44 HPTER 5 nlyti Trigonometry 5.6 The Lw of osines Wht you ll lern out Deriving the Lw of osines Solving Tringles (SS, SSS) Tringle re nd Heron s Formul pplitions... nd why The Lw of osines is n importnt extension of the Pythgoren Theorem, with mny pplitions. Deriving the Lw of osines Hving seen the Lw of Sines, you will proly not e surprised to lern tht there is Lw of osines. There re mny suh prllels in mthemtis. Wht you might find surprising is tht the Lw of osines hs solutely no resemlne to the Lw of Sines. Insted, it resemles the Pythgoren Theorem. In ft, the Lw of osines is often lled the generlized Pythgoren Theorem euse it ontins tht lssi theorem s speil se. Lw of osines Let e ny tringle with sides nd ngles leled in the usul wy (Figure 5.). Then = + - os = + - os = + - os We derive only the first of the three equtions, sine the other two re derived in extly the sme wy. Set the tringle in oordinte plne so tht the ngle tht ppers in the formul (in this se, ) is t the origin in stndrd position, with side long the positive x-xis. Depending on whether ngle is right (Figure 5.3), ute (Figure 5.3), or otuse (Figure 5.3), the point will e on the y-xis, in QI, or in QII. y y y (x, y) (x, y) (x, y) (, 0) x (, 0) x (, 0) x FIGURE 5. tringle with the usul leling (ngles,, ; opposite sides,, ). () () FIGURE 5.3 Three ses for proving the Lw of osines. () In eh of these three ses, is point on the terminl side of ngle in stndrd position, t distne from the origin. Denote the oordintes of y (x, y). y our definitions for trigonometri funtions of ny ngle (Setion 4.3), we n onlude tht nd therefore x = os nd y = sin, x = os nd y = sin.
SETION 5.6 The Lw of osines 443 Now set equl to the distne from to using the distne formul: = 1x - + 1y - 0 = 1x - + y = 1 os - + 1 sin Distne formul Squre oth sides. Sustitution = os - os + + sin = 1os + sin + - os = + - os Pythgoren identity Solving Tringles (SS, SSS) While the Lw of Sines is the tool we use to solve tringles in the S nd S ses, the Lw of osines is the required tool for SS nd SSS. (oth methods n e used in the SS se, ut rememer tht there might e 0, 1, or tringles.) 5 0 11 FIGURE 5.4 tringle with two sides nd n inluded ngle known. (Exmple 1) EXMPLE 1 Solving Tringle (SS) Solve given tht = 11, = 5, nd = 0. (See Figure 5.4.) SOLUTION = + - os = 11 + 5-11115 os 0 = 4.6338 Á = 14.6338 Á L 6.5 We ould now use either the Lw of osines or the Lw of Sines to find one of the two unknown ngles. s generl rule, it is etter to use the Lw of osines to find ngles, sine the rosine funtion will distinguish otuse ngles from ute ngles. = + - os os = 5 + 16.59 Á - 11 1516.59 Á L 144.8 = 180-144.8-0 = 15. So the six prts of the tringle re: 11 = 5 + 16.59 Á - 1516.59 Á os = os -1 5 + 16.59 Á - 11 1516.59 Á = 144.8 = 15. = 11 = 5 7 5 = 0 L 6.5 Now try Exerise 1. 9 FIGURE 5.5 tringle with three sides known. (Exmple ) EXMPLE Solving Tringle (SSS) Solve if = 9, = 7, nd = 5. (See Figure 5.5.)
444 HPTER 5 nlyti Trigonometry SOLUTION We use the Lw of osines to find two of the ngles. The third ngle n e found y sutrtion from 180. = + - os = + - os 9 = 7 + 5-1715os 7 = 9 + 5-1915os 70 os = -7 90 os = 57 = os -1 1-0.1 = os -1 157/90 L 95.7 L 50.7 Then = 180-95.7-50.7 = 33.6. Now try Exerise 3. Tringle re nd Heron s Formul The sme prts tht determine tringle lso determine its re. If the prts hppen to e two sides nd n inluded ngle (SS), we get simple re formul in terms of those three prts tht does not require finding n ltitude. Oserve in Figure 5.3 (used in explining the Lw of osines) tht eh tringle hs se nd ltitude y = sin. pplying the stndrd re formul, we hve re = 1 1se1height = 1 11 sin = 1 sin. This is tully three formuls in one, s it does not mtter whih side we use s the se. re of Tringle re = 1 sin = 1 sin = 1 sin 9 θ 9 FIGURE 5.6 regulr otgon insried inside irle of rdius 9 inhes. (Exmple 3) Heron s Formul The formul is nmed fter Heron of lexndri, whose proof of the formul is the oldest on reord, ut nient ri sholrs limed to hve known it from the works of rhimedes of Syruse enturies erlier. rhimedes (. 87 1..E.) is onsidered to e the gretest mthemtiin of ll ntiquity. EXMPLE 3 Finding the re of Regulr Polygon Find the re of regulr otgon (8 equl sides, 8 equl ngles) insried inside irle of rdius 9 inhes. SOLUTION Figure 5.6 shows tht we n split the otgon into 8 ongruent tringles. Eh tringle hs two 9-inh sides with n inluded ngle of u = 360 /8 = 45. The re of eh tringle is re = 11/1919 sin 45 = 181/) sin 45 = 811/4. Therefore, the re of the otgon is re = 8 re = 161 L 9 squre inhes. Now try Exerise 31. There is lso n re formul tht n e used when the three sides of the tringle re known. lthough Heron proved this theorem using only lssil geometri methods, we prove it, s most people do tody, y using the tools of trigonometry.
SETION 5.6 The Lw of osines 445 THEOREM Heron s Formul Let,, nd e the sides of, nd let s denote the semiperimeter. Then the re of is given y re = s1s - 1s - 1s -. Proof re = 1 sin 41re = sin 161re) = 4 sin = 4 11 - os = 4-4 os = 4-1 os = 4-1 + - = 1-1 + - 1 + 1 + - Pythgoren identity Lw of osines Differene of squres = 1-1 - + 11 + + - = 1 - ( - 11 + - = 1-1 - 1 + ( - 11 + - 11 + + Differene of squres = 1 - + 1 + - 1 + - 1 + + = 1s - 1s - 1s - 1s 161re = 16s1s - 1s - 1s - 1re = s1s - 1s - 1s - re = s1s - 1s - 1s - 1 + + )/. s = + + EXMPLE 4 Using Heron s Formul Find the re of tringle with sides 13, 15, 18. SOLUTION First we ompute the semiperimeter: s = 113 + 15 + 18/ = 3. Then we use Heron s Formul re = 13 13-1313 - 1513-18 = 13 # 10 # 8 # 5 = 1900-013. The pproximte re is 96 squre units. Now try Exerise 1. pplitions We end this setion with few pplitions. EXMPLE 5 Mesuring sell Dimond The ses on sell dimond re 90 feet prt, nd the front edge of the pither s ruer is 60.5 feet from the k orner of home plte. Find the distne from the enter of the front edge of the pither s ruer to the fr orner of first se. (ontinued)
446 HPTER 5 nlyti Trigonometry Third se Seond se 60.5 ft 45 (Home plte) 90 ft (Fir FIGURE 5.7 The dimond-shped prt of sell dimond. (Exmple 5) Pltoni Solids The regulr tetrhedron in Exmple 6 is one of only 5 regulr solids (solids with fes tht re ongruent polygons hving equl ngles nd equl sides). The others re the ue (6 squre fes), the othedron (8 tringulr fes), the dodehedron (1 pentgonl fes), nd the ioshedron (0 tringulr fes). Referred to s the Pltoni solids, Plto did not disover them, ut they re fetured in his osmology s eing the stuff of whih everything in the universe is mde. The Pltoni universe itself is dodehedron, fvorite symol of the Pythgorens. SOLUTION Figure 5.7 shows first se s, the pither s ruer s, nd home plte s. The distne we seek is side in. y the Lw of osines, = 60.5 + 90-160.5190 os 45 = 60.5 + 90-160.5190 os 45 L 63.7 The distne from first se to the pither s ruer is out 63.7 feet. Now try Exerise 37. EXMPLE 6 Mesuring Dihedrl ngle (Solid Geometry) regulr tetrhedron is solid with four fes, eh of whih is n equilterl tringle. Find the mesure of the dihedrl ngle formed long the ommon edge of two interseting fes of regulr tetrhedron with edges of length. SOLUTION Figure 5.8 shows the tetrhedron. Point is the midpoint of edge DE, nd nd re the endpoints of the opposite edge. The mesure of is the sme s the mesure of the dihedrl ngle formed long edge DE, so we will find the mesure of. euse oth D nd D re 30-60 - 90 tringles, nd oth hve length 13. If we pply the Lw of osines to, we otin os 1 = 1 3 = 113 + 113-13 13 os 1 = os -1 1 3 L 70.53 3 D 1 3 1 E The dihedrl ngle hs the sme mesure s, pproximtely 70.53. (We hose sides of length for omputtionl onveniene, ut in ft this is the mesure of dihedrl ngle in regulr tetrhedron of ny size.) Now try Exerise 43. FIGURE 5.8 The mesure of is the sme s the mesure of ny dihedrl ngle formed y two of the tetrhedron s fes. (Exmple 6) EXPLORTION 1 Estimting rege of Plot of Lnd Jim nd rr re house hunting nd need to estimte the size of n irregulr djent lot tht is desried y the owner s little more thn n re. With rr sttioned t orner of the plot, Jim strts t nother orner nd wlks stright line towrd her, ounting his pes. They then shift orners nd Jim pes gin, until they hve reorded the dimensions of the lot (in pes) s in Figure 5.9. They lter mesure Jim s pe s. feet. Wht is the pproximte rege of the lot? 1. Use Heron s Formul to find the re in squre pes.. onvert the re to squre feet, using the mesure of Jim s pe. 3. There re 580 feet in mile. onvert the re to squre miles. 4. There re 640 squre res in squre mile. onvert the re to res. 5. Is there good reson to dout the owner s estimte of the rege of the lot? 6. Would Jim nd rr e le to modify their system to estimte the re of n irregulr lot with five stright sides?
SETION 5.6 The Lw of osines 447 10 81 11 86 115 FIGURE 5.9 Dimensions (in pes) of n irregulr plot of lnd. (Explortion 1) hpter Opener Prolem (from pge 403) Prolem: euse deer require food, wter, over for protetion from wether nd predtors, nd living spe for helthy survivl, there re nturl limits to the numer of deer tht given plot of lnd n support. Deer popultions in ntionl prks verge 14 nimls per squre kilometer. If tringulr region with sides of 3 kilometers, 4 kilometers, nd 6 kilometers hs popultion of 50 deer, how lose is the popultion on this lnd to the verge ntionl prk popultion? Solution: We n find the re of the lnd region 6 km 3 km 4 km y using Heron s Formul with s = 13 + 4 + 6/ = 13/ nd re = 1s1s - 1s - 1s - = 13 13-313 - 413-6 = 13 7 5 1 L 5.3, so the re of the lnd region is 5.3 km. If this lnd were to support 14 deer/km, it would hve 15.3 Á km 114 deer/km = 74.7 L 75 deer. Thus, the lnd supports 5 deer less thn the verge. QUIK REVIEW 5.6 (For help, go to Setions.4 nd 4.7.) Exerise numers with gry kground indite prolems tht the uthors hve designed to e solved without lultor. In Exerises 1 4, find n ngle etween 0 nd 180 tht is solution to the eqution. 1. os = 3/5. os = -0.3 3. os = -0.68 4. 3 os = 1.9 In Exerises 5 nd 6, solve the eqution (in terms of x nd y) for () os nd (), 0 180. 5. 9 = x + y - xy os 6. y = x + 5-10 os In Exerises 7 10, find qudrti polynomil with rel oeffiients tht stisfies the given ondition. 7. Hs two positive zeros 8. Hs one positive nd one negtive zero 9. Hs no rel zeros 10. Hs extly one positive zero
448 HPTER 5 nlyti Trigonometry SETION 5.6 EXERISES In Exerises 1 4, solve the tringle. 1.. 8 131 13 3. 4. 4 19 7 In Exerises 5 16, solve the tringle. 5. = 55, = 1, = 7 6. = 35, = 43, = 19 7. = 1, = 1, = 95 8. =, = 31, = 8 9. = 1, = 5, = 4 10. = 1, = 5, = 8 11. = 3., = 7.6, = 6.4 1. = 9.8, = 1, = 3 13. = 4, = 7, = 10 14. = 57, = 11, = 10 15. = 63, = 8.6, = 11.1 16. = 71, = 9.3, = 8.5 In Exerises 17 0, find the re of the tringle. 17. = 47, = 3 ft, = 19 ft 18. = 5, = 14 m, = 1 m 19. = 101, = 10 m, = m 0. = 11, = 1.8 in., = 5.1 in. In Exerises 1 8, deide whether tringle n e formed with the given side lengths. If so, use Heron s Formul to find the re of the tringle. 1. = 4, = 5, = 8. = 5, = 9, = 7 3. = 3, = 5, = 8 4. = 3, = 19, = 1 5. = 19.3, =.5, = 31 6. = 8., = 1.5, = 8 7. = 33.4, = 8.5, =.3 8. = 18., = 17.1, = 1.3 9. Find the rdin mesure of the lrgest ngle in the tringle with sides of 4, 5, nd 6. 30. prllelogrm hs sides of 18 nd 6 ft, nd n ngle of 39. Find the shorter digonl. 1 4 8 14 35 17 31. Find the re of regulr hexgon insried in irle of rdius 1 inhes. 3. Find the re of regulr nongon (9 sides) insried in irle of rdius 10 inhes. 33. Find the re of regulr hexgon irumsried out irle of rdius 1 inhes. [Hint: Strt y finding the distne from vertex of the hexgon to the enter of the irle.] 34. Find the re of regulr nongon (9 sides) irumsried out irle of rdius 10 inhes. 35. Mesuring Distne Indiretly Jun wnts to find the distne etween two points nd on opposite sides of uilding. He lotes point tht is 110 ft from nd 160 ft from, s illustrted in the figure. If the ngle t is 54, find distne. 110 ft 160 ft 36. Designing sell Field 54 () Find the distne from the enter of the front edge of the pither s ruer to the fr orner of seond se. How does this distne ompre with the distne from the pither s ruer to first se? (See Exmple 5.) () Find in. Third se Seond se 60.5 ft 45 (Home plte) (First se) 37. Designing Softll Field In softll, djent ses re 60 ft prt. The distne from the enter of the front edge of the pither s ruer to the fr orner of home plte is 40 ft. () Find the distne from the enter of the pither s ruer to the fr orner of first se. () Find the distne from the enter of the pither s ruer to the fr orner of seond se. () Find in. 90 ft
SETION 5.6 The Lw of osines 449 38. Surveyor s lultions Tony must find the distne from to on opposite sides of lke. He lotes point tht is 860 ft from nd 175 ft from. He mesures the ngle t to e 78. Find distne. Third se Seond se 40 ft 45 (Home plte) (First se) 39. onstrution Engineering mnufturer is designing the roof truss tht is modeled in the figure shown. () Find the mesure of E. () If F = 1 ft, find the length DF. () Find the length EF. 6 ft 860 ft 9 ft D E 36 ft 40. Nvigtion Two irplnes flying together in formtion tke off in different diretions. One flies due est t 350 mph, nd the other flies est-northest t 380 mph. How fr prt re the two irplnes hr fter they seprte, ssuming tht they fly t the sme ltitude? 41. Footll Kik The plyer witing to reeive kikoff stnds t the 5 yrd line (point ) s the ll is eing kiked 65 yd up the field from the opponent s 30 yrd line. The kiked ll trvels 73 yd t n ngle of 8 to the right of the reeiver, s shown in the figure (point ). Find the distne the reeiver runs to th the ll. Gol line F 60 ft 78 Gol line 175 ft 4. Group tivity rhiteturl Design uilding Inspetor Julie Wng heks uilding in the shpe F of regulr otgon, eh side 0 ft long. She heks tht the ontrtor G hs loted the orners of the foundtion orretly y mesuring severl of the digonls. lulte wht the lengths of digonls H, H, nd HD should e. 43. onneting Trigonometry nd Geometry is 1 ft insried in retngulr ox whose sides re 1,, nd 3 ft long s shown. Find the mesure ft of. 44. Group tivity onneting Trigonometry nd Geometry ue hs edges of length ft. ft Point is the midpoint of n edge. Find the mesure of. Stndrdized Test Questions 45. True or Flse If is ny tringle with sides nd ngles leled in the usul wy, then + 7 os. Justify your nswer. 46. True or Flse If,, nd u re two sides nd n inluded ngle of prllelogrm, the re of the prllelogrm is sin u. Justify your nswer. You my use grphing lultor when nswering these questions. 47. Multiple hoie Wht is the re of regulr dodegon (1-sided figure) insried in irle of rdius 1? () 47 () 43 () 437 (D) 44 (E) 447 48. Multiple hoie The re of tringle with sides 7, 8, nd 9 is () 6115. () 115. () 1613. (D) 1713. (E) 1813. 49. Multiple hoie Two ots strt t the sme point nd speed wy long ourses tht form 110 ngle. If one ot trvels t 4 miles per hour nd the other ot trvels t 3 miles per hour, how fr prt re the ots fter 30 minutes? () 1 miles () miles () 3 miles (D) 4 miles (E) 5 miles 50. Multiple hoie Wht is the mesure of the smllest ngle in tringle with sides 1, 17, nd 5? () 1 () () 3 (D) 4 (E) 5 E 3 ft D H 0 ft 0 ft 0 ft 0 ft K 65 yd 8 73 yd 160 ft Explortions 51. Find the re of regulr polygon with n sides insried inside irle of rdius r. (Express your nswer in terms of n nd r.) 0 10 0 30 40 50 40 30 0 10 0
450 HPTER 5 nlyti Trigonometry os 5. () Prove the identity: = + -. () Prove the (tougher) identity: os + os + os = + +. [Hint: Use the identity in prt (), long with its other vritions.] 53. Nvigtion Two ships leve ommon port t 8:00.M. nd trvel t onstnt rte of speed. Eh ship keeps log showing its distne from port nd its distne from the other ship. Portions of the logs from lter tht morning for oth ships re shown in the following tles. Nut mi Nut mi Nut mi Nut mi from from from from Time port ship Time port ship 9:00 15.1 8.7 9:00 1.4 8.7 10:00 30. 17.3 11:00 37. 6.0 () How fst is eh ship trveling? (Express your nswer in knots, whih re nutil miles per hour.) () Wht is the ngle of intersetion of the ourses of the two ships? () How fr prt re the ships t 1:00 noon if they mintin the sme ourses nd speeds? Extending the Ides 54. Prove tht the re of tringle n e found with the formul re = sin sin. sin 55. segment of irle is the region enlosed etween hord of irle nd the r interepted y the hord. Find the re of segment interepted y 7-inh hord in irle of rdius 5 inhes. 5 5 7 HPTER 5 Key Ides Properties, Theorems, nd Formuls Reiprol Identities 404 Quotient Identities 404 Pythgoren Identities 405 ofuntion Identities 406 Odd-Even Identities 407 Sum/Differene Identities 4 44 Doule-ngle Identities 48 Power-Reduing Identities 48 Hlf-ngle Identities 430 Lw of Sines 434 Lw of osines 44 Tringle re 444 Heron s Formul 445 Proedures Strtegies for Proving n Identity 413 415 HPTER 5 Review Exerises Exerise numers with gry kground indite prolems tht the uthors hve designed to e solved without lultor. The olletion of exerises mrked in red ould e used s hpter test. In Exerises 1 nd, write the expression s the sine, osine, or tngent of n ngle. 1. sin 100 os 100 tn 40. 1 - tn 40 In Exerises 3 nd 4, simplify the expression to single term. Support your nswer grphilly. 3. 11 - sin u + 4 sin u os u 4. 1-4 sin x os x In Exerises 5, prove the identity. 5. os 3x = 4 os 3 x - 3 os x 6. os x - os x = sin x - sin x 7. tn x - sin x = sin x tn x