694 hpter 6 dditionl Topics in Trigonometry ETIO 6.2 The Lw of osines Ojectives Use the Lw of osines to solve olique tringles. olve pplied prolems using the Lw of osines. Use Heron s formul to fi nd the re of tringle. P leontologists use trigonometry to study the movements mde y dinosurs millions of yers go. Figure 6.13, sed on dt collected t Dinosur Vlley tte Prk in Glen Rose, Texs, shows footprints mde y two-footed crnivorous (met-eting) dinosur nd the hindfeet of herivorous (plnt-eting) dinosur. For ech dinosur, the figure indictes the pce nd the stride. The pce is the distnce from the left footprint to the right footprint, nd vice vers. The stride is the distnce from the left footprint to the next left footprint or from the right footprint to the next right footprint. lso shown in Figure 6.13 is the pce ngle, designted y u. otice tht neither dinosur moves with pce ngle of 180, mening tht the footprints re directly in line. The footprints show zig-zig pttern tht is numericlly descried y the pce ngle. dinosur tht is n efficient wlker hs pce ngle close to 180, minimizing zig-zg motion nd mximizing forwrd motion. Pce: 2.9 ft Pce: 3.6 ft tride: 5.78 ft u Pce ngle u tride: 5.2 ft Pce: 3.0 ft Pce: 3.2 ft rnivore FIGURE 6.13 Dinosur Footprints ource: Glen J. Kun, n Overview of Dinosur Trcking Herivore How cn we determine the pce ngles for the crnivore nd the herivore in Figure 6.13? Prolems such s this, in which we know the mesures of three sides of tringle nd we need to find the mesurement of missing ngle, cnnot e solved y the Lw of ines. To numericlly descrie which dinosur in Figure 6.13 mde more forwrd progress with ech step, we turn to the Lw of osines. The Lw of osines nd Its Derivtion We now look t nother reltionship tht exists mong the sides nd ngles in n olique tringle. The Lw of osines is used to solve tringles in which two sides nd the included ngle () re known, or those in which three sides () re known.
ection 6.2 The Lw of osines 695 DIOVERY Wht hppens to the Lw of osines c 2 = 2 + 2-2 cos if = 90? Wht fmilir theorem do you otin? The Lw of osines If,, nd re the mesures of the ngles of tringle, nd,, nd c re the lengths of the sides opposite these ngles, then 2 = 2 + c 2-2c cos 2 = 2 + c 2-2c cos c 2 = 2 + 2-2 cos. The squre of side of tringle equls the sum of the squres of the other two sides minus twice their product times the cosine of their included ngle. y x = (0, 0) c = (c, 0) FIGURE 6.14 = (x, y) Use the Lw of osines to solve olique tringles. y x To prove the Lw of osines, we plce tringle in rectngulr coordinte system. Figure 6.14 shows tringle with three cute ngles. The vertex is t the origin nd side c lies long the positive x@xis. The coordintes of re (x, y). Using the right tringle tht contins ngle, we pply the definitions of the cosine nd the sine. co s = x sin = y x = cos y = sin Multiply oth sides of ech eqution y nd solve for x nd y, respectively. Thus, the coordintes of re (x, y) = ( cos, sin ). lthough tringle in Figure 6.14 shows ngle s n cute ngle, if were otuse, the coordintes of would still e ( cos, sin ). This mens tht our proof pplies to oth kinds of olique tringles. We now pply the distnce formul to the side of the tringle with length. otice tht is the distnce from (x, y) to (c, 0). = 2(x - c) 2 + (y - 0) 2 Use the distnce formul. 2 = (x - c) 2 + y 2 qure oth sides of the eqution. 2 = ( cos - c) 2 + ( sin ) 2 x = cos nd y = sin. 2 = 2 cos 2-2c cos + c 2 + 2 sin 2 2 = 2 sin 2 + 2 cos 2 + c 2-2c cos 2 = 2 (sin 2 + cos 2 ) + c 2-2c cos qure the two expressions. Rerrnge terms. Fctor 2 from the first two terms. 2 = 2 + c 2-2c cos sin 2 + cos 2 = 1 The resulting eqution is one of the three formuls for the Lw of osines. The other two formuls re derived in similr mnner. olving Olique Tringles If you re given two sides nd n included ngle () of n olique tringle, none of the three rtios in the Lw of ines is known. This mens tht we do not egin solving the tringle using the Lw of ines. Insted, we pply the Lw of osines nd the following procedure: olving n Tringle 1. Use the Lw of osines to find the side opposite the given ngle. 2. Use the Lw of ines to find the ngle opposite the shorter of the two given sides. This ngle is lwys cute. 3. Find the third ngle y sutrcting the mesure of the given ngle nd the ngle found in step 2 from 180.
696 hpter 6 dditionl Topics in Trigonometry = 20 60 c = 30 FIGURE 6.15 olving n tringle EXMPLE 1 olving n Tringle olve the tringle in Figure 6.15 with = 60, = 20, nd c = 30. Round lengths of sides to the nerest tenth nd ngle mesures to the nerest degree. OLUTIO We re given two sides nd n included ngle. Therefore, we pply the three-step procedure for solving n tringle. tep 1 Use the Lw of osines to find the side opposite the given ngle. Thus, we will find. In this exmple, we know the exct vlue of cos 60 : cos 60 = 0.5. If the exct vlue of the cosine is not ville, you cn clculte 2 + c 2 2c cos in one step with clcultor. 2 = 2 + c 2-2c cos pply the Lw of osines to find. 2 = 20 2 + 30 2-2(20)(30) cos 60 = 20, c = 30, nd = 60. = 400 + 900-1200(0.5) Perform the indicted opertions. = 700 = 2700 26.5 Tke the squre root of oth sides nd solve for. tep 2 Use the Lw of ines to find the ngle opposite the shorter of the two given sides. This ngle is lwys cute. The shorter of the two given sides is = 20. Thus, we will find cute ngle. sin = sin 20 sin = 2700 sin 60 pply the Lw of ines. We re given = 20 nd = 60. Use the vlue of, 2700, from step 1. 2700 sin = 20 sin 60 ross multiply: If = c, then d = c. d sin = 20 sin 60 2700 0.6547 Divide y 2700 nd solve for sin. 41 Find sin -1 0.6547 using clcultor. tep 3 Find the third ngle. utrct the mesure of the given ngle nd the ngle found in step 2 from 180. = 180 - - 180-60 - 41 = 79 = 7 120 The solution is 26.5, 41, nd 79. FIGURE 6.16 heck Point 1 olve the tringle shown in Figure 6.16 with = 120, = 7, nd. Round s in Exmple 1. If you re given three sides of tringle (), solving the tringle involves finding the three ngles. We use the following procedure: olving n Tringle 1. Use the Lw of osines to find the ngle opposite the longest side. 2. Use the Lw of ines to find either of the two remining cute ngles. 3. Find the third ngle y sutrcting the mesures of the ngles found in steps 1 nd 2 from 180.
EXMPLE 2 olving n Tringle ection 6.2 The Lw of osines 697 olve tringle if = 6, = 9, nd c = 4. Round ngle mesures to the nerest degree. c = 4 = 9 = 6 FIGURE 6.17 olving n tringle GRET QUETIO! In tep 2, do I hve to use the Lw of ines to find either of the remining ngles? o. You cn lso use the Lw of osines to find either ngle. However, it is simpler to use the Lw of ines. ecuse the lrgest ngle hs een found, the remining ngles must e cute. Thus, there is no need to e concerned out two possile tringles or n miguous cse. olve pplied prolems using the Lw of osines. W =? E = 650 miles FIGURE 6.18 66 26 c = 600 miles OLUTIO We re given three sides. Therefore, we pply the three-step procedure for solving n tringle. The tringle is shown in Figure 6.17. tep 1 Use the Lw of osines to find the ngle opposite the longest side. The longest side is = 9. Thus, we will find ngle. 2 = 2 + c 2-2c cos pply the Lw of osines to find. 2c cos = 2 + c 2-2 olve for cos. cos = 2 + c 2-2 2c cos = 62 + 4 2-9 2 2 # 6 # = - 29 4 48 = 6, = 9, nd c = 4. Using clcultor, cos -1 1 29 482 53. ecuse cos is negtive, is n otuse ngle. Thus, 180-53 = 127. ecuse the domin of y = cos -1 x is [0, p], you cn use clcultor to find cos -1 1-29 482 127. tep 2 Use the Lw of ines to find either of the two remining cute ngles. We will find ngle. sin = sin 6 sin = 9 sin 127 pply the Lw of ines. We re given = 6 nd = 9. We found tht 127. 9 sin = 6 sin 127 ross multiply. 6 sin 127 sin = 9 0.5324 Divide y 9 nd solve for sin. 32 Find sin -1 0.5324 using clcultor. tep 3 Find the third ngle. utrct the mesures of the ngles found in steps 1 nd 2 from 180. = 180 - - 180-127 - 32 = 21 The solution is 127, 32, nd 21. heck Point 2 olve tringle if = 8, = 10, nd c = 5. Round ngle mesures to the nerest degree. pplictions of the Lw of osines pplied prolems involving nd tringles cn e solved using the Lw of osines. EXMPLE 3 n ppliction of the Lw of osines Two irplnes leve n irport t the sme time on different runwys. One flies on ering of 66 W t 325 miles per hour. The other irplne flies on ering of 26 W t 300 miles per hour. How fr prt will the irplnes e fter two hours? OLUTIO fter two hours, the plne flying t 325 miles per hour trvels 325 # 2 miles, or 650 miles. imilrly, the plne flying t 300 miles per hour trvels 600 miles. The sitution is illustrted in Figure 6.18.
698 hpter 6 dditionl Topics in Trigonometry W =? E = 650 miles 66 26 c = 600 miles Let = the distnce etween the plnes fter two hours. We cn use northsouth line to find ngle in tringle. Thus, = 180-66 - 26 = 88. We now hve = 650, c = 600, nd = 88. We use the Lw of osines to find in this sitution. 2 = 2 + c 2-2c cos 2 = 650 2 + 600 2-2(650)(600) cos 88 755,278 pply the Lw of osines. ustitute: = 650, c = 600, nd = 88. Use clcultor. 2755,278 869 Tke the squre root nd solve for. FIGURE 6.18 (repeted) fter two hours, the plnes re pproximtely 869 miles prt. heck Point 3 Two irplnes leve n irport t the sme time on different runwys. One flies directly north t 400 miles per hour. The other irplne flies on ering of 75 E t 350 miles per hour. How fr prt will the irplnes e fter two hours? Use Heron s formul to fi nd the re of tringle. Heron s Formul pproximtely 2000 yers go, the Greek mthemticin Heron of lexndri derived formul for the re of tringle in terms of the lengths of its sides. more modern derivtion uses the Lw of osines nd cn e found in the ppendix. Heron s Formul for the re of Tringle The re of tringle with sides,, nd c is r e = 2s(s - )(s - )(s - c), where s is one-hlf its perimeter: s = 1 2 ( + + c). EXMPLE 4 Using Heron s Formul Find the re of the tringle with = 12 yrds, = 16 yrds, nd c = 24 yrds. Round to the nerest squre yrd. OLUTIO egin y clculting one-hlf the perimeter: s = 1 2 ( + + c) = 1 2 (12 + 16 + 24) = 26. Use Heron s formul to find the re: r e = 2s(s - )(s - )(s - c) = 226(26-12)(26-16)(26-24) = 27280 85. The re of the tringle is pproximtely 85 squre yrds. heck Point 4 Find the re of the tringle with = 6 meters, = 16 meters, nd c = 18 meters. Round to the nerest squre meter.
ection 6.2 The Lw of osines 699 OEPT D VOULRY HEK Fill in ech lnk so tht the resulting sttement is true. 1. If,, nd re the mesures of the ngles of tringle, nd,, nd c re the lengths of the sides opposite these ngles, then the Lw of osines sttes tht 2 =. 2. To solve n olique tringle given two sides nd n included ngle (), the first step is to find the missing using the Lw of. Then we use the Lw of to find the ngle opposite the shorter of the two given sides. This ngle is lwys. The third ngle is found y sutrcting the mesure of the given ngle nd the ngle found in the second step from. 3. To solve n olique tringle given three sides (), the first step is to find the ngle opposite the longest side using the Lw of. Then we find either of the two remining cute ngles using the Lw of. 4. Heron s formul for the re of tringle with sides,, nd c is re =, where s =. EXERIE ET 6.2 Prctice Exercises 7. In Exercises 1 8, solve ech tringle. Round lengths of sides to the nerest tenth nd ngle mesures to the nerest degree. 1. = 4 = 4 = 6 2. 46 8. c = 3 = 6 = 16 = 10 3. 32 4. 5. 6. 96 = 4 = 6 c = 6 22 c = 15 = 8 = 6 = 12 = 10 c = 16 In Exercises 9 24, solve ech tringle. Round lengths to the nerest tenth nd ngle mesures to the nerest degree. 9. = 5, = 7, = 42 10. = 10, = 3, = 15 11. = 5, c = 3, = 102 12. = 4, c = 1, = 100 13. = 6, c = 5, = 50 14. = 4, c = 7, = 55 15. = 5, c = 2, = 90 16. = 7, c = 3, = 90 17. = 5, = 7, c = 1 0 18. = 4, = 6, c = 9 19. = 3, = 9, 20. = 4, = 7, c = 6 21. = 3, = 3, c = 3 22. = 5, = 5, c = 5 23. = 63, = 22, c = 5 0 24. = 66, = 25, c = 4 5
700 hpter 6 dditionl Topics in Trigonometry In Exercises 25 30, use Heron s formul to find the re of ech tringle. Round to the nerest squre unit. 25. = 4 feet, = 4 feet, c = 2 feet 26. = 5 feet, = 5 feet, c = 4 feet 27. = 14 meters, = 12 meters, c = 4 me t e r s 28. = 16 meters, = 10 meters, me t e r s 29. = 11 yrds, = 9 yrds, c = 7 yrds 30. = 13 yrds, = 9 yrds, c = 5 yrds Prctice Plus In Exercises 31 32, solve ech tringle. Round lengths of sides to the nerest tenth nd ngle mesures to the nerest degree. 39. Two ships leve hror t the sme time. One ship trvels on ering of 12 W t 14 miles per hour. The other ship trvels on ering of 75 E t 10 miles per hour. How fr prt will the ships e fter three hours? Round to the nerest tenth of mile. 40. plne leves irport nd trvels 580 miles to irport on ering of 34 E. The plne lter leves irport nd trvels to irport 400 miles wy on ering of 74 E. Find the distnce from irport to irport to the nerest tenth of mile. 41. Find the distnce cross the lke from to, to the nerest yrd, using the mesurements shown in the figure. 31. 15 = 8 35 = 13 c 32. 35 = 2 c 50 = 3 In Exercises 33 34, the three circles re rrnged so tht they touch ech other, s shown in the figure. Use the given rdii for the circles with centers,, nd, respectively, to solve tringle. Round ngle mesures to the nerest degree. 140 yd 80 160 yd 42. To find the distnce cross protected cove t lke, surveyor mkes the mesurements shown in the figure. Use these mesurements to find the distnce from to to the nerest yrd. 65 yrds 80 105 yrds 33. 5.0, 4.0, 3.5 34. 7.5, 4.3, 3.0 In Exercises 35 36, the three given points re the vertices of tringle. olve ech tringle, rounding lengths of sides to the nerest tenth nd ngle mesures to the nerest degree. The digrm shows three islnds in Florid y. You rent ot nd pln to visit ech of these remote islnds. Use the digrm to solve Exercises 43 44. 35. (0, 0), (-3, 4), (3, -1 ) 36. (0, 0), (4, -3), (1, -5 ) ppliction Exercises 37. Use Figure 6.13 on pge 694 to find the pce ngle, to the nerest degree, for the crnivore. Does the ngle indicte tht this dinosur ws n efficient wlker? Descrie your nswer. 38. Use Figure 6.13 on pge 694 to find the pce ngle, to the nerest degree, for the herivore. Does the ngle indicte tht this dinosur ws n efficient wlker? Descrie your nswer. Islnd Islnd 5 miles 6 miles 7 miles W E Islnd
ection 6.2 The Lw of osines 701 43. If you re on islnd, on wht ering should you nvigte to go to islnd? 44. If you re on islnd, on wht ering should you nvigte to go to islnd? 45. You re on fishing ot tht leves its pier nd heds est. fter trveling for 25 miles, there is report wrning of rough ses directly south. The cptin turns the ot nd follows ering of 40 W for 13.5 miles.. t this time, how fr re you from the ot s pier? Round to the nerest tenth of mile.. Wht ering could the ot hve originlly tken to rrive t this spot? 48. The figure shows 200-foot tower on the side of hill tht forms 5 ngle with the horizontl. Find the length of ech of the two guy wires tht re nchored 150 feet uphill nd downhill from the tower s se nd extend to the top of the tower. Round to the nerest tenth of foot. 200 ft W E 25 miles 40 13.5 miles 46. You re on fishing ot tht leves its pier nd heds est. fter trveling for 30 miles, there is report wrning of rough ses directly south. The cptin turns the ot nd follows ering of 45 W for 12 miles.. t this time, how fr re you from the ot s pier? Round to the nerest tenth of mile.. Wht ering could the ot hve originlly tken to rrive t this spot? 47. The figure shows 400-foot tower on the side of hill tht forms 7 ngle with the horizontl. Find the length of ech of the two guy wires tht re nchored 80 feet uphill nd downhill from the tower s se nd extend to the top of the tower. Round to the nerest tenth of foot. 400 ft 80 ft 80 ft 7 150 ft 150 ft 49. Mjor Legue sell dimond hs four ses forming squre whose sides mesure 90 feet ech. The pitcher s mound is 60.5 feet from home plte on line joining home plte nd second se. Find the distnce from the pitcher s mound to first se. Round to the nerest tenth of foot. 50. Little Legue sell dimond hs four ses forming squre whose sides mesure 60 feet ech. The pitcher s mound is 46 feet from home plte on line joining home plte nd second se. Find the distnce from the pitcher s mound to third se. Round to the nerest tenth of foot. 51. piece of commercil rel estte is priced t $3.50 per squre foot. Find the cost, to the nerest dollr, of tringulr lot mesuring 240 feet y 300 feet y 420 feet. 52. piece of commercil rel estte is priced t $4.50 per squre foot. Find the cost, to the nerest dollr, of tringulr lot mesuring 320 feet y 510 feet y 410 feet. Writing in Mthemtics 53. Without using symols, stte the Lw of osines in your own words. 54. Why cn t the Lw of ines e used in the first step to solve n tringle? 55. Descrie strtegy for solving n tringle. 56. Descrie strtegy for solving n tringle. 57. Under wht conditions would you use Heron s formul to find the re of tringle? 58. Descrie n pplied prolem tht cn e solved using the Lw of osines ut not the Lw of ines. 59. The pitcher on Little Legue tem is studying ngles in geometry nd hs question. och, suppose I m on the pitcher s mound fcing home plte. I ctch fly ll hit in my direction. If I turn to fce first se nd throw the ll, through how mny degrees should I turn for direct throw? Use the informtion given in Exercise 50 nd write n nswer to the pitcher s question. Without getting too technicl, descrie to the pitcher how you otined this ngle. 60. Explin why the Pythgoren Theorem is specil cse of the Lw of osines. 5
702 hpter 6 dditionl Topics in Trigonometry riticl Thinking Exercises Mke ense? In Exercises 61 64, determine whether ech sttement mkes sense or does not mke sense, nd explin your resoning. 61. The Lw of osines is similr to the Lw of ines, with ll the sines replced with cosines. 62. If I know the mesures of ll three ngles of n olique tringle, neither the Lw of ines nor the Lw of osines cn e used to find the length of side. 63. I noticed tht for right tringle, the Lw of osines reduces to the Pythgoren Theorem. 64. olving n tringle, I do not hve to e concerned out the miguous cse when using the Lw of ines. 65. The lengths of the digonls of prllelogrm re 20 inches nd 30 inches. The digonls intersect t n ngle of 35. Find the lengths of the prllelogrm s sides. (Hint: Digonls of prllelogrm isect one nother.) 66. Use the figure to solve tringle. Round lengths of sides to the nerest tenth nd ngle mesures to the nerest degree. 17 21.5 26 13.5 67. The minute hnd nd the hour hnd of clock hve lengths m inches nd h inches, respectively. Determine the distnce etween the tips of the hnds t 10:00 in terms of m nd h. Group Exercise 68. The group should design five originl prolems tht cn e solved using the Lws of ines nd osines. t lest two prolems should e solved using the Lw of ines, one should e the miguous cse, nd t lest two prolems should e solved using the Lw of osines. t lest one prolem should e n ppliction prolem using the Lw of ines nd t lest one prolem should involve n ppliction using the Lw of osines. The group should turn in oth the prolems nd their solutions. Preview Exercises Exercises 69 71 will help you prepre for the mteril covered in the next section. 69. Grph: y = 3. 70. Grph: x 2 + (y - 1) 2 = 1. 71. omplete the squre nd write the eqution in stndrd form: x 2 + 6x + y 2 = 0. Then give the center nd rdius of the circle, nd grph the eqution. 37.5 ETIO 6.3 Polr oordintes Ojectives Plot points in the polr coordinte system. Find multiple sets of polr coordintes for given point. onvert point from polr to rectngulr coordintes. onvert point from rectngulr to polr coordintes. onvert n eqution from rectngulr to polr coordintes. onvert n eqution from polr to rectngulr coordintes. p i f l ' h o q w 2 u 4 d k z j p utterflies re mong the most celerted of ll insects. It s hrd not to notice their eutiful colors nd grceful flight. Their symmetry cn e explored with trigonometric functions nd system for plotting points clled the polr coordinte system. In mny cses, polr coordintes re simpler nd esier to use thn rectngulr coordintes. Plotting Points in the Polr oordinte ystem The foundtion of the polr coordinte system is horizontl ry tht extends to the right. The ry is clled the polr xis nd is shown in Figure 6.19 t the top of the next pge. The endpoint of the ry is clled the pole. 0