5.5 The Law of Sines

Transcription

1 434 HPTER 5 nlyti Trigonometry 5.5 Te Lw of Sines Wt you ll lern out Deriving te Lw of Sines Solving Tringles (S, S) Te miguous se (SS) pplitions... nd wy Te Lw of Sines is powerful extension of te tringle ongruene teorems of Euliden geometry. Deriving te Lw of Sines Rell from geometry tt tringle s six prts (tree sides (S), tree ngles ()), ut tt its size nd spe n e ompletely determined y fixing only tree of tose prts, provided tey re te rigt tree. Tese treesomes tt determine tringle ongruene re known y teir ronyms: S, S, SS, nd SSS. Te oter two ronyms represent mtups tt don t quite work: determines similrity only, wile SS does not even determine similrity. Wit trigonometry we n find te oter prts of te tringle one ongruene is estlised. Te tools we need re te Lw of Sines nd te Lw of osines, te sujets of our lst two trigonometri setions. Te Lw of Sines sttes tt te rtio of te sine of n ngle to te lengt of its opposite side is te sme for ll tree ngles of ny tringle. Lw of Sines In ny wit ngles,, nd opposite sides,, nd, respetively, te following eqution is true: sin sin sin. Te derivtion of te Lw of Sines refers to te two tringles in Figure 5.13, in e of wi we ve drwn n ltitude to side. Rigt tringle trigonometry pplied to eiter of te tringles in Figure 5.13 tells us tt In te ute tringle on te top, sin. sin, wile in te otuse tringle on te ottom, sin 1p - 2. ut sin 1p - 2 sin, so in eiter se sin. FIGURE 5.13 Te Lw of Sines. Solving for in ot equtions yields sin sin. Te eqution sin sin is equivlent to If we were to drw n ltitude to side nd repet te sme steps s ove, we would re te onlusion tt Putting te results togeter, sin sin sin sin sin. sin. sin.

2 SETION 5.5 Te Lw of Sines FIGURE 5.14 tringle determined y S. (Exmple 1) FIGURE 5.15 Te digrm for prt 1. (Explortion 1) FIGURE 5.1 Te digrm for prt 2. (Explortion 1) FIGURE 5.17 Te digrm for prts 3 5. (Explortion 1) Solving Tringles (S, S) Two ngles nd side of tringle, in ny order, determine te size nd spe of tringle ompletely. Of ourse, two ngles of tringle determine te tird, so we relly get one of te missing tree prts for free. We solve for te remining two prts (te unknown sides) wit te Lw of Sines. EXMPLE 1 Solving Tringle Given Two ngles nd Side Solve given tt 3, 48, nd 8. (See Figure 5.14.) SOLUTION First, we note tt We ten pply te Lw of Sines: sin sin sin nd sin sin 3 sin 48 sin 3 sin sin 48 8 sin 9 sin 3 sin 3 L L Te six prts of te tringle re: L L Now try Exerise 1. Te miguous se (SS) Wile two ngles nd side of tringle re lwys suffiient to determine its size nd spe, te sme nnot e sid for two sides nd n ngle. Perps unexpetedly, it depends on were tt ngle is. If te ngle is inluded etween te two sides (te SS se), ten te tringle is uniquely determined up to ongruene. If te ngle is opposite one of te sides (te SS se), ten tere migt e one, two, or zero tringles determined. Solving tringle in te SS se involves te Lw of osines nd will e ndled in te next setion. Solving tringle in te SS se is done wit te Lw of Sines, ut wit n eye towrd te possiilities, s seen in te following Explortion. EXPLORTION 1 Determining te Numer of Tringles We wis to onstrut given ngle, side, nd side. 1. Suppose is otuse nd tt side is s sown in Figure To omplete te tringle, side must determine point on te dotted orizontl line (wi extends infinitely to te left). Explin from te piture wy unique tringle is determined if 7, ut no tringle is determined if. 2. Suppose is ute nd tt side is s sown in Figure 5.1. To omplete te tringle, side must determine point on te dotted orizontl line (wi extends infinitely to te rigt). Explin from te piture wy unique tringle is determined if, ut no tringle is determined if. 3. Suppose is ute nd tt side is s sown in Figure If 7 7, ten we n form tringle s sown. Find seond point on te dotted orizontl line tt gives side of te sme lengt, ut determines different tringle. (Tis is te miguous se. ) 4. Explin wy sin is te sme in ot tringles in te miguous se. (Tis is wy te Lw of Sines is lso miguous in tis se.) 5. Explin from Figure 5.17 wy unique tringle is determined if Ú.

3 43 HPTER 5 nlyti Trigonometry Now tt we know wt n ppen, let us try te lger FIGURE 5.18 tringle determined y SS. (Exmple 2) () () FIGURE 5.19 Two tringles determined y te sme SS vlues. (Exmple 3) EXMPLE 2 Solving Tringle Given Two Sides nd n ngle Solve given tt 7,, nd 2.3. (See Figure 5.18.) SOLUTION y drwing resonle sket (Figure 5.18), we n ssure ourselves tt tis is not te miguous se. (In ft, tis is te se desried in step 5 of Explortion 1.) egin y solving for te ute ngle, using te Lw of Sines: sin sin Lw of Sines. sin 2.3 sin 7 sin 2.3 sin 7 sin -1 sin Round to mt ury of given ngle. Ten, find te otuse ngle y sutrtion: Finlly, find side : Te six prts of te tringle re: sin sin sin 2.3 sin sin sin 2.3 L L 11.9 EXMPLE 3 Hndling te miguous se Solve given tt, 7, nd 30. Now try Exerise 9. SOLUTION y drwing resonle sket (Figure 5.19), we see tt two tringles re possile wit te given informtion. We keep tis in mind s we proeed. We egin y using te Lw of Sines to find ngle. sin sin Lw of Sines sin 30 sin 7 7 sin 30 sin sin -1 7 sin Round to mt ury of given ngle.

4 SETION 5.5 Te Lw of Sines 437 Notie tt te lultor gve us one vlue for, not two. Tt is euse we used te funtion sin -1, wi nnot give two output vlues for te sme input vlue. Indeed, te funtion sin -1 will never give n otuse ngle, wi is wy we ose to strt wit te ute ngle in Exmple 2. In tis se, te lultor s found te ngle sown in Figure Find te otuse ngle y sutrtion: Finlly, find side : sin sin 30.0 sin sin sin sin 30 L 10.9 So, under te ssumption tt ngle is ute (see Figure 5.19), te six prts of te tringle re: L 10.9 If ngle is otuse, ten we n see from Figure 5.19 tt it s mesure y sutrtion, te ute ngle We ten reompute : sin 5.7 sin 30 L 1.2 Sustitute 5.7 for in erlier omputtion. So, under te ssumption tt ngle is otuse (see Figure 5.19), te six prts of te tringle re: L 1.2 Now try Exerise 19. pplitions Mny prolems involving ngles nd distnes n e solved y superimposing tringle onto te sitution nd solving te tringle. N mi 48 FIGURE 5.20 Determining te lotion of fire. (Exmple 4) N EXMPLE 4 Loting Fire Forest Rnger ris Jonson t rnger sttion sigts fire in te diretion 32 est of nort. Rnger Rik Torpe t rnger sttion, 10 miles due est of, sigts te sme fire on line 48 west of nort. Find te distne from e rnger sttion to te fire. SOLUTION Let represent te lotion of te fire. sket (Figure 5.20) sows te superimposed tringle,, in wi ngles nd nd teir inluded side () re known. Tis is setup for te Lw of Sines. (ontinued)

5 438 HPTER 5 nlyti Trigonometry Note tt nd y sutrtion, we find tt sin sin nd sin 58 sin sin 58 sin 80 L 8. sin sin Lw of Sines sin 42 sin sin 42 sin 80 L.8 Te fire is out.8 miles from rnger sttion nd out 8. miles from rnger sttion. Now try Exerise FIGURE 5.21 telepone pole on slope. (Exmple 5) EXMPLE 5 Finding te Heigt of Pole rod slopes 10 ove te orizontl, nd vertil telepone pole stnds eside te rod. Te ngle of elevtion of te Sun is 2, nd te pole sts 14.5-foot sdow downill long te rod. Find te eigt of te telepone pole. SOLUTION Tis is n interesting vrition on typil pplition of rigt tringle trigonometry. Te slope of te rod elimintes te onvenient rigt ngle, ut we n still solve te prolem y solving tringle. Figure 5.21 sows te superimposed tringle,. little preliminry geometry is required to find te mesure of ngles nd. Due to te slope of te rod, ngle is 10 less tn te ngle of elevtion of te Sun nd ngle is 10 more tn rigt ngle. Tt is, Terefore, sin sin sin 52 sin sin 52 sin 28 L Lw of Sines Round to mt ury of input. Te pole is pproximtely 24.3 feet ig. Now try Exerise 39. QUIK REVIEW 5.5 (For elp, go to Setions 4.2 nd 4.7.) Exerise numers wit gry kground indite prolems tt te utors ve designed to e solved witout lultor. In Exerises 1 4, solve te eqution / /d for te given vrile d In Exerises 5 nd, evlute te expression. 7 sin 48 9 sin sin 23 sin 14 In Exerises 7 10, solve for te ngle x. 7. sin x 0.3, 0 x sin x 0.3, 90 x sin x -0.7, 180 x sin x -0.7, 270 x 30

6 SETION 5.5 Te Lw of Sines 439 SETION 5.5 EXERISES In Exerises 1 4, solve te tringle () 19 5 () In Exerises 5 8, solve te tringle , 30, , 2, , 70, , 103, 12 In Exerises 9 12, solve te tringle , 17, , 32, , 14, , 4, In Exerises 13 18, stte weter te given mesurements determine zero, one, or two tringles , 2, , 17, , 17, , 24, , 18, , 14, 2 In Exerises 19 22, two tringles n e formed using te given mesurements. Solve ot tringles , 1, , 21, , 19, , 11, Determine te vlues of tt will produe te given numer of tringles if 10 nd 42. () Two tringles () One tringle () Zero tringles 24. Determine te vlues of tt will produe te given numer of tringles if 12 nd 53. () Two tringles () One tringle () Zero tringles In Exerises 25 nd 2, deide weter te tringle n e solved using te Lw of Sines. If so, solve it. If not, explin wy not () () In Exerises 27 3, respond in one of te following wys: () Stte, nnot e solved wit te Lw of Sines. () Stte, No tringle is formed. () Solve te tringle , 8, , 8, , 15, , 12, , 18, , 22, , 49, , 13, , 8, , 19, Surveying nyon Two mrkers nd on te sme side of nyon rim re 5 ft prt. tird mrker, loted ross te rim, is positioned so tt 72 nd ft () Find te distne etween nd. () Find te distne etween te two nyon rims. (ssume tey re prllel.) 38. Weter Foresting N N Two meteorologists re 25 mi prt loted on n est-west rod. Te meteorologist t point sigts torndo 38 est of nort. Te meteorologist t point sigts te sme torndo 53 west of nort. 25 mi Find te distne from e meteorologist to te torndo. lso find te distne etween te torndo nd te rod.

7 440 HPTER 5 nlyti Trigonometry 39. Engineering Design vertil flgpole stnds eside rod tt slopes t n ngle of 15 wit te orizontl. Wen te ngle of elevtion of te Sun is 2, te flgpole sts 1-ft sdow downill long te rod. Find te eigt of te flgpole ft Horizontl 20 mi S (sip) 40. ltitude Oservers 2.32 mi prt see ot-ir lloon diretly etween tem ut t te ngles of elevtion sown in te figure. Find te ltitude of te lloon mi 41. Reduing ir Resistne 4-ft irfoil tted to te of truk redues wind resistne. If te ngle etween te irfoil nd te top is 18 nd ngle is 10, find te lengt of vertil re positioned s sown in te figure. 4 ft Using Mesurement Dt geometry lss is divided into ten tems, e of wi is given yrdstik nd protrtor to find te distne from point on te edge of pond to tree t point on te opposite sore. fter tey mrk points nd wit stkes, e tem uses protrtor to mesure ngles nd nd yrdstik to mesure distne. Teir mesurements re given in te tle Use te dt to find te lss s est estimte for te distne. 42. Group tivity Ferris Weel Design Ferris weel s 1 evenly sped rs. Te distne etween djent irs is 15.5 ft. Find te rdius of te weel (to te nerest 0.1 ft). 43. Finding Heigt Two oservers re 00 ft prt on opposite sides of flgpole. Te ngles of elevtion from te oservers to te top of te pole re 19 nd 21. Find te eigt of te flgpole. 44. Finding Heigt Two oservers re 400 ft prt on opposite sides of tree. Te ngles of elevtion from te oservers to te top of te tree re 15 nd 20. Find te eigt of te tree. 45. Finding Distne Two ligtouses nd re known to e extly 20 mi prt on nort-sout line. sip s ptin t S mesures S to e 33. rdio opertor t mesures S to e 52. Find te distne from te sip to e ligtouse. Stndrdized Test Questions 47. True or Flse Te rtio of te sines of ny two ngles in tringle equls te rtio of te lengts of teir opposite sides. Justify your nswer. 48. True or Flse Te perimeter of tringle wit two 10-in sides nd two 40 ngles is greter tn 3. Justify your nswer. You my use grping lultor wen nswering tese questions. 49. Multiple oie Te lengt x in te tringle sown t te rigt is () 8.. () () (D) (E) x

8 SETION 5.5 Te Lw of Sines Multiple oie Wi of te following tree tringle prts do not neessrily determine te oter tree prts? () S () S () SS (D) SS (E) SSS 51. Multiple oie Te sortest side of tringle wit ngles 50, 0, nd 70 s lengt 9.0. Wt is te lengt of te longest side? () 11.0 () 11.5 () 12.0 (D) 12.5 (E) Multiple oie How mny nonongruent tringles n e formed if 5, 0, nd 8? () None () One () Two (D) Tree (E) Infinitely mny Explortions 53. Writing to Lern () Sow tt tere re infinitely mny tringles wit given if te sum of te tree positive ngles is 180. () Give tree exmples of tringles were 30, 0, nd 90. () Give tree exmples were Use te Lw of Sines nd te ofuntion identities to derive te following formuls from rigt tringle trigonometry: () sin opp () os dj () tn opp yp yp dj () In terms of te given ngle nd te given lengt, stte te onditions on lengt tt will result in unique tringle eing formed. (d) In terms of te given ngle nd te given lengt, stte te onditions on lengt tt will result in two possile tringles eing formed. Extending te Ides 5. Solve tis tringle ssuming tt is otuse. [Hint: Drw perpendiulr from to te line troug nd.] 57. Pilot lultions Towers nd re known to e 4.1 mi prt on level ground. pilot mesures te ngles of depression to te towers to e 3.5 nd 25, respetively, s sown in te figure. Find distnes nd nd te eigt of te irplne mi Wrpping up Explortion 1 Refer to Figures 5.1 nd 5.17 in Explortion 1 of tis setion. () Express in terms of ngle nd lengt. () In terms of te given ngle nd te given lengt, stte te onditions on lengt tt will result in no tringle eing formed.