13.5 pply te Lw of Sines TEKS.1,.4, 2.4.; P.3.E efore Now You solved rigt tringles. You will solve tringles tt ve no rigt ngle. Wy? So you n find te distne etween frwy ojets, s in Ex. 44. Key Voulry lw of sines In Lesson 13.1, you solved rigt tringles. To solve tringle wit no rigt ngle, you need to know te lengt of t lest one side nd ny two oter prts of te tringle. Te lw of sines n e used to solve tringles wen two ngles nd te lengt of ny side re known (S or S ses), or wen te lengts of two sides nd n ngle opposite one of te two sides re known (SS se). KEY ONEPT For Your Noteook Lw of Sines Te lw of sines n e written in eiter of te following forms for n wit sides of lengt,, nd. sin } 5} sin 5 sin } } 5} 5} sin sin sin E XMPLE 1 Solve tringle for te S or S se Solve n wit 5 1078, 5 258, nd 5 15. 5 15 First find te ngle: 5 18082107822585488. y te lw of sines, you n write } 5} 15 5 }. sin 488 sin 258 sin 1078 1078 258 } 5 15 Write two equtions, e } sin 488 sin 258 wit one vrile. } sin 1078 5 15 } sin 258 5 15 sin 488 } Solve for e vrile. 5 15 sin 1078 } sin 258 sin 258 ø 26.4 Use lultor. ø 33.9 In n, 5 488, ø 26.4, nd ø 33.9. GUIDED PRTIE for Exmple 1 Solve n. 1. 5 348, 5 1008, 5 8 2. 5 518, 5 448, 5 11 882 pter 13 Trigonometri Rtios nd Funtions
DESRIE SES euse te SS se n result in 0, 1, or 2 tringles, it is lled te miguous se. SS SE Two ngles nd one side (S or S) determine extly one tringle. Two sides nd n ngle opposite one of te sides (SS) my determine no tringle, one tringle, or two tringles. KEY ONEPT For Your Noteook Possile Tringles in te SS se onsider tringle in wi you re given,, nd. y fixing side nd ngle, you n sket te possile positions of side to figure out ow mny tringles n e formed. In te digrms elow, note tt 5 sin. is otuse. is ute. No tringle > No tringle 5 One tringle > One tringle < < Two tringles > One tringle E XMPLE 2 Solve te SS se wit one solution Solve n wit 5 1158, 5 20, nd 5 11. First mke sket. euse is otuse nd te side opposite is longer tn te given djent side, you know tt only one tringle n e formed. Use te lw of sines to find. sin } 5 sin 1158 } Lw of sines 11 20 5 20 5 11 1158 sin 5 11 sin 1158 } 20 ø 0.4985 Multiply e side y 11. ø 29.98 Use inverse sine funtion. You ten know tt ø 18082 11582 29.985 35.18. Use te lw of sines gin to find te remining side lengt of te tringle. } 5} 20 Lw of sines sin 35.18 sin 1158 5 20 sin 35.18 } sin 1158 Multiply e side y sin 35.18. ø 12.7 Use lultor. In n, ø 29.98, ø 35.18, nd ø 12.7. 13.5 pply te Lw of Sines 883
E XMPLE 3 Exmine te SS se wit no solution Solve n wit 5 518, 5 3.5, nd 5 5. egin y drwing orizontl line. On one end form 518 ngle () nd drw segment 5 units long ( }, or ). t vertex, drw segment 3.5 units long (). You n see tt needs to e t lest 5 sin 518 ø 3.9 units long to re te orizontl side nd form tringle. So, it is not possile to drw te indited tringle. 5 5 518 5 3.5 5 sin 518 3.9 E XMPLE 4 Solve te SS se wit two solutions Solve n wit 5 408, 5 13, nd 5 16. First mke sket. euse sin 5 16 sin 408 ø 10.3, nd 10.3 < 13 < 16 ( < < ), two tringles n e formed. Tringle 1 Tringle 2 5 16 408 5 13 5 16 408 5 13 Use te lw of sines to find te possile mesures of. sin } 5 sin 408 } Lw of sines 16 13 sin 5 16 sin 408 } 13 ø 0.7911 Use lultor. Tere re two ngles etween 08 nd 1808 for wi sin ø 0.7911. One is ute nd te oter is otuse. Use your lultor to find te ute ngle: sin 21 0.7911 ø 52.38. Te otuse ngle s 52.38 s referene ngle, so its mesure is 18082 52.385 127.78. Terefore, ø 52.38 or ø 127.78. Now find te remining ngle nd side lengt for e tringle. Tringle 1 Tringle 2 ø 18082408252.38587.78 ø 180824082127.78512.38 } 5 13 } } 5} 13 sin 87.78 sin 408 sin 12.38 sin 408 5 13 sin 87.78 } ø 20.2 5 13 sin 12.38 } ø 4.3 sin 408 sin 408 In Tringle 1, ø 52.38, ø 87.78, In Tringle 2, ø 127.78, ø 12.38, nd ø 20.2. nd ø 4.3. t lsszone.om 884 pter 13 Trigonometri Rtios nd Funtions
GUIDED PRTIE for Exmples 2, 3, nd 4 Solve n. 3. 5 1228, 5 18, 5 12 4. 5 368, 5 9, 5 12 5. 5 508, 5 2.8, 5 4 6. 5 1058, 5 13, 5 6 RE OF TRINGLE You n use te following result to find te re of tringle wen you know te lengts of two sides nd te mesure of te inluded ngle. Tis result n lso e used to derive te lw of sines (see Exerise 42). KEY ONEPT For Your Noteook re of Tringle Te re of ny tringle is given y one lf te produt of te lengts of two sides times te sine of teir inluded ngle. For n sown, tere re tree wys to lulte te re: re 5 1 } 2 sin re 5 1 } 2 sin re 5 1 } 2 sin E XMPLE 5 Find te re of tringle IOLOGY lk-neked stilts re irds tt live trougout Florid nd surrounding res ut reed mostly in te tringulr region sown on te mp. Find te re of tis region. 223 mi 54.2 125 mi Te re of te region is: re 5 1 } 2 sin Write re formul. 5 1 } 2 (125)(223) sin 54.28 Sustitute. ø 11,300 Use lultor. Te re of te region is out 11,300 squre miles. GUIDED PRTIE for Exmple 5 Find te re of n wit te given side lengts nd inluded ngle. 7. 5 10, 5 14, 5 468 8. 5 19, 5 8, 5 758 9. 5 11, 5 7, 5 1208 10. 5 20, 5 24, 5 878 13.5 pply te Lw of Sines 885
13.5 EXERISES SKILL PRTIE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 13, 31, nd 45 5 TKS PRTIE ND RESONING Exs. 28, 41, 47, 48, 50, nd 51 5 MULTIPLE REPRESENTTIONS Ex. 45 1. VOULRY Wt informtion do you need to use te lw of sines? 2. WRITING Suppose,, nd re given for n were < 908. Under wt onditions would you ve no tringle? one tringle? two tringles? EXMPLES 1, 2, 3, nd 4 on pp. 882 884 for Exs. 3 28 IDENTIFYING SES Stte te se (S, S, or SS) pplile to te given mesurements. Ten deide weter te mesurements determine one tringle, two tringles, or no tringle. 3. 5 1128, 5 9, 5 4 4. 5 408, 5 758, 5 20 5. 5 528, 5 32, 5 42 6. 5 378, 5 8, 5 14 7. 5 288, 5 648, 5 55 8. 5 1498, 5 7, 5 10 9. 5 348, 5 5, 5 16 10. 5 708, 5 85, 5 88 11. 5 488, 5 28, 5 20 SOLVING TRINGLES Solve n. 12. 858 19 478 13. 16 25 1048 14. 818 26 338 15. 738 34 428 16. 17 14 568 17. 248 43 618 SOLVING TRINGLES Solve n. (Hint: Some of te tringles ve no solution nd some ve two solutions.) 18. 5 738, 5 18, 5 11 19. 5 268, 5 358, 5 13 20. 5 1028, 5 438, 5 21 21. 5 388, 5 19, 5 25 22. 5 558, 5 648, 5 34 23. 5 1148, 5 15, 5 10 24. 5 988, 5 29, 5 33 25. 5 498, 5 328, 5 44 26. 5 218, 5 17, 5 32 27. ERROR NLYSIS Desrie nd orret te error in finding te mesure of ngle in te tringle elow. 558 5 6 sin } 5 sin 558 } 6 5 sin 5 6 sin 558 } ø 0.9830 5 ø 79.48 28. TKS RESONING Wt is te side lengt in n if 5 328, 5 678, nd 5 31 ft? 16.6 ft 28.9 ft 33.3 ft D 57.8 ft 886 pter 13 Trigonometri Rtios nd Funtions
EXMPLE 5 on p. 885 for Exs. 29 41 FINDING RE Find te re of n wit te given side lengts nd inluded ngle. 29. 5 1248, 5 9, 5 11 30. 5 688, 5 13, 5 7 31. 5 348, 5 29, 5 36 32. 5 798, 5 25, 5 17 33. 5 578, 5 9, 5 5 34. 5 968, 5 7, 5 15 35. 5 1308, 5 23, 5 20 36. 5 608, 5 19, 5 14 37. 5 298, 5 38, 5 31 FINDING RE Find te re of n. 38. 15 39. 838 24 40. 16 20 578 13 598 668 41. TKS RESONING Wt is te re of n if 5 528, 5 29, nd 5 24? 274 units 2 348 units 2 548 units 2 D 696 units 2 42. HLLENGE Using te tringle sown t te rigt s referene, derive te formuls for te re of tringle given on pge 885. Ten use te re formuls to derive te lw of sines. PROLEM SOLVING EXMPLE 1 on p. 882 for Ex. 43 43. LIFEGURDS Two lifegurds re wting windsurfer. Use te informtion in te digrm to find te distne from e lifegurd to te windsurfer. 40 300 ft 45 EXMPLE 2 on p. 883 for Ex. 44 44. NEW YORK ITY You re on te oservtion dek of te Empire Stte uilding looking t te rysler uilding. Wen you turn 1458 lokwise, you see te Sttue of Lierty. You know tt te rysler uilding nd te Empire Stte uilding re out 0.6 mile prt nd tt te rysler uilding nd te Sttue of Lierty re out 5.7 miles prt. Estimte te distne etween te Empire Stte uilding nd te Sttue of Lierty. EXMPLE 5 on p. 885 for Exs. 45 46 45. MULTIPLE REPRESENTTIONS You re fertilizing tringulr grden. One side of te grden is 62 feet long nd noter side is 54 feet long. Te ngle opposite te 62 foot side is 588.. Drwing Digrm Drw digrm to represent tis sitution.. Solving Tringle Use te lw of sines to solve te tringle you drew in prt ().. pplying Formul One g of fertilizer overs n re of 200 squre feet. How mny gs of fertilizer will you need to over te entire grden? 13.5 pply te Lw of Sines 887
46. MULTI-STEP PROLEM Qudrilterl D sown t te rigt is kite.. Find te re of n D.. Find te re of nd.. Wt is te re of te kite? 8.5 in. 14 in. 548 328 8.5 in. 14 in. D 47. TKS RESONING uilding is onstruted on top of liff tt is 300 meters ig. person stnding on level ground elow te liff oserves tt te ngle of elevtion to te top of te uilding is 728, nd te ngle of elevtion to te top of te liff is 638.. How fr wy is te person from te se of te liff?. Desrie two different metods you n use to find te eigt of te uilding. Use one of tese metods to find te uilding s eigt. 48. TKS RESONING Use grping lultor to explore ow te inluded ngle in te formuls on pge 885 ffets tringle s re.. Model oose lengts for two sides of te tringle. Let x represent te mesure (in degrees) of te inluded ngle. Write n eqution tt gives te tringle s re y s funtion of x.. Grping lultor Enter te eqution from prt () into grping lultor. Use te tle feture to exmine vlues of te re for 08 < x8 < 1808. Does te re lwys inrese s x inreses? Explin.. Interpret Wt vlue of x mximizes te tringle s re? Wt is te mximum re, nd ow is it relted to te side lengts you ose in prt ()? 49. HLLENGE Te distne etween Merury nd te sun is pproximtely 36 million miles. Te distne etween Ert nd te sun is pproximtely 93 million miles. If on ertin dy te ngle (mesured from Ert) etween te sun nd Merury is 228, wt re te possile distnes etween Merury nd Ert? MIXED REVIEW FOR TKS TKS PRTIE t lsszone.om REVIEW Skills Review Hndook p. 991; TKS Workook 50. TKS PRTIE my uses 100 of te tiles sown to tile squre room. Wt is te perimeter of te room? TKS Oj. 10 24 ft 36 ft 32 ft D 80 ft 0.8 ft 0.8 ft REVIEW Skills Review Hndook p. 1008; TKS Workook 51. TKS PRTIE Te ges of te first 20 people entering n musement prk re 12, 16, 35, 24, 40, 48, 15, 18, 20, 50, 38, 14, 11, 28, 18, 19, 26, 15, 16, nd 21. Wt is n eptle set of intervls to use wen mking istogrm of te ges? TKS Oj. 9 F 10 20, 21 40, nd 41 50 G 10 15, 16 20, 21 30, nd 31 50 H 11 20, 21 30, 31 40, nd 41 50 J 15 25, 26 35, 36 45, nd 46 55 888 EXTR PRTIE for Lesson 13.5, p. 1022 ONLINE QUIZ t lsszone.om