3 3 3 3 3D 3E 3F 3G 3H Review of right-ngled tringles erings Using the sine rule to find side lengths Using the sine rule to find ngles re of tringle Using the osine rule to find side lengths Using the osine rule to find ngles Rdil surveys pplitions of trigonometry Syllus referene Mesurement 6 pplitions of trigonometry This hpter will over finding side lengths nd ngles in right-ngled tringles nd non-right-ngled tringles, finding res of tringles, nd pplying these methods to prolems involving erings nd rdil surveys.
Re you Redy? eookplus Digitl do SkillSHEET 3.1 do-1320 Right-ngled trigonometry finding side length eookplus Digitl do SkillSHEET 3.2 do-1321 Using the inverse trigonometri rtios eookplus Digitl do SkillSHEET 3.4 do-1323 Right-ngled trigonometry finding n ngle Try the questions elow. If you hve diffiulty with ny of them, extr help n e otined y ompleting the mthing SkillSHEET. Either lik on the SkillSHEET ion next to the question on the Mths Quest HS ourse eookplus ion. Right-ngled trigonometry finding side length 1 In eh of the following find the length of the side mrked with the pronumerls orret to 2 deiml ples. 63 16 m Using the inverse trigonometri rtios d 40 25.2 km 2 Find ngle q, where q is ute, orret to the nerest degree. sin q = 0.7 tn q = 1.5 os q = 0.8 Right-ngled trigonometry finding n ngle 71 f 19.5 m 3 In eh of the following find the size of the ngle mrked with the pronumerls orret to the nerest degree. 8 m 20 m 46.1 mm eookplus 16 m 25 m 31.2 mm Digitl do SkillSHEET 3.5 do-1324 onverting nutil miles to kilometres onverting nutil miles to kilometres 4 Use 1 nutil mile = 1.852 km to onvert: 4 nutil miles to kilometres 1.2 nutil miles to metres 50 kilometres to nutil miles d 3560 metres to nutil miles. eookplus Digitl do SkillSHEET 3.6 do-1325 ngle sum of tringle eookplus Digitl do SkillSHEET 3.7 do-1326 Solving frtionl equtions ngle sum of tringle 5 Find the ngle mrked with the pronumerl in eh of the following. 58 Solving frtionl equtions 63 71 132 41 6 Solve eh of the following equtions, where pproprite give your nswer orret to 2 deiml ples. x 5 = 3 x 3 = 4 8 x 95. = d 9 2 36. 24. x = 5 82 mths Quest Generl mths hs ourse
3 Review of right-ngled tringles Previously we hve studied right-ngled tringles nd disovered tht we n lulte side length of tringle when given the length of one other side nd one of the ute ngles. To do this we need to use one of the formuls for the three trigonometri rtios. sin q = opposite side hypotenuse djent side os q = hypotenuse opposite side tn q = djent side Worked Exmple 1 Find the length of the side mrked x in the figure on the right (orret to 1 deiml ple). 42 x Think Write 29.2 m Method 1: Tehnology-free 1 Lel the sides of the digrm. 42 dj hyp x 29.2 m opp 2 hoose the sine rtio nd write the formul. sin q = oppositeside hypotenuse 3 Sustitute for the opposite side nd hypotenuse. 29. 2 sin 42 = x 4 Mke x the sujet of the formul. x sin 42 = 29.2 x = 29. 2 sin 42 5 lulte the vlue of x. x = 43.6 m Method 2: Tehnology-enled 1 From the MEU selet EQU. hpter 3 pplitions of trigonometry 83
2 Press 3 (SOLV). 3 Delete ny eqution, enter the eqution sin 42 = 29.2 X nd press w. ote: Your lultor my disply different vlue of X t this stge. This is just the lst vlue of X stored in the lultor s memory. 4 Press 6 (SOLV) to solve the eqution. The sme formuls n e used to lulte the size of n ngle if we re given two side lengths in the tringle. WoRked exmple 2 lulte the size of the ngle mrked q in the figure on the right (orret to the nerest degree). 35 mm 47 mm 84 mths Quest Generl mths hs ourse
Think Method 1: Tehnology-free WRiTe 1 Lel the sides of the tringle. Opposite = 47 mm djent = 35 mm 2 hoose the tngent rtio nd write the formul. tn q = 3 Sustitute for the opposite side nd the djent side. tn q = 47 35 opposite side djentside 4 Mke q the sujet of the formul. q = tn- 1 47 35 5 lulte q. q = 53 Method 2: Tehnology-enled 1 From the MEU selet EQU. 2 Press 3 (SOLV). 3 Delete ny existing eqution, then enter the eqution tn X = 47$35 nd press w. ote: Your lultor my disply different vlue of X t this stge. This is just the lst vlue of X stored in the lultor s memory. 4 Press 6 (SOLV) to solve the eqution. Using these results, we re le to solve prolems tht involve more thn one right-ngled tringle. WoRked exmple 3 Greg stnds 70 m from the se of uilding nd mesures the ngle of elevtion to the top of the uilding s eing 35. Julie is stnding 40 m from the se of the uilding on the other side of the uilding s shown in the figure on the right. lulte the height of the uilding, orret to 2 deiml ples. lulte the ngle of elevtion of the top of the uilding tht Julie would mesure, orret to the nerest degree. h 40 m 70 m 35 eookplus Tutoril int-2415 Worked exmple 3 hpter 3 pplitions of trigonometry 85
Think 1 Drw the tringle showing the ngle of elevtion from where Greg is stnding nd lel the sides. Write h 2 hoose the tngent rtio nd write the formul. tn q = 3 Sustitute for q nd the djent side. tn35 = h 70 70 m 35 opposite side djentside 4 Mke h the sujet of the formul. h = 70 tn 35 5 lulte the vlue of h. h = 49.01 m 1 Drw the tringle from where Julie is stnding nd lel the sides. 49.01 m 40 m opposite side 2 hoose the tngent rtio nd write the formul. tn q = djentside 3 49. 01 Sustitute for the opposite side nd the tn q = djent side. 40 4 Mke q the sujet of the formul. q = tn - 1 49. 01 40 5 lulte q, orret to the nerest degree. q = 51 REMEMER 1. 2. 3. 4. 5. The formuls for the three trigonometri rtios re: opposite side sin q = hypotenuse djent side os q = hypotenuse opposite side tn q = djent side To lulte the length of side we need to e given one side length nd one ute ngle. To lulte the size of n ngle we need to e given two side lengths. Mny prolems involve solving two or more right-ngled tringles. fter sustitution, the vlue of the unknown n e found using the eqution solver on grphis lultor. 86 Mths Quest Generl Mths HS ourse
exerise 3 eookplus Digitl do SkillSHEET 3.1 do-1320 Right-ngled trigonometry finding side length Review of right-ngled tringles 1 We 1 lulte the length of the side mrked with the pronumerls in eh of the following, orret to 1 deiml ple. 13.2 m 23 38 142 mm d 314 mm 61 d 11.4 m 50 e 9.1 m e 17 f f 5 19.2 m eookplus Digitl do SkillSHEET 3.2 do-1321 Using the inverse trigonometri rtios eookplus Digitl do SkillSHEET 3.3 do-1322 Rounding ngles to the nerest degree eookplus Digitl do SkillSHEET 3.4 do-1323 Right-ngled trigonometry finding n ngle 2 We2 lulte the size of eh of the ngles mrked with the pronumerls, orret to the nerest degree. 9.5 m 113 m 36 mm 71 mm 11.4 m 3 The ngle of depression from the top of liff to ot siling 100 m 100 m offshore is 32. lulte the height of the liff, orret 32 to the nerest metre. 4 ndrew wlks 5 km from point P to point Q. t the sme time in wlks from P to R suh tht PQ is perpendiulr to PR. Given tht PQR = 28 : drw digrm of DPQR lulte the distne wlked y in, orret to the nerest metre lulte the distne tht ndrew would need to wlk in stright line to meet in, orret to the nerest metre. 5 lighthouse is 40 m tll nd the eon top the lighthouse is sighted y ship 150 m from shore, s shown in the figure 40 m on the right. lulte the ngle of elevtion t whih the lighthouse is sighted from the ship, orret to the nerest degree. 150 m 61 m h hpter 3 pplitions of trigonometry 87
6 From point 65 m ove the ground, seond point is sighted on the ground t distne of 239 m. Drw digrm of this sitution. lulte the ngle of depression from the first point to the seond point. 7 WE3 Slly nd Tim re oth sighting the top of uilding, s shown in the figure on the right. Slly is 40 m from the h se of the uilding nd sights the ngle of elevtion to the top of the uilding s 35. Tim is 60 m from the se of the uilding. 35 lulte the height of the uilding, or ret to 2 deiml 40 m 60 m ples. lulte the ngle of elevtion t whih Tim will sight S the uilding. 8 George nd Diego re oth flying kite from the sme point. George s kite is flying on 50 m of string nd the string mkes 70 ngle with the ground. Diego s kite is flying on 60 m piee of string nd is t the sme height s George s kite, s shown in the figure on the right. lulte the ngle tht the string from Diego s kite mkes with the ground. Give your nswer orret to the nerest degree. Further development 9 The shdow st y sttue 2 metres tll is 0.6 metres. The ngle of the sun to the ground is losest to: 17 18 72 D 73 10 In the digrm find x (to 1 deiml ple), nd q (to the nerest degree). 11 The sun is overhed, sting shdow of length 90 m from 1.75 m srerow, whih is no longer stnding upright. Determine the ngle (to the nerest degree) tht the srerow mkes with the ground. 12 kite is hovering in strong winds, 10 m vertilly ove the ground. It is eing held in ple y tut 12-m length of rope from the kite to the ground. Find the ngle (to the nerest degree) tht the rope mkes with the ground. 13 rmp joins two points, nd. The horizontl distne etween nd is 1.4 m, nd is 30 m vertilly ove the level of. Find the length of the rmp (in metres to 2 deiml ples). Find (to the nerest degree) the ngle tht the rmp mkes with the horizontl. 14 hirlift follows diret line from mountin pek (ltitude 1400 m) to ridge (ltitude 960 m). If the horizontl distne etween the pek nd the ridge is 510 m, find the ngle of desent (to the nerest degree) from one to the other. 70 G 50 m 60 m x 60 8 m 6.5 m D 3 erings ering is n ngle used to desrie diretion. erings re used in nvigtion nd re ommon pplition of trigonometry to prtil situtions. We n therefore pply our trigonometril formuls to mke lultions sed upon these erings. There re two types of ering tht we need to e le to work with: ompss erings nd true erings. 88 Mths Quest Generl Mths HS ourse
eookplus Intertivity int-0190 vigtion nd speifitions of lotions ompss erings ompss erings use the four points of the ompss. With ompss erings there re four min diretions: north, south, est nd west. In etween eh of these min diretions there re four others: north-est, south-est, south-west nd north-west. Eh of these diretions is t 45 to two of the four min diretions. Trigonometry n then e used to solve prolems out distnes nd ngles using these eight si diretions. W W SW S E SE E WoRked exmple 4 ship () is 10 nutil miles due est of lighthouse. seond ship () ers SE of the lighthouse nd is due south of the first ship. lulte the distne of the seond ship from the lighthouse, orret to 1 deiml ple. Think WRiTe 1 Drw digrm lelling the sides of the tringle. dj L 10 M 45 2 hoose the osine rtio nd write the formul. os q = dj hyp 10 3 Sustitute for q nd the djent side. os 45 = x hyp x opp 4 Mke x the sujet of the eqution. x os 45 = 10 10 x = os 42 5 lulte the vlue of x, orret to 1 deiml ple. = 14.1 M 6 Give written nswer. The seond ship is 14.1 nutil miles from the lighthouse. These eight ompss points do not llow us to mke lultions out more preise diretions. For this reson n lterntive method of desriing erings is needed for ny diretion other thn these si eight points. True erings true ering is n ngle mesured from north in lokwise diretion. s there re 360 in revolution, ll true erings re represented s three-digit numer etween 000 nd 360. For exmple, est is t ering of 090, south hs ering of 180 nd west 270. 270 090 180 hpter 3 pplitions of trigonometry 89
When given informtion out ering, we n solve prolems using trigonometry y onstruting right-ngled tringle. s most questions involving er ings re in prolem form, digrm is neessry to solve the prolem nd n nswer in words should e given. Worked Exmple 5 ship sils on ering of 130 for distne of 10 nutil miles. lulte how fr south of its strting point the ship is, orret to 2 deiml ples. Think Write 1 Drw digrm ompleting right ngled tringle nd lel the sides. 130 dj x 50 hyp 10 M 2 hoose the osine rtio nd write the formul. os q = dj hyp 3 Sustitute for q nd the hypotenuse. os50 = x 10 4 Mke x the sujet of the eqution. x = 10 os 50 5 lulte. x = 6.43 M 6 Give written nswer. The ship is 6.43 nutil miles south of its strting point. opp We n lso use our methods of lulting ngles to mke lultions out erings. fter solving the right-ngled tringle, however, we need to provide the nswer s ering. Worked Exmple 6 On hike Lis wlked south for 3.5 km nd then turned west for 1.2 km. lulte Lis s ering from her strting point. Think Write 1 Drw digrm nd lel the sides of the tringle. hyp 3.5 km dj 1.2 km opp 2 hoose the tngent rtio nd write the formul. tn q = opp dj 3 Sustitute for the opposite nd djent sides nd simplify. 12. tn q = 35. = 0. 3429 90 Mths Quest Generl Mths HS ourse
4 Mke q the sujet of the eqution. q = tn -1 (0.3429) 5 lulte q. = 19 6 From the digrm we n see the ngle lies etween south nd west. South hs ering of 180, nd so we must dd 19 to 180 to lulte the true ering. ering = 180 + 19 = 199 7 Give written nswer. Lis is t ering of 199 from her strting point. WoRked exmple 7 Soldiers on reonnissne set off on return journey from their se mp. The journey onsists of three legs. The first leg is on ering of 150 T for 3 km; the seond is on ering of 220 T for 5 km. Find the diretion (to the nerest minute) nd distne (orret to 2 deiml ples) of the third leg y whih the group returns to its se mp. eookplus Tutoril int-0473 Worked exmple 7 Think WRiTe/dRW 1 Drw digrm of the journey nd indite or superimpose suitle tringle. se mp 150 3 km 220 T 5 km 2 Identify the side of the tringle to e found. Redrw simple tringle with the most importnt informtion provided. 150 30 30 3 km 220 x 110 40 30 5 km 110 40 hpter 3 pplitions of trigonometry 91
3 Identify tht the prolem requires the use of the osine rule, s you re given two sides nd the ngle in etween. 4 Sustitute the known vlues into the osine rule nd evlute. = 3 km = 5 km = 110 = x km 2 = 2 + 2-2 os () x 2 = 3 2 + 5 2-2 3 5 os (110 ) x 2 = 44.260 604 x = 44. 260604 = 6.65 5 For diretion, we need to find the ngle etween the diretion of the seond nd third legs using the sine or osine rules. 40 40 = 3 = 5 = 6.65 or 44. 260604 6 Sustitute the known vlues into the rerrnged osine rule. ote: Use the most urte form of the length of side. 7 lulte the ngle of the turn from the north ering. 8 Write the nswer in orret units nd to the required level of ury. os () = 2 2 + 2-2 os () = 5 2 260604 2 +44 4. 604-3 2 5 5 44. 260604 os () = 0.9058 = 25.07 = 25 4 q = 40-25 4 = 14 56 ering is 14 56 E. The distne overed in the finl leg is 6.65 km, orret to 2 deiml ples, on ering of 14 56 E, orret to the nerest minute. RememeR 1. erings re used to desrie diretion. We hve used two types of erings. ompss erings use the four min points of the ompss, north, south, est nd west, s well s the four middle diretions, north-est, north-west, south-est nd south-west. True erings desrie more speifi diretion y using three-digit ngle, whih is mesured from north in lokwise diretion. 2. ering questions re usully given in written form so you will need to drw digrm to extrt ll the informtion from the question. 92 mths Quest Generl mths hs ourse
3. Red refully to see if the question is sking you to find side or n ngle. 4. lwys give written nswer to worded questions. 5. Use 1 M = 1.852 km to onvert etween nutil miles nd kilometres. Exerise 3 erings 1 Speify the following diretions s ompss erings. 25 W E W E 10 S d S W 310 E W 12 E e S f S W 12 E W 12 E S 2 Speify the following diretions s true erings. S W 45 E W 1 22 2 E S S hpter 3 pplitions of trigonometry 93
d W 135 E W 1 67 2 E e S f S W E W 1 1122 E SW 22 1 2 eookplus Digitl do SkillSHEET 3.5 do-1324 onverting nutil miles to kilometres S 3 We4 rod runs due north. hiker leves the rod nd wlks for 4.2 km in W diretion. Drw digrm of this sitution. How fr due est must the hiker wlk to get k to the rod? (Give your nswer orret to 3 deiml ples.) 4 driver heds due south for 34 km, then turns left nd drives until he is SE of his strting point. Drw digrm to show the driver s journey. lulte the distne the driver trvelled in n esterly diretion from his strting point. 5 Two ots, nd, sil from port. heds due west, while heds W for distne of 43 nutil miles, where it drops nhor. ot drops nhor due south of ot. Drw digrm showing the positions of ots nd. lulte the distne etween ots nd in nutil miles, orret to 1 deiml ple. lulte the distne in kilometres etween nd. 6 m true ering of 315 is equivlent to ompss ering of: E W SE D SW 7 m ompss ering of SE is equivlent to true ering of: 045 135 225 D 315 8 We5 Two hikers, drin nd ertrnd, set out on wlk. drin wlks 5 km due north to point,, nd ertrnd wlks on ering of 052 to point,. ertrnd lets off flre nd drin noties ertrnd is now due est of him, s shown in the digrm on the right. lulte the distne etween the two hikers, orret to 1 deiml ple. S d 5 km 52 94 mths Quest Generl mths hs ourse
9 yht sights lighthouse on ering of 060. fter siling nother eight nutil miles due north, the yht is due west of the lighthouse. Drw digrm of this sitution. lulte the distne from the yht to the lighthouse when it is due west of it (orret to 1 deiml ple). 10 n eroplne tkes off from n irport nd flies on ering of 220 for distne of 570 km. lulte how fr south of the irport the eroplne is (orret to the nerest kilometre). 11 mping ground is due est of r prk. Eden nd Jeff wlk 3.8 km due south from the mping ground until the r prk is on ering of 290. Drw digrm showing the r prk, the mping ground, nd Eden nd Jeff s position. lulte the distne Eden nd Jeff need to wlk diretly k to the r prk, orret to 1 deiml ple. 12 M ship is on ering of 070 from lighthouse. The ering of the lighthouse from the ship will e: 070 160 200 D 250 13 M mping ground is SW of r prk. The ering of the r prk from the mping ground will e: E W SE D SW 14 WE6 serh prty leves its se nd hed 4 km due west efore turning south for 3.5 km. Drw digrm of this sitution. lulte the true ering of the serh prty from its se, orret to the nerest degree. 15 WE7 ship is two nutil miles due west of hrour. yht tht sils 6.5 nutil miles from tht hrour is due north of the ship. lulte the true ering (orret to the nerest degree) of the ourse on whih the yht sils from the hrour. Further development 16 For eh of the following, find how fr north/south nd est/west position is from position O. 200 m 40 E 100 m 200 m 50 E 100 m 75 E O 240 T 25 m 30 km 160 T O d O 750 m 17 The distnes overed in yhting regtt re shown in the digrm. Find (to the nerest metre): how fr south the yht is from the strting point how fr west the yht is from the strting point the distne from the strting point d the diretion of the finl leg to return to the strting point. O 10 W S60 E 150 m 50 W 0.5 km 40 W 1.5 km 3 km S15 W Strting point hpter 3 pplitions of trigonometry 95
18 ptin ook siled from ook Islnd due north for 100 km. He then hnged diretion nd siled west to deserted islnd tht is W of ook islnd. How fr ws ptin ook from ook Islnd? Wht distne ould hve een sved if ptin ook hd siled diretly? 19 digrm representing the journey y hot-ir lloon is shown. The lloonist reorded the first leg of the journey s 15 km SE. The seond leg ws lso 15 km. Find the distne for the finl leg of the lloonist s journey. 20 golfer is iming for hole tht is 190 metres on ering of 220 from the tee. She hits the ll 8 off entre to point tht is on ering of 310 from the hole. How fr does she hve to hit the ll to reh the hole? Strt Trigonometri rtios for otuse ngles Mny non-right-ngled tringles hve one otuse ngle. In the following setions we will e solving non-right-ngled tringles nd will need to investigte the trigonometri rtios for otuse ngles. 1 Use your lultor to give eh of the following, orret to 3 deiml ples. sin 100 os 100 tn 100 d sin 135 e os 135 f tn 135 g sin 179 h os 179 i tn 179 2 Whih of the nswers to question 1 re positive nd whih re negtive? 3 lulte the sine, osine nd tngent of severl other otuse ngles nd see if the estlished pttern ontinues. 4 n you develop rule for the sign of trigonometri rtios of otuse ngles? SW 15 km 15 km 3 Using the sine rule to find side lengths Finding side lengths The trigonometry we hve studied so fr hs een pplile to only right-ngled tringles. The sine rule llows us to lulte the lengths of sides nd the size of ngles in non-rightngled tringles. onsider the tringle drwn on the right. The sine rule sttes tht in ny tringle,, the rtio of eh side to the sine of its opposite ngle will e equl. = = sin sin sin Derivtion of the sine rule, nd represent the three ngles in the tringle nd, nd represent the three sides, rememering tht eh side is nmed with the lower-se letter of the opposite vertex. onstrut line from to point, D, perpendiulr to. D is the perpendiulr height of the tringle, h. h D 96 Mths Quest Generl Mths HS ourse
ow onsider DD nd DD seprtely. h h D D Use the formul for the sine rtio: sin q = opp hyp sin q = opp hyp sin = h sin = h h = sin We re now le to equte these two expressions for h. sin = sin Dividing oth sides y sin sin we get: sin sin = sin sin sin sin h = sin = sin sin Similrly, we re le to show tht eh of these is lso equl to. Try it! sin This formul llows us to lulte the length of side in ny tringle if we re given the length of one other side nd two ngles. When using the formul we need to use only two prts of it. Worked Exmple 8 lulte the length of the side mrked x in the tringle on the right, orret to 1 deiml ple. 80 16 m 40 x Think Write Method 1: Tehnology-free 1 Write the formul. 2 Sustitute = x, = 16, = 80 nd = 40. 3 Mke x the sujet of the eqution y multiplying y sin 80. = sin sin x 16 sin80 = sin 40 16 sin 80 x = sin 40 4 lulte. x = 24.5 m hpter 3 pplitions of trigonometry 97
Method 2: Tehnology-enled 1 From the MEU selet EQU. 2 Press 3 (SOLV). 3 Delete ny existing eqution, enter the eqution X sin 80 = 16 sin 40, nd then press w. ote: Your lultor my disply different vlue of X t this stge. This is just the lst vlue of X stored in the lultor s memory. 4 Press 6 (SOLV) to solve the eqution. WoRked exmple 9 Find the unknown length, x m, in the tringle t right (to 1 deiml ple). 30 130 x 7 m eookplus Tutoril int-0465 Worked exmple 9 Think WRiTe/diSply 1 Drw the tringle. ssume it is non-rightngled. 2 Lel the tringle ppropritely for the sine rule. 30 130 = 7 m = x 3 onfirm tht it is the sine rule tht n e used s you hve the ngle opposite to the unknown side nd known side ngle rtio. 4 Sustitute known vlues into the two rtios. = = sin( ) sin ( ) sin( ) = x = 130 = 7 m = 30 x sin( 130 ) = 7 sin( 30 ) 5 Isolte x nd evlute. x = 7 sin( 130 ) sin( 30 ) x = 10.7246 x = 10.7 6 Write the nswer. The unknown length is 10.7 m, orret to 1 deiml ple. 98 mths Quest Generl mths hs ourse
ote: Some questions my sk for you to give the nswer in form other thn numer nd s suh the grphis lultor method n not e used. For exmple, the question ove 16 sin 80 ould e worded to, sy, show x =, in whih se you must mnipulte the eqution sin 40 to rrive t the desired expression. To use the sine rule we need to know the ngle opposite the side we re finding nd the ngle opposite the side we re given. In some ses these re not the ngles we re given. In suh ses we need to use the ft tht the ngles in tringle dd to 180 to lulte the required ngle. Worked Exmple 10 lulte the length of the side lelled m in the figure on the right, orret to 4 signifint figures. m 65 Think Write 75 16 m 1 lulte the size of ngle. = 180-65 - 75 = 40 2 Write the formul. = sin sin 3 Sustitute = 16, = m, = 65 nd = 40. 16 sin65 = m sin 40 4 Mke m the sujet of the eqution. 16 sin 40 m = sin 65 5 lulte. = 11.35 m s mentioned in the previous investigtion, we need to pply the sine rule to otuse-ngled tringles. In suh exmples the method used is extly the sme with the su stitution of n otuse ngle. Using the sine rule llows us to solve numer of more omplex prolems. s with our erlier trigonometry prolems, we egin eh with digrm nd give written nswer to eh. Worked Exmple 11 Georg looks south nd oserves n eroplne t n ngle of elevtion of 60. Henriett is 20 km south of where Georg is nd she fes north to see the eroplne t n ngle of elevtion of 75. lulte the distne of the eroplne from Henriett s oservtion point, to the nerest metre. x G 60 75 20 km H hpter 3 pplitions of trigonometry 99
Think WRiTe 1 lulte the size of GH. = 180-60 - 75 = 45 2 Write the formul. g = sing sin 3 Sustitute g = x, = 20, G = 60 nd H = 75. x 20 sin60 = sin 45 20 sin 60 4 Mke x the sujet. x = sin 45 5 lulte. x = 24.495 km 6 Give written nswer. The distne of the eroplne from Henriett s oservtion point is 24.495 km. RememeR 1. 2. 3. 4. 5. The sine rule formul is = =. sin sin sin The sine rule is used to find side in ny tringle when we re given the length of one other side nd two ngles. We need to use only two prts of the sine rule formul. For written prolems, egin y drwing digrm nd finish y giving written nswer. You n use the eqution solver on grphis lultor to find the vlue of the unknown fter sustituting into the formul. exerise 3 eookplus Digitl do SkillSHEET 3.6 do-1325 ngle sum of tringle Using the sine rule to find side lengths 1 Write down the sine rule formul s it pplies to eh of the tringles elow. X Z Y P R Q 100 mths Quest Generl mths hs ourse
eookplus Digitl do SkillSHEET 3.7 do-1326 Solving frtionl equtions 2 We 8, 9 Use the sine rule to lulte the length of the side mrked with the pronumerl in eh of the following, orret to 3 signifint figures. L R x 16 m 50 45 1.9 km 63 59 M q T 84 52 t 89 mm S 3 We 10 In eh of the following, use the sine rule to lulte the length of the side mrked with the pronumerl, orret to 1 deiml ple, y first finding the size of the third ngle. G x H 74 74 80 m 85 y I 18.2 mm M 62 35.3 m 4 m Look t the figure drwn on the right. Whih of the following expressions gives the vlue of m? 42 sin 117 m = sin 28 42 sin 28 m = sin 117 42 sin 117 m = sin 35 42 sin 35 D m = sin 117 5 m Look t the figure drwn on the right. Whih of the following expressions gives the vlue of n? 42 sin 117 n = sin 28 42 sin 28 n = sin 117 42 sin 117 n = sin 35 42 sin 35 D n = sin 117 6 is tringle in whih = 9 m, = 54 nd = 62. lulte the length of side, orret to 1 deiml ple. 7 XYZ is tringle in whih y = 19.2 m, XYZ = 42 nd XZY = 28. lulte x, orret to 3 signifint figures. 8 We 11 X nd Y re two trees, 30 m prt on one side of river. Z is tree on the opposite side of the river, s shown in the digrm elow. Z P 19.4 km 27 42 m 35 28 m n 35 28 42 m X 59 72 30 m Y It is found tht XYZ = 72 nd YXZ = 59. lulte the distne XZ, orret to 1 deiml ple. hpter 3 pplitions of trigonometry 101
9 From point, M, the ngle of elevtion to the top of uilding,, is 34. From point,, 20 m loser to the uilding, the ngle of elevtion is 49. Drw digrm of this sitution. lulte the distne, orret to 1 deiml ple. lulte the height of the uilding, orret to the nerest metre. 10 Look t the figure on the right. 80 sin 30 Show tht XY n e given y the expression sin 40. Show tht h n e found using the 80 sin30 sin 70 expression. sin 40 lulte h, orret to 1 deiml ple. W 30 70 80 m X Y h Z Further development 11 Steel trusses re used to support hevy gte t the entrne to shipping yrd. The struts in the truss shown re eh mde from 0.8 m steel lengths nd re welded t the ontt points with the upper nd lower setions of the truss. On the lower setion of the truss, wht is the distne (to the nerest entimetre) etween eh pir of onseutive welds? Wht is the height (to the nerest entimetre) of the truss? 12 seni flight leves Town nd flies west of north for the 80-km diret journey to Town. t Town the plne turns 92 to the right to fly est of north to Town. From here the plne turns 129 to the right nd flies the 103 km stright k to Town. Find the distne (to the nerest km) of the diret flight from Town to Town. 13 Two lighthouses re 17 kilometres prt on n est west line. From lighthouse, ship is seen on ering of 130. From lighthouse, the sme ship is spotted on ering of S20 W. Whih lighthouse is the ship loser to? How fr is tht lighthouse from the ship? Two lighthouses re 25 kilometres prt on south north line. From lighthouse, ship is reported on ering of 082 T. The sme ship is deteted from lighthouse on ering of 165 T. Whih lighthouse is loser to the ship nd how fr is tht lighthouse from the ship? Two fire-spotting towers re 33 km prt on n est west line. From Tower fire is spotted on ering of 63 E, while from Tower the sme fire is spotted on ering of 290 T. How fr wy from the nerer tower is the fire? Whih tower is this? 14 Two lighthouses re 25 km prt on north south line. The northern lighthouse spots ship on ering of S60 E. The southern lighthouse spots the sme ship on ering of 050 T. Find the distne from the northern lighthouse to the ship. Find the distne from the southern lighthouse to the ship. 15 light irrft hs stryed into mjor ir orridor. It hs een deteted y two ir trffi ontrol towers. Tower 1 hs the light irrft on ering of 315 T. Tower 2 hs the light irrft on ering of north. The two towers re 300 kilometres prt on E line s shown. How fr is the light plne from eh tower? 130 130 130 Tower 1 0.8 m 300 km Tower 2 E 102 Mths Quest Generl Mths HS ourse
3D Using the sine rule to find ngles Finding ngles Using the sine rule result, we re le to lulte ngle sizes s well. To do this, we need to e given the length of two sides nd the ngle opposite one of them. For simpliity, in solving the tringle we invert the sine rule formul when we re using it to find n ngle. The formul is written: sin sin sin = = Your formul sheet hs the sine rule to find side length. You need to invert this formul when finding n ngle. s with finding side lengths, we use only two prts of the formul. Worked Exmple 12 Find the size of the ngle, q, in the figure on the right, orret to the nerest degree. 6 m 110 20 m Think Method 1: Tehnology-free Write 1 Write the formul. sin sin = 2 Sustitute = 110, = q, = 20 nd = 6. sin110 sin q = 20 6 3 Mke sin q the sujet of the eqution. 6sin 110 sin q = 20 4 lulte vlue for sin q. sin q = 0.2819 5 lulte sin -1 (0.2819) to find q. q = 16 Method 2: Tehnology-enled 1 From the MEU selet EQU. 2 Press 3 (SOLV). 3 Delete ny existing eqution, enter the eqution sin 110 20 = sin X 6 nd press w. ote: Your lultor my disply different vlue of X t this stge. This is just the lst vlue of X stored in the lultors memory. hpter 3 pplitions of trigonometry 103
4 Press 6 (SOLV) to solve the eqution. ote: When using the grphis lultor, you do not need to rememer to invert the sine rule. If you enter 20 sin 110 = 6 sin x, the grphis lultor will still solve the eqution. s with finding side lengths, some questions will e prolems tht require you to drw digrm to extrt the required informtion nd then write the nswer. WoRked exmple 13 From point, P, ship (S) is sighted 12.4 km from P on ering of 137. point, Q, is due south of P nd is distne of 31.2 km from the ship. lulte the ering of the ship from Q, orret to the nerest degree. Think WRiTe eookplus Tutoril int-2416 Worked exmple 13 1 Drw digrm. P 137 12.4 km 43 S 31.2 km Q 2 Write the formul. sinq sin P = q p 3 Sustitute for p, q nd P. sin Q sin 43 = 12. 4 31. 2 4 Mke sin Q the sujet. 12. 4sin 43 sin Q = 31. 2 5 lulte vlue for sin Q. sin Q = 0.271 6 lulte sin -1 (0.271) to find Q. Q = 16 7 Give written nswer. The ering of the ship from Q is 016. RememeR sin sin sin 1. The sine rule formul for finding n ngle is = =. 2. The formul sheet gives the sine rule in the form used to find side. You hve to invert the formul when finding ngles. 3. We n use this formul when we re given two sides nd the ngle opposite one of them. 4. Worded questions should egin with digrm nd finish with written nswer. 104 mths Quest Generl mths hs ourse
Exerise 3D Using the sine rule to find ngles 1 WE 12 Find the size of the ngle mrked with pronumerl in eh of the following, orret to the nerest degree. P L 100 32 m φ 46 m Q 60 18.9 m 29.5 m R 79 mm 153 mm 117 α M d V e X f 170 27 mm 23.6 km 23.6 km 16.5 m 27.6 m 156 mm U 75 W 86 Y β Z 2 M Whih of the sttements elow give the orret vlue for sin q? 13 sin 36 sin q = 7 36 sin 13 sin q = 7 7sin 36 sin q = 13 7sin 13 D sin q = 36 3 M In whih of the tringles elow is the informtion insuffiient to use the sine rule? 14.8 m 12.7 m 7 13 36 57 12.6 m 16.2 m 45 115 12.9 m 6.2 m D 12.7 m 8.7 m 9 4 In Questions find the size of the ngle mrked, orret to the nerest degree. 60 32 m 41 m 44 m 65 m 31 23 4.9 m 3.6 m 5 In DPQR, q = 12 m, r = 16 m nd PRQ = 56. Find the size of PQR, orret to the nerest degree. 6 In DKLM, LM = 4.2 m, KL = 5.6 m nd KML = 27. Find the size of LKM, orret to the nerest degree. hpter 3 pplitions of trigonometry 105
eookplus Digitl do WorkSHEET 3.1 do-1327 7 We 13, nd re three towns mrked on mp. Judy lultes tht the distne etween nd is 45 km nd the distne etween nd is 32 km. is 45. lulte, orret to the nerest degree. 8 surveyor mrks three points X, Y nd Z in the ground. The surveyor mesures XY to e 13.7 m nd X Z to e 14.2 m. XYZ is 60. lulte X Z Y to the nerest degree. lulte Y X Z to the nerest degree. 9 Two wires support flgpole. The first wire is 8 m long nd mkes 65 ngle with the ground. The seond wire is 9 m long. Find the ngle tht the seond wire mkes with the ground. 10 onstrut suitle tringle from the following instrutions nd find ll unknown sides nd ngles. The smllest side is 17 m nd one of the other sides is 25 m. The smllest ngle is 32. 11 yht sils the three-leg ourse shown. 15 km The smllest ngle etween ny two legs within the ourse, to the nerest degree, is: 78 18 km 34 55 45 D 78 13 km 12 The orret expression for ngle S in the given tringle is: - sin sin 1 40 41 S 30 41 30 40 - sin sin 1 30 41 40 - sin sin 1 41 41 30 - D sin 1 30 40 sin 41 13 Two rmy mps nd re on the sme est west line. Rdio tower T is loted 20 km from mp, SE of mp. The tower is distne of 15 km from mp. Find the ering of the rdio tower, T, from mp. 14 en is plnning to hike in the mountins out of the snow seson. From position in front of the ski lodge, en n see the hirlift sttion nd the strt of the ski run in the distne. He notes tht in moving his eye from the ski lodge sttion to the strt of the run, the ngle is 34. en then wlks in stright line to the hirlift sttion, distne of 365 m, turns nd wlks the 230 m stright line distne to the strt of the ski run. From here, wht ngle (to the nerest degree) would he note etween the ski lodge nd the hirlift sttion? 106 mths Quest Generl mths hs ourse
15 golfer is teeing off on the 1st hole. The distne nd diretion to the green is 410 metres on ering of 190. If the tee shot of the plyer ws 250 metres on ering of 220, nd if the distne etween where her tee shot lnds nd the green is 230.36 m, wht diretion should she im for? (Give the diretion to the nerest degree.) 3e re of tringle You should e fmilir with finding the re of tringle using the formul re = 1 h. In this formul, is the se of the 2 tringle nd h is the perpendiulr height. This formul n t e used in tringles where we do not know the perpendiulr height. Trigonometry llows us to find the re of suh tringles when we re given the length of two sides nd the inluded ngle. onsider the tringle drwn on the right. In this tringle: h D re = 1 h [] 1 2 ( = se of tringle, h = height) ow onsider DD. Sine this tringle is right ngled: sin = opp hyp Sustituting for h in [1]: sin = h h = sin re = 1 sin 2 This eomes the formul for the re of tringle. There re three equivlent formuls for the re of tringle. re = re = re = 1 2 1 2 1 2 sin sin sin The formul sheet gives the first version of this formul only. The others re n dpttion of the sme rule. These formuls llow us to find the re of ny tringle where we re given the length of two sides nd the inluded ngle. The inluded ngle is the ngle etween the two given sides. The formul hosen should e the one tht uses the ngle you hve een given. WoRked exmple 14 Find the re of the tringle t elow (to 2 deiml ples). eookplus Tutoril int-0469 Worked exmple 14 9 m 37 6 m hpter 3 pplitions of trigonometry 107
Think 1 Identify the shpe s tringle with two known sides nd the ngle in etween. 2 Identify nd write down the vlues of the two sides, nd, nd the ngle in etween them,. 3 Identify the pproprite formul nd sustitute the known vlues into it. WRiTe/diSply = 9 37 = 6 = 9 = 37 = 6 re tringle = 1 sin () 2 = 1 2 6 9 sin (37 ) = 16.249 4 Write the nswer in orret units. The re of the tringle is 16.25 m 2, orret to 2 deiml ples. s with ll other trigonometry we n use this formul to solve prtil prolems. WoRked exmple 15 Two pths diverge t n ngle of 72. The pths lengths re 45 m nd 76 m respetively. lulte the re etween the two pths, orret to the nerest squre metre. Think WRiTe 1 Drw digrm. 45 m 72 76 m 2 Write the formul. re = 1 sin 2 3 Sustitute = 45, = 76 nd = 72. = 1 45 76 sin 72 2 4 lulte. = 1626 m 2 5 Give written nswer. The re etween the pths is 1626 m 2. RememeR 1. The re of tringle n e found when you re given the length of two sides nd n inluded ngle. 2. The formuls to use re: re = 1 sin 2 re = 1 sin 2 re = 1 sin 2 3. Where possile you should still use re = 1 2 h. 4. egin worded prolems with digrm nd finish them with written nswer. 108 mths Quest Generl mths hs ourse
Exerise 3E re of tringle 1 Write down the formul for the re of tringle in terms of eh of the tringles drwn elow. Write the formul using the oldfed ngle. X Y Z G M 2 For eh of the tringles drwn elow, stte whether the re would e est found using the formul re = 1 2 sin or re = 1 2 h. 6 m 60 1.9 m 8.3 m 12 m 6.2 m d 2.6 m 9.1 m 60 12.4 m 3 WE 14 Find the re of eh of the following tringles, orret to 1 deiml ple. 11 m 196 mm 207 mm 117 mm 40 12 m 10 120 92 mm 4 Use either re = 1 sin or re = 1 h to find the re of eh of the following tringles. 2 2 Where neessry, give your nswer orret to 1 deiml ple. 19 m 32 m 38 m 14 m 66 38 m 32 m hpter 3 pplitions of trigonometry 109
5 M In whih of the following tringles n the formul re = 1 sin not e used to find 2 the re of the tringle? 4 m 60 9 m 4 m 9 m D 4 m 75 9 m 120 4 m 9 m 6 M The re of the tringle on the right (orret to 1 deiml ple) is: 4.4 m 2 14.7 m 2 17.1 m 2 D 20.5 m 2 5 m 7 m 78 6 m 7 In DPQR, p = 4.3 m, q = 1.8 m nd PRQ = 87. lulte the re of DPQR, orret to 4 signifint figures. 8 The figure on the right is of prllelogrm, D. opy the digrm into your workook nd drw the digonl on your digrm. y onsidering the prllelogrm s two equl tringles, lulte its re, orret to 1 deiml ple. 2.5 m 70 D 5.2 m 9 On the right is digrm of pentgon insried in irle of rdius 5 m. lulte the size of eh of the ngles mde t the entre. lulte the re of the pentgon, orret to the nerest squre entimetre. 10 WE 15 surveyor sights the four orners of lok of lnd nd mkes the following noteook entry. lulte the re of the lok of lnd, orret to the nerest squre metre. 15 m 18 m 90 80 70 120 20 m 25 m 110 Mths Quest Generl Mths HS ourse
Further development 11 Find the re of n equilterl tringle with side length of 10 m. 12 4 m 6 m 30 50 The orret expression for the re of the shpe ove is: 1 2 1 2 1 2 6 4 sin 80 6 4 os 80 6 4 sin 30 D 1 2 6 4 sin 100 13 The orret expression for the re of the otgon shown is: 11 sin 45 88 sin 67.5 88 sin 45 D 11 sin 67.5 14 onsider the tringle drwn elow. 4 5.5 15 m 105 18 m Find the re of the tringle. Use your nswer to prt () together with the formul = 1 2 distne of the point from the line. h to find the shortest hpter 3 pplitions of trigonometry 111
15 The tringle PQR hs side lengths PQ = 15 m, QR = 22 m nd PQR = 75. Find the re of the tringle. etty drws the tringle y mistke with PQR = 105. Show tht etty will still get the orret nswer. Explin why the sme nswer for the re of the tringle is otined. 16 Penny is mking tringulr disply se s shown in the digrm. The two sides re to e 30 m in length. Find the ngle etween the two sides tht will mximise the re of the tringulr ross-setion. 30 m 30 m 3F Using the osine rule to find side lengths Finding side lengths When given the length of one side nd two ngles in tringle, we n use the sine rule to find nother side length. However, in mny ses we do not hve this informtion nd need nother method of lulting the side lengths. The osine rule llows us to lulte the length of the third side of tringle when we re given the length of the other two sides nd the inluded ngle. 2 = 2 + 2-2os 2 = 2 + 2-2os 2 = 2 + 2-2os The formul sheet gives the third version of this formul only. The others re n dpttion of the sme rule. It is importnt to notie tht the formul is given in terms of 2, 2 or 2. This mens tht to find the vlue of, or we need to tke the squre root of our lultion. Derivtion of the osine rule onsider D on the right. In this tringle, h is the perpendiulr height of the tringle nd meets t D. We will let D = x, nd therefore D = - x. Using Pythgors theorem on DD: 2 = ( - x) 2 + h 2 2 = 2-2x + x 2 + h 2 [1] From DD: 2 = x 2 + h 2 Therefore: h 2 = 2 - x 2 Sustituting for h 2 in [1]: 2 = 2-2x + x 2 + 2 - x 2 2 = 2-2x + 2 [2] x h D x ow in DD: Therefore: os = x x = os Sustituting for x in [2]: 2 = 2-2( os ) + 2 2 = 2 + 2-2 os This eomes the formul for the osine rule. similr formul n e used for finding sides nd. You my like to try it for yourself. 112 Mths Quest Generl Mths HS ourse
1 Strt with D nd drw perpendiulr line from to. 2 Use this digrm nd follow the method shown to otin the following version of the osine rule: 2 = 2 + 2-2 os. 3 n you otin 2 = 2 + 2-2 os? Worked Exmple 16 Find the length of the side mrked in the tringle on the right, orret to 1 deiml ple. 70 10 m 12 m Think Write Method 1: Tehnology-free 1 Write the formul with 2 s the sujet. 2 = 2 + 2-2 os 2 Sustitute = 12, = 10 nd = 70. = 12 2 + 10 2-2 12 10 os 70 3 lulte the vlue of 2. = 161.915 4 Find y tking the squre root of 2. = 161. 915 = 12. 7m Method 2: Tehnology-enled 1 From the MEU selet EQU. 2 Press 3 (SOLV). 3 Delete ny existing eqution, enter 2 = 12 2 + 10 2-2 12 10 os 70 nd then press w. 4 Press 6 (SOLV) to solve the eqution. hpter 3 pplitions of trigonometry 113
s with sine rule questions, we n pply the osine rule to otuse-ngled tringles. You should rell from the erlier investigtion tht the osine rtio of n otuse ngle is negtive. The method of solution remins unhnged. WoRked exmple 17 Find the unknown length (to 2 deiml ples), x, in the tringle t right. Think 7 m x 80 6 m WRiTe eookplus Tutoril int-0468 Worked exmple 17 1 Identify the tringle s non-right-ngled. 2 Lel the tringle ppropritely for the sine rule or osine rule. = 7 80 = 6 = x 3 Identify tht it is the osine rule tht is required s you hve the two sides nd the ngle in etween. 4 Sustitute the known vlues into the osine rule formul. = 6 = 80 = 7 = x 2 = 2 + 2-2 os () x 2 = 6 2 + 7 2-2 6 7 os (80 ) x 2 = 36 + 49-84 os (80 ) x 2 = 70.4136 5 Rememer to get the squre root vlue, x. x = 70. 4136 = 8.391 6 Evlute the length nd inlude units with the nswer. x = 8.39 The unknown length is 8.39 m, orret to 2 deiml ples. The osine rule lso llows us to solve wider rnge of prtil prolems. The importnt prt of solving suh prolems is mrking the orret informtion on your digrm. If you n identify two side lengths nd the inluded ngle, you n use the osine rule. WoRked exmple 18 surveyor stnding t point, X, sights point, M, 50 m wy nd point,, 80 m wy. If the ngle etween the lines XM nd X is 45, lulte the distne etween the points M nd, orret to 1 deiml ple. Think 1 Drw digrm nd mrk ll given informtion on it. WRiTe X 50 m 45 M 80 m 114 mths Quest Generl mths hs ourse
2 Write the formul with x 2 s the sujet. x 2 = m 2 + n 2-2mn os X 3 Sustitute m = 80, n = 50 nd X = 45. = 80 2 + 50 2-2 80 50 os 45 4 lulte the vlue of x 2. = 3243.15 5 lulte x y tking the squre root of x 2. x = 3243. 15 6 Give written nswer. = 56.9 m REMEMER 1. To use the osine rule to find side length, you need to e given the length of two sides nd the inluded ngle. 2. The osine rule formuls re: 2 = 2 + 2-2 os 2 = 2 + 2-2 os 2 = 2 + 2-2 os. 3. In the solution to osine rule questions, your finl nswer is found y tking the squre root of the lultion. 4. egin worded questions y drwing digrm nd finish them y giving written nswer. Exerise 3F Using the osine rule to find side lengths 1 Write down the osine rule formul s it pplies to eh of the tringles elow. In eh se, mke the oldfed pronumerl the sujet. P L n M r q m l Q R p 2 WE 16 Find the length of the side mrked with pronumerl in eh of the following, orret to 3 signifint figures. P X x 12 m r 13 m 12 m 60 12 m 35 42 14 m Q 21 m R Y x Z 3 WE 17 In eh of the following otuse-ngled tringles, find the length of the side mrked with the pronumerl, orret to 1 deiml ple. X R 114 m 112 m 110 q 6.1 m Z 130 160 63 mm P x 9.7 m 43 mm Q Y hpter 3 pplitions of trigonometry 115
4 M In whih of the following tringles re we unle to use the osine rule to find x? 14.8 m 16.4 m x 16.2 m 132 32 18.2 m 63 8.3 km 9.6 km D 10.5 m x 9.7 m x 63 x 5 M Look t the tringle drwn on the right. The vlue of x, orret to 1 deiml ple, is: 7.2 m 7.3 m 52.4 m D 52.5 m 8 m 50 9 m 6 M Lieng is sked to find the vlue of, orret to 1 deiml ple, in the figure drwn on the right. elow is Lieng s solution. Line 1: 2 = 12 2 + 8 2-2 12 8 os 60 60 Line 2: = 144 + 64-192 os 60 8 m Line 3: = 208-192 os 60 Line 4: = 16 os 60 Line 5: = 8 Line 6: = 2.8 m Lieng s solution is inorret. In whih line did she mke her error? Line 2 Line 3 Line 4 d Line 5 7 In D, = 14 m, = 25 m nd = 29. lulte, orret to 1 deiml ple. 8 In DPQR, PQ = 234 mm, QR = 981 mm nd PQR = 128. Find the length of side PR, orret to 3 signifint figures. 9 WE 18 Len nd Morg wlk seprte pths tht diverge from one nother t n ngle of 48. fter three hours Len hs wlked 7.9 km nd Morg 8.6 km. Find the distne etween the two wlkers t this time, orret to the nerest metre. 10 riketer is fielding 20 m from the tsmn nd t n ngle of 35 to the pith. The tsmn hits ll 55 m nd stright ehind the owler. How fr must the fieldsmn run to field the ll? (Give your nswer to the nerest metre.) 11 The sides of prllelogrm re 5.3 m nd 11.3 m. The sides meet t ngles of 134 nd 46. Drw digrm of the prllelogrm showing this informtion nd mrk oth digonls on it. lulte the length of the shorter digonl, orret to 1 deiml ple. lulte the length of the long digonl, orret to 1 deiml ple. 12 The ord supporting piture frme is 58 m long. It is hung over single hook in the entre of the ord nd the ord then mkes n ngle of 145 s shown in the figure on the right. lulte the length of the king of the piture frme, to the nerest entimetre. 58 m 145? x 12 m 116 Mths Quest Generl Mths HS ourse
Further development 13 During siling re, the ots followed ourse s shown. Find the length, x, of its third leg (to 1 deiml ple). 10 km 105 7 km 14 Two irles, with rdii 4 m nd 7 m, overlp slightly s shown. If the ngle etween the two rdii tht meet t the point of intersetion of the irumferenes is 110, find the distne etween the entres of the irles (to 1 deiml ple). 15 Two hikers set out from the sme point. One wlks 30 E for 1500 metres nd the other wlks S40 E for 1200 metres. How fr prt to the nerest metre re the two hikers? 4 m x 7 m 110 16 n dvertising lloon is tthed to two ropes 120 m nd 100 m long. The shorter rope mkes 70 ngle with the ground nd is tthed to the ottom of the lloon. The longer rope mkes n 80 ngle with the horizontl nd is tthed to the top of the lloon. How tll is the lloon? Give your nswer orret to 2 deiml ples. 17 plne tkes off t 10.00 m from n irfield nd flies t 120 km/h on ering of 325. seond plne tkes off from the sme irfield nd flies on ering of 100 t speed of 90 km/h. How fr prt re the plnes t 10.25 m? 3g Using the osine rule to find ngles Finding ngles We n use the osine rule to find the size of the ngles within tringle. onsider the osine rule formul. 2 = 2 + 2-2 os We now mke os the sujet of this formul. 2 = 2 + 2-2 os 2 + 2 os = 2 + 2 2 os = 2 + 2-2 os = 2 + 2-2 2 In this form, we n use the osine rule to find the size of n ngle if we re given ll three side lengths. We should e le to write the osine rule in three forms depending upon whih ngle we wish to find. os + - = 2 os + - = 2 2 2 os + = - 2 2 2 2 2 2 2 gin, the formul sheet gives the third version of this formul only. The others re n dpttion of the sme rule. 2 hpter 3 pplitions of trigonometry 117
Worked Exmple 19 Find the size of ngle in the tringle on the right, orret to the nerest degree. 7 m 5 m 9 m Think Write Method 1: Tehnology-free 1 Write the formul with os s the sujet. os + - = 2 2 2 2 9 + 7-5 2 Sustitute = 9, = 5 nd = 7. os = 2 9 7 2 2 2 3 lulte the vlue of os. 105 os = 126 = 0. 8333 4 Mke the sujet of the eqution. = os -1 (0.8333) 5 lulte. = 34 Method 2: Tehnology-enled 1 From the MEU selet EQU. 2 Press 3 (SOLV). 3 Delete ny existing eqution, enter the eqution os = (9 2 + 7 2-5 2 ) (2 9 7), nd then press w. 4 Press 6 (SOLV) to solve the eqution. Your formul sheet will give you two versions of the osine rule, one for finding side length nd one for finding n ngle. When using the eqution solver it does not mtter whih version you use to find side or n ngle. 118 Mths Quest Generl Mths HS ourse
Try using the solver on the eqution 5 2 = 9 2 + 7 2-2 9 7 os. s we found erlier, the osine rtio for n otuse ngle will e negtive. So, when we get negtive result to the lultion for the osine rtio, this mens tht the ngle we re finding is otuse. Your lultor will give the otuse ngle when we tke the inverse. Worked Exmple 20 Find the size of ngle Q in the tringle on the right, orret to the nerest degree. 4 m Q 3 m P 6 m R Think Write 1 Write the formul with os Q s the sujet. os Q p + r - = q 2pr 2 2 2 3 + 4-6 2 Sustitute p = 3, q = 6 nd r = 4. os Q = 2 4 3 2 2 2 3 lulte the vlue of os Q. os Q = - 11 24 =-0. 4583 4 Mke Q the sujet of the eqution. Q = os -1 (-0.4583) 5 lulte Q. Q = 117 In some osine rule questions, you need to work out whih ngle you need to find. For exmple, you ould e sked to lulte the size of the lrgest ngle in tringle. To do this you do not need to lulte ll three ngles. The lrgest ngle in ny tringle will e the one opposite the longest side. Similrly, the smllest ngle will lie opposite the shortest side. Worked Exmple 21 Find the size of the lrgest ngle in the tringle drwn on the right. R 3.4 m 4.9 m S 5.7 m T Think Write 1 ST is the longest side, therefore ngle R is the lrgest ngle. 2 Write the formul with os R the sujet. os R s + t - = r 2st 2 2 2 3 49. + 34. -57. Sustitute r = 5.7, s = 4.9 nd t = 3.4. os R = 2 4. 9 3. 4 2 2 2 hpter 3 pplitions of trigonometry 119
4 lulte the vlue of os R. 308. os R = 33. 32 = 0. 0924 5 Mke R the sujet of the eqution. R = os -1 (0.0924) 6 lulte R. R = 85 7 Give written nswer. The lrgest ngle in the tringle is 85. Mny prolems tht require you to find n ngle re solved using the osine rule. s lwys, these egin with digrm nd re finished off y giving written nswer. Worked Exmple 22 Two pths diverge from point,. The first pth goes for 1.25 km to point,. The seond pth goes for 1.4 km to point,. nd re extly 2 km prt. Find the ngle t whih the two pths diverge. Think Write 1 Drw digrm. 1.25 km 2 km 1.4 km 2 Write the formul with os s the sujet. os 2 + = 2-2 2 3 14. + 125. -2 Sustitute = 2, = 1.4 nd = 1.25. os = 2 1. 4 1. 25 2 2 2 4 lulte the vlue of os.. os = - 0 4775 35. =-0. 1364 5 Mke the sujet of the eqution. = os -1 (- 0.1364) 6 lulte the vlue of. = 98 7 Give written nswer. The rods diverge t n ngle of 98. REMEMER 1. The osine rule formuls re: os 2 + = 2-2 2 os 2 + = 2-2 2 os 2 + = 2-2 2 120 Mths Quest Generl Mths HS ourse
2. If the vlue of the osine rtio is negtive, the ngle is otuse. 3. In ny tringle, the lrgest ngle lies opposite the lrgest side nd the smllest ngle lies opposite the smllest side. 4. Worded prolems egin with digrm nd end with written nswer. Exerise 3G Using the osine rule to find ngles 1 For eh of the following, write the osine rule formul s it pplies to the tringle drwn with the oldfed ngle s the sujet. P P M R Q 2 WE 19 Find the size of the ngle mrked with the pronumerl in eh of the following tringles, orret to the nerest degree. 8 m 11 m 3.2 m 2.8 m M 4.5 m 5.4 m 13 m 4.0 m 6.2 m 3 WE20 In eh of the otuse-ngled tringles elow find the size of the ngle mrked with the pronumerl, to the nerest degree. O 6 m 8 m 11 m 4.2 m α 9.6 m 6.1 m 12.9 m 9.2 m 4.2 m 4 M Look t the figure drwn elow. 3 m 5 m 7 m Whih of the following orretly represents the vlue of os q? 3 + 7-5 os q = 2 3 7 2 2 2 3 + 5-7 os q = 2 3 5 2 2 2 3 + 7-5 os q = 2 5 7 2 2 2 5 + 7-3 D os q = 2 5 7 2 2 2 hpter 3 pplitions of trigonometry 121
5 m In whih of the following is the ngle q otuse? 3 m 4 m 3 m 4 m 5 m 4 m 3 m 4 m D 3 m 4 m 4 m 6 m 6 In DPQR, p = 7 m, q = 9 m nd r = 6 m. Find QRP, orret to the nerest degree. 7 In DKLM, k = 85 mm, l = 145 mm nd m = 197 mm. Find the size of the smllest ngle, orret to the nerest degree. 8 We 21 lulte the size of ll three ngles (orret to the nerest degree) in tringle with side lengths 12 m, 14 m nd 17 m. 9 WXYZ is prllelogrm. WX = 9.2 m nd XY = 13.6 m. The digonl WY = 14 m. Drw digrm of the prllelogrm. lulte the size of WXY, orret to the nerest degree. 10 We 22 Two rods diverge from point, P. The first rod is 5 km long nd leds to point, Q. The seond rod is 8 km long nd leds to point, R. The distne etween Q nd R is 4.6 km. lulte the ngle t whih the two rods diverge. 11 soer gol is 8 m wide. plyer is diretly in front of the gol suh tht he is 12 m from eh post. Within wht ngle must he kik the ll to sore gol? seond plyer tkes n ngled shot. This plyer is 12 m from the nerest post nd 17 m from the fr post. Within wht ngle must this plyer kik to sore gol? 122 mths Quest Generl mths hs ourse
12 The king of piture frme is 50 m long nd is hung over piture hook y ord 52 m long s shown in the figure on the right. lulte the ngle mde y the ord t the piture hook. 52 m Further development 13 Mri yles 12 km in diretion 68 W nd then 7 km in diretion of 34 E. How fr is she from her strting point? Wht is the ering of the strting point from her finishing point? 50 m 14 grden ed is in the shpe of tringle, with sides of length 3 m, 4.5 m nd 5.2 m. lulte the smllest ngle. Hene, find the re of the grden. 15 hokey gol is 3 m wide. When Sophie is 7 m from one post nd 5.2 m from the other, she shoots for gol. Within wht ngle must the shot e mde if it is to sore gol? 16 plne flies in diretion of 70 E for 80 km nd then on ering of S10 W for 150 km. How fr is the plne from its strting point? Wht diretion is the plne from its strting point? 17 Three irles of rdii 5 m, 6 m nd 8 m re positioned so tht they just touh one nother. Their entres form the verties of tringle. Find the lrgest ngle in the tringle. 18 From the top of vertil liff 68 m high, n oserver noties yht t se. The ngle of depression to the yht is 47. The yht sils diretly wy from the liff, nd fter 10 min the ngle of depression is 15. How fst is the yht siling? 3H Rdil surveys In the preliminry ourse we exmined the offset survey. In this survey method n re is mesured y drwing trverse line nd mesuring offsets t right ngles to the trv erse line. euse the offset survey reted right-ngled tringles, the length of eh oundry ould e lulted using Pythgors theorem nd the re ould e lu lted using the formul re = 1 2 h. n lterntive survey method to this is rdil survey. One type of rdil survey is the plne tle rdil survey. The following steps re tken in plne tle survey. 1. tle is pled in the entre of the field to e surveyed, eh orner of the field is sighted nd line is ruled on the pper long the line of sight. 2. The distne from the plne tle to eh orner is then mesured. 20 m 25 m 28 m 26 m hpter 3 pplitions of trigonometry 123
3. The ngle etween eh rdil line is then mesured nd the rdil lines joined to omplete the digrm. 20 m 115 60 115 70 25 m 28 m 26 m The field will then e divided into tringles. The length of eh side of the field n then e lulted y using the osine rule. The perimeter of the field is then found y dding the lengths of eh side. Worked Exmple 23 The figure on the right is plne tle survey of lok of lnd. lulte the perimeter of the lok of lnd, orret to the nerest metre. 23 m 110 125 40 85 X 28 m 17 m 30 m D Think 1 pply the osine rule in DX to lulte the length of. Write For DX: x 2 = 2 + 2-2 os X = 30 2 + 23 2-2 30 23 os 110 = 1900.99 x = 43.6 m The length of is 43.6 m. 124 Mths Quest Generl Mths HS ourse
2 pply the osine rule in DX to lulte the length of. 3 pply the osine rule in DXD to lulte the length of D. 4 pply the osine rule in DDX to lulte the length of D. 5 lulte the perimeter y dding the length of eh side nd rounding the nswer to the nerest metre. For DX: x 2 = 2 + 2-2 os X = 17 2 + 30 2-2 17 30 os 40 = 407.63 x = 20.2 m The length of is 20.2 m. For DXD: x 2 = 2 + d 2-2d os X = 28 2 + 17 2-2 28 17 os 85 = 990.03 x = 31.5 m The length of D is 31.5 m. For DDX: x 2 = d 2 + 2-2d os X = 23 2 + 28 2-2 23 28 os 125 = 2051.77 x = 45.3 m The length of D is 45.3 m. Perimeter = 43.6 + 20.2 + 31.5 + 45.3 = 140.6 m = 141 m (orret to the nerest metre) similr pproh is used to lulte the re of suh field. The re of eh tringle is found using the formul re = 1 sin. The totl re is then found y dding the re of eh 2 tringle. Worked Exmple 24 lulte the re of the field on the right. Give your nswer orret to the nerest squre metre. 52 m 96 144 120 48 m X 67 m Think Write 1 lulte the re of DX. For DX: re = 1 sin X 2 = 1 2 48 52 sin 96 = 1241.2 m 2 2 lulte the re of DX. For DX: re = 1 sin X 2 67 48 sin 120 = 1 2 = 1392.6 m 2 hpter 3 pplitions of trigonometry 125
3 lulte the re of DX. For DX: re = 1 sin X 2 52 67 sin 144 4 lulte the totl re y dding the re of eh tringle. = 1 2 = 1023.9 m 2 Totl re = 1241.2 + 1392.6 + 1023.9 = 3657.7 m 2 = 3658 m 2 (orret to the nerest m 2 ) n lterntive to the plne tle rdil survey is the ompss rdil survey. In this survey the ering of eh rdil line is lulted nd this ering is used to lulte the ngle etween eh rdil, s in the worked exmple elow. The method of lu lting the perimeter nd re of the field is then the sme s for the plne tle rdil survey. Worked Exmple 25 The figure on the right shows ompss rdil survey of lok of lnd. lulte the size of X. Hene, lulte the distne, orret to the nerest metre. 338 D 239 58 m 55 m X 49 m 57 m 067 114 Think is 22 west of orth, is 67 est of orth. Write 22 + 67 = 89 1 Write the osine rule formul. For DX: x 2 = 2 + 2-2 os X 2 Sustitute for, nd X. = 49 2 + 58 2-2 49 58 os 89 3 lulte the vlue of x 2. = 5665.8 4 lulte x. x = 75 m (orret to the nerest metre) 5 Write your nswer. The distne is 75 m. REMEMER 1. In rdil survey, rdil lines re drwn nd mesured from point in the entre of n re. 2. In plne tle rdil survey, rdil lines re drwn on tle y sighting eh orner of the field. The length of eh line nd the ngle etween the lines is then mesured. 3. ompss rdil survey is similr ut the ering of eh rdil line is mesured. 4. Eh survey divides the re into tringles nd the length of eh oundry n e lulted using the osine rule. 5. The re of eh tringle n e lulted using the formul re = 1 2 sin. 126 Mths Quest Generl Mths HS ourse
100 m Exerise 3H Rdil surveys 1 WE23 The figure elow is plne tle rdil survey of lok of lnd. Use the osine rule to lulte the perimeter of the lok of lnd, orret to the nerest metre. 15 m 20 m 100 70 80 110 10 m 25 m 2 lulte the perimeter of eh of the following res, orret to the nerest metre. 60 m 45 m 100 m 114 m 121 m 55 m 70 85 80 125 45 m 95 115 150 80 m 75 m 92 m 90 40 60 140 30 78 m 89 m 3 WE24 The figure on the right is plne tle survey of lok of lnd. lulte the re of the lok, orret to the nerest squre metre. 90 m 160 60 100 40 80 m 110 m 100 m 4 For eh of the plne tle surveys shown in question 2 lulte the re, orret to the nerest squre metre. 5 WE25 The figure on the right is ompss rdil survey 315 of field. 040 lulte the size of X. Hene, use the osine rule to lulte the distne, X orret to the nerest metre. 50 m 60 m 70 m 6 lulte the perimeter of the field given y the ompss rdil survey on the right. Give your nswer orret to the nerest metre. 350 40 m 170 D 110 250 30 m 90 m 110 hpter 3 pplitions of trigonometry 127
7 lulte the perimeter of eh of the ompss rdil surveys shown elow. 327 030 339 215 53 m 24 m 42 m 38 m 097 114 m 99 m 108 m 319 020 052 226 72 m 29 m 63 m 38 m 49 m 085 eookplus 196 170 Digitl do WorkSHEET 3.2 do-1328 8 For eh of the ompss rdil surveys in question 7 lulte the re, orret to the nerest squre metre. Further development 9 The figure elow is tht of tringulr field. 240 000 20 m 30 m 35 m Find the re of the field orret to 2 deiml ples. Find the perimeter of the field orret to 2 deiml ples. 120 n lterntive formul for the re of tringle is = s( s-)( s-)( s-), where, nd re the side lengths of the tringle nd s = + +. Use this formul nd 2 ompre the result with tht otined in question 9. 10 D is squre field. O is point in the entre of the squre suh tht it is 10 metres from eh orner. Use the osine rule to find eh side length. Find the re of tringle O nd find the re of the squre y multiplying this result y the numer of tringles tht mke up the squre. ompre the nswer to prt to the re of the squre found using = l 2. 11 surveyor, t point S sights two trees, nd, on the opposite side of river s eing distne of 52 m nd 64 m wy on erings of 320 nd 030 respetively. Find the distne orret to 2 deiml ples. Find the re of the tringle S orret to 2 deiml ples. Use your nswers to nd to find the width of the river. 12 Json is t point. Pet is 100 m from Json on ering 090 t point X. Json oserves tower t point on ering 050 nd tree t ering of 150 t Y. Pet notes the ering of the tower s 310 nd tree s 240. Use this informtion to drw 1 : 1000 sketh. Use this digrm to find the length of: i X ii X iii Y iv Y v XY n oserver is t X. Give the ering of: i ii iii Y d lulte the re of XY to the nerest 100 m 2. 128 mths Quest Generl mths hs ourse
13 meron sets up plne tle t O nd from this point mkes the following notes out the surrounding fetures to E. Feture Distne to feture ering of feture from O 25 m 045 35 m 120 20 m 180 D 40 m 250 E 65 m 325 Wht surveying method is meron using? Drw net sketh representing this informtion. Use 1 : 1000 sle. n oserver is t E. From E, wht would you expet to e the erings of,, nd D? d Estimte the re enlosed y the perimeter linking fetures of to E. Express to the nerest 10 m 2. 14 Mrgret ples plne tle diretly over point nd sights, 60 m from her on ering of 020, nd D, 50 m wy on ering of 100. She moves the plne tle to nd notes the ering of, 70 m from her, to e 090. Drw net 1 : 1000 sketh showing this informtion. Mrgret then moves the pln tle to. Wht ering is expeted of: i D ii? Wht is the distne from to: i ii D? d Determine the re of D to the nerest 100 m 2. onduting rdil survey hoose n pproprite re in or ner your shool to ondut rdil survey. 1 Set up tle in the entre of the re nd tpe lrge piee of pper to the tle. 2 Mrk point in the middle of the piee of pper nd sight eh orner of the field from this point, ruling line from the point in tht diretion. 3 Use tpe or trundle wheel to mesure the distne from the tle to eh orner of the field. 4 Use your protrtor to mesure the ngle etween eh rdil line. 5 lulte the re nd the perimeter of the field. hpter 3 pplitions of trigonometry 129
Summry Review of right-ngled tringles The formuls to e used when solving right-ngled tringles re: sin q = os q = opposite side hypotenuse djent side hypotenuse opposite side tn q = djent side To lulte side length, you need to e given the length of one other side nd one ngle. To lulte the size of n ngle, you need to e given two side lengths. If question is given s prolem, egin y drwing digrm nd give written nswer. erings erings re mesure of diretion. ompss ering uses the four min points of the ompss, north, south, est nd west, s well s the four intermedite diretions, north-est, north-west, south-est, south-west. More speifi diretions re given using true erings. true ering desries diretion s three-digit ngle tken in lokwise diretion from north. Most ering questions will require you to drw digrm to egin the question nd require written nswer. Sine rule The sine rule llows us to lulte sides nd ngles in non-right-ngled tringles. When finding side length you need to e given the length of one other side nd two ngles. The sine rule formul is sin = sin = sin. When finding n ngle you need to e given two side lengths nd one ngle. sin sin sin The sine rule formul when finding n ngle is = =. re of tringle When you do not know the perpendiulr height of tringle, you n lulte the re using the formul re = 1 2 sin. To lulte the re using this formul, you need to e given the length of two sides nd the inluded ngle. osine rule The osine rule llows you to lulte the length of sides nd size of ngles of non-right-ngled tringles where you re unle to use the sine rule. To find side length using the osine rule, you need to e given the length of two sides nd the inluded ngle nd use the formul 2 = 2 + 2-2 os. To find n ngle using the osine rule, you need to e given the length of ll three sides nd use the formul os 2 + = 2-2. 2 130 Mths Quest Generl Mths HS ourse
Surveying plne tle rdil survey sights eh orner of field nd drws rdil line in tht diretion. This divides the field into tringles. The length of eh rdil line nd the ngle etween rdil lines re then mesured. The osine rule n then e used to lulte the length of eh oundry. The formul re = 1 2 sin n e then used to lulte the re of the field. ompss rdil survey tkes the ering of eh rdil line nd this is then used to lulte the ngles etween them. 338 067 58 m 49 m 20 m 115 60 115 70 26 m 25 m 28 m D 239 55 m X 57 m 114 hpter 3 pplitions of trigonometry 131
hpter review Multiple hoie 1 In the figure elow, whih of the following will give the vlue of x? 64 13 m 4 The figure elow is ompss rdil survey. X is: 80 35 305 55 85 X D 135 x 36 174 13 sin 36 x = sin 64 13 sin 64 x = sin 36 13 sin 64 x = sin 80 13 sin 80 D x = sin 64 2 In the figure elow, whih of the following will give the vlue of os q? 6 m 7 m Short nswer 1 Find the length of the side mrked with the pronumerl in eh of the right-ngled tringles elow, orret to 1 deiml ple. 17.2 m x 60 7.9 m 42 km t 72 6 + 7-8 os q = 2 6 7 6 + 8-7 os q = 2 6 8 7 + 8-6 os q = 2 7 8 6 + 7-8 D os q = 2 7 8 2 2 2 2 2 2 2 2 2 2 2 2 8 m 3 Murie wlks 3 km on true ering of 225. To return to his strting point he must wlk on ompss ering of: north-est north-west south-est D south-west 45 m 2 In eh of the following right-ngled tringles, find the size of the ngle mrked with the pronumerl, orret to the nerest degree. 16 m 35 m φ 9 m 24.8 m 20.1 km 8.3 km α 132 Mths Quest Generl Mths HS ourse
3 n eroplne t n ltitude of 2500 m sights ship t n ngle of depression of 39. lulte, to the nerest metre, the horizontl distne from the eroplne to the ship. 4 When yht is 500 m from shore, the top of liff is sighted t n ngle of elevtion of 12. 4.6 km d 31 28 136 mm e 117 19 9 In DXYZ: x = 9.2 m, XYZ = 56 nd YXZ = 38. Find y, orret to 1 deiml ple. 10 Use the sine rule to lulte the size of the ngle mrked with pronumerl, orret to the nerest degree. 8 m 9 m lulte the height of the liff, orret to the nerest metre. lulte wht the ngle of elevtion of the top of the liff will e when the yht is 200 m from shore. 5 Two irrft re pprohing n irport. The Qnts plne (Q) is 40 km due north of the runwy (R), while Jetstr plne (J) is due est of the Qnts plne nd north-est of the runwy. lulte the distne of the Jetstr plne from the runwy. (Give your nswer orret to the nerest metre.) 6 r rlly requires rs to trvel for 25 km on ering of 240. The rs re then required to trvel due north until they re due west of the strting point. lulte the distne from the rs to the strting point. (Give your nswer orret to 1 deiml ple.) 7 yht sils due west for 45 nutil miles efore turning north for 23 nutil miles. lulte the ering of the yht from its strting point. On wht ering must the yht sil to return to its strting point? 8 Use the sine rule to lulte eh of the sides mrked with pronumerl, orret to 3 signifint figures. 70 4.6 m 20 63 α 7.1 m 9 φ 1.2 m 9.7 m 123 4.1 m 11 In D: = 46 m, = 37 m nd = 72. Find the re of the tringle, orret to the nerest squre entimetre. 12 Find the re of tringulr field with two sides of 80 m nd 98 m, whih meet t n ngle of 130 (orret to the nerest hundred squre metres). 13 Use the osine rule to find eh of the following unknown sides, orret to 3 signifint figures. 50 9 m 11 m hpter 3 pplitions of trigonometry 133
4.2 m 5.3 m 5.7 m 7.9 m 6.2 m 117 4.6 m 128 6.9 m 14 In DLM: LM = 63 m, M = 84 m nd LM = 68. Find the length of L, orret to 1 deiml ple. 15 During stunt show two eroplnes fly side y side until they suddenly diverge t n ngle of 160. fter oth plnes hve flown 500 m wht is the distne etween the plnes, orret to the nerest metre? 9 m 7 m 15 m 17 In DXYZ: x = 8.3 m, y = 12.45 m nd z = 7.2 m. Find YZX, to the nerest degree. 18 Two wooden fenes re 50 m nd 80 m long respetively. Their ends re onneted y red wire fene 44 m long. Find the ngle t whih the two wooden fenes meet. 19 The figure elow is plne tle rdil survey of field. 60 m 80 120 50 110 30 m 40 m 70 m Use the osine rule to lulte the perimeter of the field. lulte the re of the field. 20 The figure elow is ompss rdil survey. 340 16 Use the osine rule to find the size of the ngle in eh of the following, orret to the nerest degree. 6 m 6 m 6 m 260 150 m 90 m 140 m 190 160 m 080 lulte the perimeter of the field. lulte the re of the field. extended ReSponSe 1 The distne etween footll gol posts is 7 m. If Soon Ho is 20 m from one gol post nd 25 m from the other: drw digrm showing the gol posts nd Soon Ho s position. lulte the ngle within whih Soon Ho must kik to sore gol. (Give your nswer orret to the nerest degree.) 134 mths Quest Generl mths hs ourse