5, i j i j 57. u i j 58. u os, sin sin, os 59. u, 60. u,, 6. u 0, 6. u,, 6 (6, ) u 6 (, ) (6, ) 6 u (, ) 7. REVENUE The etor u 600, 550 gies the numers of units of two models of ellulr phones produed teleommunitions ompn. The etor 79.99, gies the pries (in dollrs) of the two models of ellulr phones, respetiel. () Find the dot produt u nd interpret the result in the ontet of the prolem. () Identif the etor opertion used to inrese the pries 5%. 7. REVENUE The etor u 0, 750 gies the numers of hmurgers nd hot dogs, respetiel, sold t fst-food stnd in one month. The etor.5,.75 gies the pries (in dollrs) of the food items. () Find the dot produt u nd interpret the result in the ontet of the prolem. () Identif the etor opertion used to inrese the pries.5%. 75. RKING LOD truk with gross weight of 0,000 pounds is prked on slope of d (see figure). ssume tht the onl fore to oerome is the fore of grit. d Weight = 0,000 l () Find the fore required to keep the truk from rolling down the hill in terms of the slope d. () Use grphing utilit to omplete the tle. d 0 5 Fore d Fore () Find the fore perpendiulr to the hill when d RKING LOD sport utilit ehile with gross weight of 500 pounds is prked on slope of 0. ssume tht the onl fore to oerome is the fore of grit. Find the fore required to keep the ehile from rolling down the hill. Find the fore perpendiulr to the hill.

41 Setion 6. Vetors nd Dot Produts WORK Determine the work done person lifting 5-newton g of sugr meters. 78. WORK Determine the work done rne lifting 00-pound r 5 feet. 79. WORK fore of 5 pounds eerted t n ngle of 0 oe the horizontl is required to slide tle ross floor (see figure). The tle is drgged 0 feet. Determine the work done in sliding the tle. 0 5 l 8. PROGRMMING Use the progrm ou wrote in Eerise 8 to find the ngle etween the gien etors. () () u 8, nd, 5 u, 6 nd, 85. PROGRMMING Gien etors u nd in omponent form, write progrm for our grphing utilit in whih the output is the omponent form of the projetion of u onto. 86. PROGRMMING Use the progrm ou wrote in Eerise 85 to find the projetion of u onto for the gien etors. () () u 5, 6 nd, u, nd, EXPLORTION 80. WORK trtor pulls log 800 meters, nd the tension in the le onneting the trtor nd log is pproimtel 5,69 newtons. The diretion of the fore is 5 oe the horizontl. pproimte the work done in pulling the log. 8. WORK One of the eents in lol strongmn ontest is to pull ement lok 00 feet. One ompetitor pulls the lok eerting fore of 50 pounds on rope tthed to the lok t n ngle of 0 with the horizontl (see figure). Find the work done in pulling the lok WORK to wgon is pulled eerting fore of 5 pounds on hndle tht mkes 0 ngle with the horizontl (see figure). Find the work done in pulling the wgon 50 feet ft 0 ft Not drwn to sle 8. PROGRMMING Gien etors u nd in omponent form, write progrm for our grphing utilit in whih the output is () u, (), nd () the ngle etween u nd. TRUE OR FLSE? In Eerises 87 nd 88, determine whether the sttement is true or flse. Justif our nswer. 87. The work W done onstnt fore F ting long the line of motion of n ojet is represented etor. 88. sliding door moes long the line of etor PQ \. If fore is pplied to the door long etor tht is orthogonl to PQ \, then no work is done. 89. PROOF Use etors to proe tht the digonls of rhomus re perpendiulr. 90. PSTONE Wht is known out, the ngle etween two nonzero etors u nd, under eh ondition (see figure)? 9. THINK OUT IT Wht n e sid out the etors u nd under eh ondition? () The projetion of u onto equls u. () The projetion of u onto equls PROOF Proe the following. u u u u θ Origin () u 0 () u > 0 () u < 0 9. PROOF Proe tht if u is unit etor nd is the ngle etween u nd i, then u os i sin j. 9. PROOF Proe tht if u is unit etor nd is the ngle etween u nd j, then u os i sin j.

42 68 hpter 6 dditionl Topis in Trigonometr 6.5 TRIGONOMETRI FORM OF OMPLEX NUMER Wht ou should lern Plot omple numers in the omple plne nd find solute lues of omple numers. Write the trigonometri forms of omple numers. Multipl nd diide omple numers written in trigonometri form. Use DeMoire s Theorem to find powers of omple numers. Find nth roots of omple numers. Wh ou should lern it You n use the trigonometri form of omple numer to perform opertions with omple numers. For instne, in Eerises on pge 77, ou n use the trigonometri forms of omple numers to help ou sole polnomil equtions. The omple Plne Just s rel numers n e represented points on the rel numer line, ou n represent omple numer z i s the point, in oordinte plne (the omple plne). The horizontl is is lled the rel is nd the ertil is is lled the imginr is, s shown in Figure 6.. FIGURE 6. (, ) or i Imginr is (, ) or + i Rel is The solute lue of the omple numer etween the origin 0, 0 nd the point,. i is defined s the distne Definition of the solute Vlue of omple Numer The solute lue of the omple numer z i is i. (, 5) 9 5 FIGURE 6.5 Imginr is Rel is If the omple numer i is rel numer (tht is, if 0), then this definition grees with tht gien for the solute lue of rel numer 0i 0. Emple Finding the solute Vlue of omple Numer Plot z 5i nd find its solute lue. The numer is plotted in Figure 6.5. It hs n solute lue of z 5 9. Now tr Eerise 9.

43 Setion 6.5 Trigonometri Form of omple Numer 69 Trigonometri Form of omple Numer (,) r Imginr is θ Rel is In Setion., ou lerned how to dd, sutrt, multipl, nd diide omple numers. To work effetiel with powers nd roots of omple numers, it is helpful to write omple numers in trigonometri form. In Figure 6.6, onsider the nonzero omple numer i. letting e the ngle from the positie rel is (mesured ounterlokwise) to the line segment onneting the origin nd the point,, ou n write r os nd where r. onsequentl, ou he i r os r sin i r sin from whih ou n otin the trigonometri form of omple numer. FIGURE 6.6 Trigonometri Form of omple Numer The trigonometri form of the omple numer z i is z ros i sin where r os, r sin, r, nd tn The numer r is the modulus of z, nd is lled n rgument of z.. The trigonometri form of omple numer is lso lled the polr form. euse there re infinitel mn hoies for, the trigonometri form of omple numer is not unique. Normll, is restrited to the interl 0 <, lthough on osion it is onenient to use < 0. Emple Writing omple Numer in Trigonometri Form Write the omple numer z i in trigonometri form. The solute lue of z is r i 6 z = π Imginr is Rel is nd the referene ngle tn is gien euse tn nd euse z i lies in Qudrnt III, ou hoose to e So, the trigonometri form is z ros. i sin. z = i os i sin. See Figure 6.7. FIGURE 6.7 Now tr Eerise 7.

44 70 hpter 6 dditionl Topis in Trigonometr Emple Writing omple Numer in Stndrd Form TEHNOLOGY grphing utilit n e used to onert omple numer in trigonometri (or polr) form to stndrd form. For speifi kestrokes, see the user s mnul for our grphing utilit. Write the omple numer in stndrd form i. euse os nd sin ou n write, z 8 os i sin i 6i. z 8 os i sin Now tr Eerise 5. Multiplition nd Diision of omple Numers The trigonometri form dpts niel to multiplition nd diision of omple numers. Suppose ou re gien two omple numers z r os i sin z z The produt of nd is gien nd z z r r os i sin os i sin r r os os sin sin isin os os sin. Using the sum nd differene formuls for osine nd sine, ou n rewrite this eqution s z z r r os i sin. z r os i sin. This estlishes the first prt of the following rule. The seond prt is left for ou to erif (see Eerise 09). Produt nd Quotient of Two omple Numers Let z nd z r os i sin r os i sin e omple numers. z z r r os i sin Produt z r os z r i sin, z 0 Quotient Note tht this rule ss tht to multipl two omple numers ou multipl moduli nd dd rguments, wheres to diide two omple numers ou diide moduli nd sutrt rguments.

45 Setion 6.5 Trigonometri Form of omple Numer 7 Emple Multipling omple Numers TEHNOLOGY Some grphing utilities n multipl nd diide omple numers in trigonometri form. If ou he ess to suh Find the produt z z of the omple numers. z os i sin z z os i sin 6 os 6 os 5 5 i sin 60 i 6i You n hek this result first onerting the omple numers to the stndrd forms z i nd z i nd then multipling lgerill, s in Setion.. z z i i i i 6i Now tr Eerise 7. 6 i sin z 8 os i sin os 6 i sin 6 6 Multipl moduli nd dd rguments. grphing utilit, use it to find z z nd z z in Emples nd 5. 6 os i sin nd re oterminl. 5 Emple 5 Diiding omple Numers Find the quotient z z of the omple numers. z os 00 i sin 00 z 8os 75 i sin 75 z os 00 i sin 00 z 8os 75 i sin 75 os00 75 i sin os 5 i sin 5 i i Now tr Eerise 5. Diide moduli nd sutrt rguments.

46 7 hpter 6 dditionl Topis in Trigonometr Powers of omple Numers The trigonometri form of omple numer is used to rise omple numer to power. To omplish this, onsider repeted use of the multiplition rule. z ros z ros i sin i sin ros i sin r os i sin i sin r os i sin z r os i sin ros z r os i sin z 5 r 5 os 5 i sin 5.. This pttern leds to DeMoire s Theorem, whih is nmed fter the Frenh mthemtiin rhm DeMoire (667 75). HISTORIL NOTE The Grnger olletion rhm DeMoire (667 75) is rememered for his work in proilit theor nd DeMoire s Theorem. His ook The Dotrine of hnes (pulished in 78) inludes the theor of reurring series nd the theor of prtil frtions. DeMoire s Theorem If z ros is omple numer nd is positie integer, then Finding Powers of omple Numer Use DeMoire s Theorem to find i. z n ros Emple 6 i sin First onert the omple numer to trigonometri form using r So, the trigonometri form is i sin nn r n os n i sin n. nd z i os i sin. Then, DeMoire s Theorem, ou he i os i sin os i sin 096os 8 i sin Now tr Eerise 69. rtn.