on-right-ngled trigonometry 4Syllus topi M6 on-right-ngled trigonometry This topi is foused on solving prolems involving right- nd non-right-ngled tringles in vriety of ontets. Outomes Solve prolems using trigonometri rtios in one or more right-ngled tringles. Solve prolems involving ngles of elevtion nd depression. Solve prolems involving ompss nd true erings. Determine the sign of trigonometry rtios involving otuse ngles. lulte the re of tringle given two sides nd n inluded ngle. Use the sine rule to find lengths nd ngles. Use the osine rule to find lengths nd ngles. Use trigonometry to solve vriety of prtil prolems. onstrut nd interpret ompss rdil surveys nd solve relted prolems. Investigte nvigtionl methods used y other ultures. Digitl Resoures for this hpter In the Intertive Tetook: Videos Litery worksheet Quik Quiz Solutions (enled y teher) Desmos widgets Spredsheets Study guide In the Online Tehing Suite: SMPLE PGES Tehing Progrm Tests Review Quiz Tehing otes Knowledge hek The Intertive Tetook provides test of prior knowledge for this hpter, nd my diret you to revision from the previous yers work.
148 hpter 4 on-right-ngled trigonometry 4 4 Right-ngled trigonometry Trigonometri rtios re defined using the sides of right-ngled tringle. The hypotenuse is opposite the right ngle, the opposite side is opposite the ngle nd the djent side is the shorter side net to. The trigonometri rtios sin, os nd tn re defined using the sides of right-ngled tringle. sin = opposite os = djent hypotenuse hypotenuse Emple 1: Finding n unknown side in right-ngled tringle Find the length of the unknown side in the tringle shown. nswer orret to three deiml ples. 35 SOLUTIO: 20 1 Identify the sides of the right-ngled tringle tht The opposite side is the unknown, the re relevnt to the question. djent side is 20. 2 Determine the rtio tht uses these sides (TO). tn = o 3 Sustitute the known vlues nd for o. tn 35 = 20 4 To mke the sujet, multiply oth sides of the 20 tn 35 = eqution y 20. = 20 tn 35 5 Press 20 tn 35 ee or =. 6 Write the nswer orret to three deiml ples. tn = opposite djent sin = o h (SOH) os = h (H) tn = o (TO) In trigonometry n ngle is usully mesured in degrees, minutes nd seonds suh s 31 05 51 (31 degrees 5 minutes 51 seonds). It is importnt to rememer tht one degree is equl to 60 minutes (1 = 60 ) nd one minute is equl to 60 seonds (1 = 60 ). lultors my lso mesure ngles in deiml degrees suh s 31.0975, nd utton like this or this DMS is provided to onvert etween deiml degrees nd degrees/minutes/seonds. Mke sure lso tht the degree mode (not rdin mode) is seleted on your lultor. RIGHT-GLED TRIGOOMETRY The mnemoni SOH H TO is pronouned s single word. SOH: Sine-Opposite-Hypotenuse H: osine-djent-hypotenuse TO: Tngent-Opposite-djent The order of the letters mthes the rtio of the sides. Hypotenuse djent = 14.00415076 = 14.004 Opposite SMPLE PGES 4
4 Right-ngled trigonometry 149 Emple 2: Finding n unknown ngle in right-ngled tringle 4 Find the ngle in the tringle shown. nswer orret to the nerest minute. SOLUTIO: 1 Identify the relevnt sides of the right-ngled tringle. 2 Determine the rtio tht uses these sides (SOH). 3 Sustitute the known vlues. 4 Mke the sujet of the eqution. 5 Press SHIFT sin -1 (13 22) ee or =. or Press SHIFT sin -1 13 22 ee or =. 6 onvert deiml degrees to degrees nd minutes using or DMS on your lultor. 7 Write the nswer orret to the nerest minute. Emple 3: pplition requiring the length of side vertil tent pole is supported y rope tied to the top of the pole nd to peg on the level ground. The peg is 2.5 m from the se of the pole nd the rope mkes n ngle of 63 to the horizontl. Wht is the length of the rope etween the peg nd the top of the tent pole? nswer orret to two deiml ples. SOLUTIO: 1 Drw digrm nd lel the required rope length s. 13 Opposite = 13, hypotenuse = 22 sin = o h sin = 13 22 = sin 1 13 ( 22) = 36.22154662 = 36 13 22 63 2.5 m 63 2.5 m SMPLE PGES 2 Identify the relevnt sides of the right-ngled tringle. djent is 2.5 m, hypotenuse is the unknown. 3 Determine the rtio tht uses these sides (H). os = h 4 Sustitute the known vlues nd for. os 63 = 2.5 4
Overmtter Pge 149 5 Multiply oth sides of the eqution y. os 63 = 2.5 6 Divide oth sides of the eqution y os 63, so is now = 2.5 os 63 the sujet of the eqution. 7 Press 2.5 os 63 ee or =. = 5.506723161 8 Write the nswer orret to two deiml ples. = 5.51 9 Write the nswer in words. Length of the rope is 5.51 m. SMPLE PGES
150 hpter 4 on-right-ngled trigonometry 4 Eerise 4 LEVEL 1 Emple 1 Emple 2 1 Find the vlue of the following trigonometri rtios, orret to two deiml ples. Your lultor must e in degree mode nd needs to hve degrees/minutes/seonds utton whih you press fter entering the degrees, so you n then enter the minutes. sin 33 os 70 2 tn 51 d 7 sin 24 e os 42 15 f tn 55 28 g 2 os 62 59 h 8 sin 15 42 2 Given the following trigonometri rtios, find the vlue of to the nerest degree. Rememer the sin 1, os 1 nd tn 1 uttons give ngles from the trigonometri rtio. sin = 0.4712 os = 0.3412 tn = 0.3682 d tn = 5 8 e sin = 1 2 f os = 1 4 3 Find the length of the unknown side in eh tringle, orret to two deiml ples. 14 28 28 43 28 30 d 35 12 6 e 24 5 40 f 75 33 58 4 Find the unknown ngle in eh tringle. nswer orret to the nerest minute. 26 21 17 19 19 48 d 7.8 SMPLE PGES 5.2 e 65.1 38.9 f 6.8 4.2
4 Right-ngled trigonometry 151 5 Find the length of the unknown side in eh tringle, orret to two deiml ples. 21 47 6 Find the unknown ngle in eh tringle. nswer orret to the nerest degree. 12 9 15 16 21 66 30 34 7 lloon is tied to string 25 m long. The other end of the string is seured y peg to the surfe of level sports field. The wind lows so tht the string forms stright line mking n ngle of 37 with the ground. Find the height of the lloon ove the ground. nswer orret to one deiml ple. 8 pole is supported y wire tht runs from the top of the pole to point on the level ground 5 m from the se of the pole. The wire mkes ngle of 42 with the ground. Find the height of the pole, orret to two deiml ples. 9 nn notied tree ws diretly opposite her on the fr nk of the river. fter she wlked 50 m long the side of the river, she found her line of sight to the tree mde n ngle of 39 with the river nk. Find the width of the river, to the nerest metre. 10 ship t nhor requires 70 m of nhor hin. If the hin is inlined t 35 to the horizontl, find the depth of the wter, orret to one deiml ple. 72 20 Tree 29 LEVEL 2 39 21 25 m 37 42 5 m 70 m 35 39 50 m SMPLE PGES Ship nn
152 hpter 4 on-right-ngled trigonometry 4 Emple 3 11 pole sts shdow of 5.4 m long. The sun s rys mke n ngle of 36 with the level ground. Find the height of the pole to the nerest metre. 12 vertil tent pole is supported y rope of length 3.6 m tied to the top of the pole nd to peg on the ground. The pole is 2 m in height. Find the ngle the rope mkes to the horizontl. nswer orret to the nerest degree. 36 5.4 m 13 3 m ldder hs its foot 1.5 m out from the se of wll. Wht ngle does the ldder mke with the ground? nswer orret to the nerest degree. 14 shooter, 80 m from trget nd level with it, ims 2 m ove the ullseye nd hits it. Wht is the ngle, to the nerest minute, tht his rifle is inlined to the line of sight from his eye to the trget? LEVEL 3 15 rope needs to e fied with one end tthed to the top of 6 m vertil pole nd the other end pegged t n ngle of 65 with the level ground. Find the required length of rope. nswer orret to one deiml ple. 16 wheelhir rmp is eing provided to llow ess to first floor shops. The first floor is 3 m ove the ground floor. The rmp requires n ngle of 20 with the horizontl. How long will the rmp e? nswer orret to two deiml ples. 17 plne mintins flight pth of 19 with the horizontl fter it tkes off. It trvels for 4 km long the flight pth. Find, orret to one deiml ple: the horizontl distne of the plne from its tke-off point. the height of the plne ove ground level. 18 Two ldders re the sme distne up the wll. The shorter ldder is 5 m long nd mkes n ngle of 50 with the ground. The longer ldder is 7 m long. Find: 7 m the distne, orret to two deiml ples, the ldders re up the wll the ngle, to the nerest degree, the longer ldder mkes with the ground. 19 pole is supported y wire tht runs from the top of the pole to point on the level ground 7.2 m from the se of the pole. The height of the pole is 5.6 m. Find the ngle, to the nerest degree, tht the wire mkes with the ground. 2 m 50 3.6 m SMPLE PGES 5 m Sun
4 ngles of elevtion nd depression 153 4 ngles of elevtion nd depression The ngle of elevtion is the ngle mesured upwrds from the horizontl. The ngle of depression is the ngle mesured downwrds from the horizontl. GLE OF ELEVTIO ngle of elevtion Horizontl The ngle of elevtion is equl to the ngle of depression s they form lternte ngles etween two prllel lines. This informtion is useful to solve some prolems. Emple 4: ngle of elevtion prk rnger mesured the top of plume of volni sh to e t n ngle of elevtion of 41. From her mp she noted tht the volno ws 7 km wy. lulte the height of the plume of volni sh. nswer orret to two deiml ples. SOLUTIO: 1 Drw digrm nd lel the required height s. GLE OF DEPRESSIO Horizontl ngle of depression 41 7 km 2 Determine the rtio (TO). tn = o 3 Sustitute the known vlues. 4 Multiply oth sides of the eqution y 7. ngle of depression tn 41 = 7 7 tn 41 = = 7 tn 41 41 7 km ngle of elevtion 5 Write the nswer orret to two deiml ples. = 6.085007165 = 6.09 6 Write the nswer in words. The height of the volni plume ws 6.09 km. 4 Volni plume SMPLE PGES
154 hpter 4 on-right-ngled trigonometry 4 Emple 5: Finding distne using ngle of depression 4 The top of liff is 85 m ove se level. Minh sw tll ship. He estimted the ngle of depression to e 17. How fr ws the ship from the se of the liff? nswer to the nerest metre. How fr is the ship in stright line from the top of the liff? nswer to the nerest metre. SOLUTIO: 1 Drw digrm nd lel the distne to the se of the liff s nd the distne to the top of the liff s y. 85 m 2 Determine the rtio (TO). tn = o 3 Sustitute the known vlues. 4 Multiply oth sides of the eqution y. 5 Divide oth sides y tn 17. 6 Write the nswer orret to nerest metre. 7 Write the nswer in words. 17 y 17 tn 17 = 85 tn 17 = 85 = 85 tn 17 = 278.022... 278 m 8 Determine the rtio (SOH). sin = o h The ship is 278 metres from the se of the liff. 9 Sustitute the known vlues. 10 Multiply oth sides of the eqution y y. 11 Divide oth sides y sin 17. 12 Write the nswer orret to nerest metre. 13 Write the nswer in words. sin 17 = 85 y y sin 17 = 85 y = 85 sin 17 = 290.7258... 291 m SMPLE PGES The ship is 291 metres from the top of the liff.
4 ngles of elevtion nd depression 155 Eerise 4 LEVEL 1 Emple 4 Emple 5 1 Luke wlked 400 m wy from the se of tll uilding, on level ground. He mesured the ngle of elevtion to the top of the uilding to e 62. Find the height of the uilding. nswer orret to the nerest metre. 2 The ngle of depression from the top of TV tower to stellite dish ner its se is 59. The dish is 70 m from the entre of the tower s se on flt lnd. Find the height of the tower. nswer orret to one deiml ple. 3 When Srh looked from the top of liff 50 m high, she notied ot t n ngle of depression of 25. How fr ws the ot from the se of the liff? nswer orret to two deiml ples. 4 The pilot of n eroplne sw n irport t se level t n ngle of depression of 13. His ltimeter showed tht the eroplne ws t height of 4000 m. Find the horizontl distne of the eroplne from the irport. nswer orret to the nerest metre. 4000 m 5 The ngle of elevtion to the top of tree is 51 t distne of 45 m from the point on level ground diretly elow the top of the tree. Wht is the height of the tree? nswer orret to one deiml ple. 6 iron ore sem 120 m slopes down t n ngle of depression from the horizontl of 38. The mine engineer wishes to sink vertil shft,, s shown. Wht is the depth of the required vertil shft? nswer orret to the nerest metre. 7 Jk mesures the ngle of elevtion to the top of tree from point on level ground s 35. Wht is the height of the tree if Jk is 50 m from the se of the tree? nswer to the nerest metre. 50 m 13 70 m 25 59 51 45 m 62 400 m 38 120 m SMPLE PGES 35 50 m
156 hpter 4 on-right-ngled trigonometry 4 8 tourist viewing Sydney Hrour from uilding 130 m ove se level oserves ferry 800 m from the se of the uilding. Find the ngle of depression. nswer orret to the nerest degree. 9 Wht would e the ngle of elevtion to the top of rdio trnsmitting tower 130 m tll nd 300 m from the oserver? nswer orret to the nerest degree. 10 Lhln oserves the top of tree t distne of 60 m from the se of the tree. The tree is 40 m high. Wht is the ngle of elevtion to the top of the tree? nswer orret to the nerest degree. 11 town is 12 km from the se of mountin. The town is lso distne of 12.011 km in stright line to the top of the mountin. Wht is the ngle of depression from the top of mountin to the town? nswer orret to the nerest tenth of degree. 12 Find, to the nerest degree, the ngle of elevtion of rilwy line tht rises 7 m for every 150 m long the trk. 13 The distne from the se of the tree is 42 m. The tree is 28 m in height. Wht is the ngle of elevtion mesured from ground level to the top of tree? nswer orret to the nerest degree. 14 heliopter is flying 850 m ove se level. It is lso 1162 m in stright line to ship. Wht is the ngle of depression from the heliopter to the ship? nswer orret to the nerest degree. 800 m 15 The ngle of elevtion to the top of tree from point on the ground is 25. The point is 22 m from the se of the tree. Find the height of the tree. nswer orret to nerest metre. 16 plne is 460 m diretly ove one end of 1200 m runwy. Find the ngle of depression to the fr end of the runwy. nswer orret to the nerest minute. 40 m 130 m 12 km 150 m 300 m 60 m 12.011 km 42 m 130 m 7 m 28 m 850 m 1162 m SMPLE PGES
4 ngles of elevtion nd depression 157 17 roket lunhing pd sts shdow 35 m long when the ngle of elevtion of the sun is 52. How high is the top of the lunhing pd? nswer orret to nerest metre. LEVEL 2 18 ommunition tower is loted on the top of hill. The ngle of elevtion to the top of the hill from n oserver 2 km wy from the se of the hill is 6. The ngle of elevtion to the top of the tower from the oserver is 8. Find, to the nerest metre, the height of the: hill hill nd the tower tower. 19 Jk is on the top of 65 m high liff. He oserves mn swimming out to se t n ngle of depression of 51. Jk lso sees ot out to se t n ngle of depression of 30. Find, to the nerest metre, the distne: of the mn from the se of the liff y of the ot from the se of the liff from the mn to the ot. 20 lighthouse stnds on the top of liff. siling ot is loted 75 m from the se of the liff. The ngle of elevtion to the top of the liff nd the top of the lighthouse is 42 nd 50 respetively. Find the height of the lighthouse, orret to the nerest metre. 21 plne is flying t n ltitude of 880 m. Kyl is stnding on the ground, she oserves the ngle of elevtion to the plne s 67 40 nd 25 seonds lter the ngle of elevtion hd hnged to 24 30. How fr hd the plne flown in tht time? nswer to the nerest metre. Wht is the speed (to the nerest km/h) of the plne? LEVEL 3 SMPLE PGES 22 The ngle of elevtion from ot out to se to the top of 350 m liff is 13. fter the ot trvels diretly towrds the liff, the ngle of elevtion from the ot is 19. How fr did the ot trvel towrds the liff? nswer orret to the nerest metre. 65 m 51 30 y 75 m 880 m
158 hpter 4 on-right-ngled trigonometry 4 4 ompss nd true erings ering is the diretion one ojet is from nother ojet or n oserver or fied point. There re two types of erings: ompss erings nd true erings. ompss erings ompss erings use the four diretions of the ompss: north, est, south nd west (, E, S nd W). The S line is vertil nd the EW line W is horizontl. In-etween these diretions re nother four diretions: north-est, south-est, south-west nd north-west (E, SE, SW nd W W). Eh of these diretions mkes n ngle of 45 with the S nd EW lines. diretion is given using ompss ering y stting the ngle either SW side of north or south. For emple, ompss ering of S50 W is found y mesuring n ngle of 50 from the south diretion towrds the west side. Emple 6: Understnding ompss ering Find the ompss ering of: from O from O. SOLUTIO: 1 Determine the qudrnt of the ompss ering. The line O is in the north/est qudrnt. 2 Find the ngle the diretion mkes with the vertil (north/south) line. 3 Write the ompss ering with or S first, then the ngle with the vertil line nd finlly either E or W. 30 4 ompss ering of from O is 30 E. SMPLE PGES 4 Determine the qudrnt of the ompss ering. The line O is in the south/est qudrnt. 5 Find the ngle the diretion mkes with the vertil (north/south) line. 6 Write the ompss ering with or S first, then the ngle with vertil line nd finlly either E or W. W O 180 120 = 60 ompss ering of from O is S60 E. S S 30 120 E SE E E
4 ompss nd true erings 159 True erings true ering is the ngle mesured lokwise from north round to the required diretion, nd it is written with the letter T fter the degree or minutes or seonds symol. True erings re sometimes lled three-figure erings euse they re written using three numers or figures. For emple, 120 T is the diretion mesured 120 lokwise from north. It is the sme ering s S60 E. The smllest true ering is 000 T nd the lrgest true ering is 360 T. The eight diretions of the ompss hve the following true erings: north is 000 T, est is 090 T, south is 180 T, west is 270 T, north-est is 045 T, south-est is 135 T, south-west is 225 T nd north-west is 315 T. The erings in the following digrms re given using oth methods. W 10 S 170 P OMPSS ERIG E diretion given y stting the ngle either side of north or south, suh s S60 E. Emple 7: Understnding true ering Find the true ering of: from O D from O. ompss ering S 10 E True ering 170 T TRUE ERIG 120 diretion given y mesuring the ngle lokwise from north to the required diretion, suh s 120 T. SOLUTIO: S 1 Find the ngle the ering mkes in the lokwise 210 diretion with the north diretion. 2 Write the true ering using this ngle. dd the letter T. from O is 210 T. 3 Write the true ering of west. 4 dd ngle etween west nd D to true ering for west. 270 T 270 + 60 = 330 5 Write the true ering using this sum. dd the letter T. D from O is 330 T. W P 30 S 330 W E D 60 O 210 lokwise from north ompss ering 30 W True ering 330 T SMPLE PGES 4 E
160 hpter 4 on-right-ngled trigonometry 4 Eerise 4 LEVEL 1 Emple 6, 7 1 Stte the ompss ering nd the true ering of eh of the following diretions. E W SE d SW 2 Stte the ompss ering nd the true ering of eh of the following diretions. d W W g j 59 33 S S 52 240 E E e h k W W 78 63 S S 283 20 SMPLE PGES E E f W W i 3 Sketh eh of these erings on seprte digrm. 10 E S 25 W 60 W d S42 E e 300 T f 105 T g 219 T h 050 T 4 ron runs distne of 7.2 km in the SE diretion. How fr est hs ron run? nswer orret to one deiml ple. l 250 98 35 W S S 55 45 SE S E E E
4 ompss nd true erings 161 5 plne is trvelling on true ering of 030 from to. Wht is the ompss ering of to? Wht is the true ering of to? Wht is the ompss ering of to? 6 The digrm shows the position of P, Q nd R reltive to S. In the digrm, R is E of S, Q is W of S nd PSR is 155. Wht is the true ering of R from S? Wht is the true ering of Q from S? Wht is the true ering of P from S? 7 The ering of E from D is 38 E, F is est of D nd DEF is 87 Find the vlues of nd y. Wht is the ompss ering of E from F? Wht is the true ering of E from F? 8 Riley trvels from X to Y for 125 km on ering of 32 W. How fr did Riley trvel due north, to the nerest kilometre? How fr did Riley trvel due west, to the nerest kilometre? 9 Mi yled for 15 km west nd then 24 km south. Wht is the vlue of to the nerest degree? Wht is Mi s true ering from her strting point? Wht is Mi s ompss ering from her strting point? 38 D Q LEVEL 2 SMPLE PGES W Y E 87 32 S P X 24 S y 15 155 R F E 10 ot sils 137 km from Port Stephens on ering of 065 T. How fr est hs the ot siled? nswer orret to one deiml ple. How fr north hs the ot siled? nswer orret to one deiml ple.
162 hpter 4 on-right-ngled trigonometry 4 11 ship sils 5 kilometres west, then 5 kilometres south. Wht is the ompss ering of the ship from its originl position? Wht is the true ering of the ship from its originl position? 12 Hrry trvelled for 8.5 km on ering of S 30 W from his home. How fr west is Hrry from home? nswer orret to two deiml ples. How fr south is Hrry from home? nswer orret to two deiml ples. Wht is the ompss ering of his home from his urrent position? LEVEL 3 13 ylist rides 15 km on ering of 290. How fr north is the ylist? nswer orret to one deiml ple. How fr west is the ylist? nswer orret to one deiml ple. Wht is the true ering of the ylist s strting position from the ylist s urrent position? 14 meli wlked from point for 1 hour in the diretion 51 E to reh point. She then wlked 2 hours heding south pst until she ws t point D. meli ws wlking t onstnt speed of 5 km/h for the entire journey. nswer the following questions, orret to two deiml ples where neessry. Wht ws the distne wlked from to? Wht ws the distne wlked from to? 51 Wht ws the distne wlked from to D? W E d lulte the distne etween nd. e lulte the distne etween nd D. D f Wht is the ompss ering needed for meli to return to her strting point? S 15 plne left from O nd trvelled 350 km in the diretion 225 T to reh P. It then hnged diretion nd trvelled due north for 500 km to reh point. Wht ws the distne from P to M? (nswer orret to two deiml ples.) Wht ws the distne from M to O? (nswer orret to two deiml ples.) O W Wht ws the distne from M to? (nswer orret to M two deiml ples.) d Wht is the ngle, orret to the nerest degree? P e Wht is the true ering of from O? S SMPLE PGES 16 Osr drives t speed of 80 km/h on ering of 125 T for 2.5 hours. How fr is Osr est of his strting position? nswer to the nerest kilometre. E
4D Trigonometry with otuse ngles 163 4D Trigonometry with otuse ngles When the ngle is ute (0 to 90 ) ll the trigonometri rtios re positive. However, when the ngle is otuse (90 to 180 ) the sine rtio is positive, the osine rtio is negtive nd the tngent rtio is negtive. These results re stored in the lultor s memory. For emple, pressing os 120 will result in 0.5, s 120 is n otuse ngle nd the osine rtio is negtive. However, y pressing sin 120 the result is 0.866, s the sine rtio for n otuse ngle is positive. UTE GLE (0 TO 90 ) OTUSE GLE (90 TO 180 ) sin positive os positive tn positive sin positive os negtive tn negtive Emple 8: Finding trigonometri rtio of n otuse ngle Find the vlue of the following otuse ngles, orret to two deiml ples. sin 164.25 tn 124 30 SOLUTIO: 1 Press sin 164.25 ee or =. sin 164.25 = 0.2714404499 = 0.27 2 Press tn 124 ee (or DMS) tn 124 30 = 1.455009029 30 ee or =. = 1.46 Emple 9: Finding n otuse ngle from trigonometri rtio Given os = 0.5178, find the vlue of to the nerest degree. Given tn = 1.32, find the vlue of to the nerest minute. SOLUTIO: 1 Press SHIFT os 1 0.5178 ee or =. os = 0.5178 = 121 SMPLE PGES 2 Press SHIFT tn 1 1.32 ee or =. 3 To find the otuse ngle, dd 180 to the negtive ngle ( 52.85). 4 onvert the nswer to minutes y using the or DMS. tn = 1.32 = 52.8533133 = 180 52.8533133 = 127.1466867 = 127 09 4D 4D
164 hpter 4 on-right-ngled trigonometry 4D Eerise 4D LEVEL 1 Emple 8 Emple 9 1 Without using lultor, stte whether these rtios re positive or negtive. tn 134 sin 92 os 153 d tn 178 e os 142 30 f sin 100 10 g os 92 46 h tn 125 54 2 Find the vlue of the following trigonometri rtios, orret to two deiml ples. sin 140 os 91 tn 115 d os 129 e tn 99 f sin 174 g os 156 h tn 168 3 Find the vlue of the following trigonometri rtios, orret to one deiml ple. sin 90 09 os 147 20 tn 173 53 d os 102 35 e sin 92 51 f tn 123 54 g tn 148 7 h os 107 17 4 Find the vlue of the following trigonometri rtios, orret to two deiml ples. 5 tn 149 3 sin 105 10 os 101 d 12 tn 122 e 3 sin 132 f 8 os 153 g 7 tn 163 h 9 os 108 5 Find the vlue of the otuse ngle to the nerest degree. os = 0.4625 tn = 0.6582 sin = 0.6291 d tn = 3 4 e os = 3 f sin = 1 8 2 6 Find the vlue of the otuse ngle to the nerest minute. tn = 1.7356 os = 0.5196 sin = 0.6456 d os = 1 5 e tn = 1 1 f sin = 4 3 7 7 Find the vlue of the otuse ngle to the nerest degree. os = 0.45 tn = 1.85 SMPLE PGES sin = 0.83 d tn = 1 6 e os = 5 f sin = 1 7 4 8 The osine rtio of n otuse ngle is 0.7. Wht is the vlue of, orret to the nerest minute?
4D Trigonometry with otuse ngles 165 9 Find the vlue of the following trigonometri rtios, orret to two deiml ples. 5 sin 36 sin 98 d g 11 sin 139 sin 28 8 sin 17 25 sin 120 5 7 sin 25 sin 138 e 6 sin 128 5 sin 8 9 h 3 sin 77 4 sin 90 19 10 Find the vlue of the otuse ngle to the nerest degree. tn = 1 3 sin = 3 2 e tn = 7 3 os = 2 5 d os = 3 2 f sin = 1 2 11 Find the vlue of the otuse ngle to the nerest minute. os = 2 tn = 11 10 2 sin = 5 4 e os = 3 11 d tn = 3 3 f sin = 10 8 12 Find the vlue of the otuse ngle to the nerest degree. tn = 2 : 3 os = 1 : 5 tn = 9 : 8 d = os 1 ( 2 : 9) e = tn 1 ( 1 : 2) f = os 1 ( 5 : 6) 4 sin 156 sin 34 f 2 sin 109 28 sin 23 1 13 Given tht os = 0.8 nd ngle is etween 90 nd 180, find the vlue of: to the nerest degree tn, orret to two deiml ples sin, orret to one deiml ple. 14 Given tht tn = 2.3 nd ngle is etween 90 nd 180, find the vlue of: to the nerest minute os, orret to three deiml ples sin, orret to four deiml ples. LEVEL 2 LEVEL 3 SMPLE PGES
166 hpter 4 on-right-ngled trigonometry 4E 4E re of tringle The re of tringle n e lulted in tringle without right ngle. It requires tht two sides nd the inluded ngle re known. The re of tringle is hlf the produt of two sides multiplied y the sine of the ngle etween the two sides (inluded ngle). This result is derived y onstruting perpendiulr line of length h from to D. re of tringle is lulted using the formul: = 1 h 2 In ΔD, sin = h h = sin h Sustituting sin for h into = 1 h 2 D = 1 sin 2 = 1 sin 2 Similrly, y onstruting perpendiulrs from nd, we n otin the other two results elow. RE OF TRIGLE = 1 sin 2 Side = 1 sin 2 = 1 sin 2 ngle Side re of tringle is hlf the produt of two sides multiplied y the sine of the ngle etween the two sides (inluded ngle). Emple 10: Finding the re of tringle Find the re of the tringle to the nerest squre entimetre. SOLUTIO: 36 1 Write the formul. = 1 sin 2 2 We re given two sides, nd the ngle etween these sides (inluded ngle). 3 Sustitute vlues for, nd. 32 20 sin 36 4 Evlute. 5 Write, orret to the nerest squre entimetre. = 1 2 = 188.0912807 = 188 m 2 20 m SMPLE PGES 32 m 4E
4E re of tringle 167 Eerise 4E LEVEL 1 Emple 10 1 Find the re of eh tringle, orret to one deiml ple. d 7 53 4 m 8 65 5 m e 26 46 19 41 m 31 28 m f 16 mm 22 109 13 mm 69 2 In tringle, side is 36 m, side is 48 m nd ngle is 68. Find the re of the tringle. nswer orret to two deiml ples. 3 In tringle XYZ, side is 4 m, side y is 7 m nd ngle Z is 34. Find the re of the tringle. nswer orret to two deiml ples. D 4 Find the re of tringle DEF if DF = 5 m, EF = 6 m nd 5 m DFE = 40. nswer orret to the nerest squre entimetre. 40 E 6 m F 5 Find the re of eh tringle, orret to one deiml ple. d 19 64 24 73 23 mm 25 mm 23 e 27 68 56 60 m 19 mm 32 m 101 30 8.2 5.1 128 4.3 f 50 2.5 m 2.4 m 36 m 1.6 m SMPLE PGES 30
168 hpter 4 on-right-ngled trigonometry 4E 6 In tringle DEF, the length of DF is 21 m, EF is 28 m, FDE is 64 nd DEF is 43. Find the re of tringle DEF to the nerest squre entimetre. 7 prllelogrm PQRS hs PS = 4 m, SR = 5 m nd PSR = 40. Wht is the re of tringle PRS? nswer orret to one deiml ple. Wht is the re of prllelogrm PQRS? nswer orret to one deiml ple. 8 drwing of frmer s property is shown elow. W 8 km X 84 7 km 114 5 km Y 7.5 km Z Wht is the re of tringle WXZ? nswer orret to one deiml ple. Wht is the re of tringle XYZ? nswer orret to one deiml ple. Find the totl re of the property in squre kilometres. nswer orret to one deiml ple. 9 The tringle hs n re of 243 m 2, = 30.1 m nd = 54 21. Wht is the length of in metres, orret to one deiml ple? 10 rhomus D hs side length of 13 m nd D = 64 35. Wht is the re of the rhomus? Give the nswer orret to the nerest squre entimetre. 21 m LEVEL 2 43 28 m LEVEL 3 SMPLE PGES 11 tringle hs n ngle of 129 56 with one rm of this ngle 15 m long. Wht is the length of the other rm of this ngle if the re of the tringle is 69.94 m 2? nswer orret to the nerest entimetre. S F P D 64 R E Q
4F The sine rule 169 4F The sine rule Trigonometry is lso pplied to non-right-ngled tringles. The sides of the tringle re nmed ording to the opposite ngle. Side is opposite ngle. Side is opposite ngle. Side is opposite ngle. The sine rule reltes the sides nd ngles in tringle. It sttes tht Side divided y the sine of ngle is equl to side divided y the sine of ngle nd is equl to side divided y the sine of ngle. This result is derived y onstruting perpendiulr line of length h from to D. In ΔD, sin = h h = sin In ΔD, sin = h h = sin h The vlues of h must e equl to eh other. sin = sin D sin = sin Similrly, y onstruting perpendiulr line from it n e shown tht sin = sin We n omine these two rules to otin the result elow. THE SIE RULE Sine rule reltes the sides nd ngles in tringle. To find side, use sin = sin = sin sin To find n ngle, use = sin = sin Sine rule is used in non-right-ngled tringle given: two sides nd n ngle opposite one of the given sides, or two ngles nd one side. SMPLE PGES
170 hpter 4 on-right-ngled trigonometry 4F Emple 11: Using the sine rule to find n unknown side Find the vlue of, orret to one deiml ple. SOLUTIO: 1 hek tht the sides nd ngles re opposite eh other. 4.1 37 50 2 Write the sine rule to find side. sin = sin 3 Sustitute the known vlues sin 50 = 4.1 ( =, = 50, = 4.1 nd = 37). sin 37 4.1 sin 50 4 Multiply oth sides of the eqution y sin 50. = sin 37 5 Evlute. = 5.218849806 6 Write the nswer orret to one deiml = 5.2 ple. Emple 12: Using the sine rule to find n unknown ngle Hnnh is stnding 4.5 m from the se of 3 m sloping wll. The ngle of elevtion to the top of the wll is 36. Find the ngle t the top of the wll, to the nerest minute. 36 4.5 m 3 m SOLUTIO: 1 hek tht the sides nd ngles re opposite eh other. 2 Write the sine rule to find n ngle. 3 Sustitute the known vlues ( = 4.5, =, = 3 nd = 36). 4 Multiply oth sides of the eqution y 4.5. sin = sin sin sin 36 = 4.5 3 4.5 sin 36 sin = 3 = 0.8816778784 = 61 51 The ngle t the top of the wll is 61 51. 37 50 SMPLE PGES 5 Evlute. 6 Write the nswer orret to the nerest minute. 7 Write the nswer in words. 4.1 4F 4F
4F The sine rule 171 Eerise 4F LEVEL 1 Emple 11 1 For eh tringle, stte the lengths of sides, nd. 12 14 11 2 Find the vlue of, orret to two deiml ples. 5 sin 65 = sin 46 d sin 40 = 8 sin 62 15 26 19 10 sin 76 = sin 25 e sin 58 = 4 sin 78 5 7 6 3 sin 100 = sin 18 f sin 21 = 29 sin 120 3 For eh tringle, find the length of the unknown side, orret to two deiml ples. d 94 45 49 30 32 73 2.2 55 48 e 43 110 29 f 4 Find the length of the unknown side in eh tringle, orret to two deiml ples. SMPLE PGES 7 60 23 10 49 25 34 94 5 71 14 62 49 46 4 64 4 54 13 115 27
172 hpter 4 on-right-ngled trigonometry 4F Emple 12 5 Given the following trigonometri rtios, find the vlue of to the nerest minute. 12 sin 71 25 sin 65 sin = sin = 18 32 24 sin 103 sin = 40 e sin sin 73 30 = 187 236 6 Find the unknown ngle. nswer to the nerest degree. d 40 23 71 30 36 56 e 8 75 53 7 Find the unknown ngle. nswer to the nerest minute 46 75 13 50 2.9 54 2 19 6 4.6 d sin sin 53 20 = 50 65 f sin sin 61 46 = 6 7 21 f 9.8 89 12 16 12.7 37 44 27 124 3 8 Find the vlue of the pronumerl in following questions, orret to two deiml ples. In tringle, = 56, = 75 nd = 15. Find. In tringle XYZ, X = 107, Y = 45 nd = 30. Find y. In tringle PQR, P = 118, Q = 34 nd p = 59.5. Find q. SMPLE PGES 9 Find the vlue of the pronumerl in following questions, orret to the nerest degree. In tringle DEF, D = 46, d = 10 nd f = 5. Find ngle F. In tringle, = 102, = 18 nd = 13. Find ngle. In tringle RST, R = 68, r = 15 nd t = 8. Find ngle T.
4F The sine rule 173 10 Find the length of the unknown side,, in eh tringle, orret to two deiml ples. 88 e 12 84 20 51 105 40 59 32 29 33 55 57 4.8 d 10 65 3 59 21 f 1.2 85 21 27 40 LEVEL 2 11 enjmin is plnning to uild tringulr grden for his dughter. The verties of the tringle re nmed PQR. He mesured PQ s 2.7 m, QR s 3.1 m nd PRQ s 57. Use the sine rule to find the size of RPQ, orret to the nerest degree. 12 Tringle hs = 71, = 69 nd = 40. The length of is 10. Wht is the vlue of, orret to one deiml ple? Wht is the vlue of, orret to one deiml ple? SMPLE PGES 13 Tringle XYZ hs sides YZ = 30 m, XY = 24 m nd YXZ = 54. Use the sine rule to: Find the size of ngle XYZ. Give your nswer to the nerest degree. Find the size of y. Give your nswer to the nerest entimetre. 14 Find the length of the longest side of tringle with ngles of 42, 55 nd 83, given tht the length of the shortest side is 8.6 m. nswer orret to one deiml ple. X 24 m Y 54 y 30 m Z
174 hpter 4 on-right-ngled trigonometry 4F 15 Sienn ws loted t X nd sw fire in the diretion 15 E. Seven kilometres to the est of X t Z, Dyln sw the fire in the diretion 50 W. How fr is X from the fire? nswer in kilometres, orret to one deiml ple. How fr is Z from the fire? nswer in kilometres, orret to one deiml ple. 16 mn t R mesures the ngle of elevtion to plne t P s 45 15. The plne trvels 15 km from P to Q. The mn then mesures the ngle of elevtion to the plne t Q s 35 42. Wht re the sizes of PRQ, PQR nd QPR? Wht is the distne from R to P? nswer in kilometres, orret to one deiml ple. Wht is the distne from R to Q? nswer in kilometres, orret to one deiml ple. 17 helse is trvelling due est from to. Unfortuntely, the rod is loked nd she mkes detour y trvelling from to distne of 30 km, on ering of 040. helse then turns nd trvels southest until she rehes. Wht re the sizes of nd? How fr did helse trvel from to? nswer orret to the nerest kilometre. Wht ws the etr distne trvelled on the detour? nswer orret to the nerest kilometre. LEVEL 3 18 Hrrison mesured the ngle of elevtion to the top of the mountin s 28. He moved 140 m loser to the mountin nd mesured the ngle of elevtion to the top of the mountin s 43. How fr in stright line is Hrrison from the top of the mountin t his new position? nswer orret to the nerest metre. 19 The ptin of ship,, sighted the smoke of volni islnd in the diretion 53 E. ptin on nother ship,, 30 km due est of the first ship sw the smoke in the diretion 42 W. How fr is ship from the volno? nswer orret to one deiml ple. How fr is ship from the volno? nswer orret to one deiml ple. The ship loser to the volno is trvelling t 20 km/h. How long will it tke for this ship to reh the volno? nswer to the nerest minute. SMPLE PGES X Q Fire P Z R
4G The osine rule 175 4G The osine rule The osine rule is nother formul tht reltes the sides nd ngles in tringle. It is pplied to prolems where three sides nd one ngle re involved. The osine rule is derived y onstruting perpendiulr line of length h from to D. Let the distne from to D e nd hene the distne from to D is ( ). In ΔD using Pythgors theorem: 2 = h 2 + 2 ➀ h 2 = 2 2 In ΔD, os = h = os ➁ In ΔD using Pythgors theorem 2 = h 2 + ( ) 2 D = h 2 + 2 2 + 2 = h 2 + 2 + 2 2 = 2 + 2 2 From ➀ 2 = 2 + 2 2 os From ➁ Similrly, y onstruting perpendiulrs from nd, we n otin the other two results elow. THE OSIE RULE To find the third side given two sides nd the inluded ngle in Δ, 2 = 2 + 2 2 os 2 = 2 + 2 2 os 2 = 2 + 2 2 os ( is opposite ) ( is opposite ) ( is opposite ) To find n ngle given three sides (rerrngements of the ove formuls): os = 2 + 2 2 2 os = 2 + 2 2 2 Emple 13: Using the osine rule to find n unknown side Find the vlue of, orret to two deiml ples. os = 2 + 2 2 2 SMPLE PGES SOLUTIO: 1 Write the osine formul to find side. 2 = 2 + 2 2 os 2 Sustitute the vlues for,, nd. 2 = 12 2 + 6 2 2 12 6 os 76 3 lulte the vlue of 2. 2 = 145.163247 4 Tke the squre root of oth sides. = 12.05 6 76 12 4G
176 hpter 4 on-right-ngled trigonometry 4G Emple 14: Using the osine rule to find n unknown ngle Find the vlue of the ngle. nswer in degrees, orret to one deiml ple. SOLUTIO: 1 Write the osine formul to find n ngle. os X = y2 + z 2 2 2 Sustitute the vlues for, y, z nd X ( = 8, y = 12, z = 11 nd X = ). 3 lulte the vlue of os. 4 Use your lultor to find. 5 Write the nswer orret to one deiml ple. Emple 15: Solving prolem using the osine rule Smuel shoots for gol when he is 2.5 m from one post nd 3.2 m from the other post. The gol is 2 m wide. Wht is the size of the ngle for Smuel to sore gol? nswer orret to the nerest minute. SOLUTIO: 2 m 3.2 m 1 Write the osine formul to find n ngle. 2 Sustitute the vlues for,, nd ( = 2, = 2.5, = 3.2 nd = ). 3 lulte the vlue of os. 4 Use your lultor to find. 5 Use your lultor to onvert deiml degrees to degrees nd minutes. 6 Write the nswer in words. 2.5 m 2yz os = (122 + 11 2 8 2 ) (2 12 11) os = 0.7613636364 = 40.41543902 = 40.4 os = 2 + 2 2 2 os = (2.52 + 3.2 2 2 2 ) (2 2.5 3.2) os = 0.780625 = 38.68216452 = 38 41 X 11 4G Y 8 12 Z SMPLE PGES 4G The size of the ngle for Smuel to sore gol is 38 41.
4G The osine rule 177 Eerise 4G LEVEL 1 Emple 13 Emple 14 1 In eh tringle, write the osine rule with 2 s the sujet of the eqution. 35 3 70 15 62 40 4 28 59 50 45 17 2 Find the vlue of the, orret to one deiml ple. 2 = 3 2 + 4 2 2 3 4 os 39 2 = 7 2 + 10 2 2 7 10 os 135 2 = 2 2 + 5 2 2 2 5 os 45 15 d 2 = 13 2 + 9 2 2 13 9 os 101 34 3 Find the length of the unknown side in eh tringle, orret to two deiml ples. 8 69 44 105 21 9 50 29 d 16 46 13 e 67 4 Find the length of the unknown side in eh tringle, orret to three deiml ples. 1.9 45 113 10 43 29 14 91 53 2.6 98 f 25 20 58 78 18 SMPLE PGES 24 56 5 In the tringle, = 24, = 22 nd = 42. Wht is the vlue of side? nswer orret to three signifint figures.
178 hpter 4 on-right-ngled trigonometry 4G Emple 15 6 Find the vlue of the ngle. nswer in degrees orret to two deiml ples. os = 72 + 9 2 6 2 os = 162 + 20 2 24 2 2 7 9 2 16 20 os = 4.92 + 3.4 2 5.7 2 d os = 32 + 4 2 6 2 2 4.9 3.4 2 3 4 7 Find the size of the unknown ngle. nswer to the nerest degree. 6 9 7 13 18 10 d 74 79 77 e 54 32 39 27 f 8.1 8 In the tringle, = 10, = 11 nd = 12. Wht re the sizes of the following ngles? nswer orret to the nerest minute. ngle ngle ngle 9 In tringle XYZ, XY = 8 m, YZ = 12 m nd XZ = 16 m. Wht is the size of ngle Y, to the nerest minute? 10 Find the size of the unknown ngle. nswer to the nerest minute. 13 17 11 20 16 24 SMPLE PGES 11 In tringle FGH, the length of GH is 14 m, FH is 9 m nd FG is 8 m. Find HFG, orret to the nerest minute. 8.7 3.2 F 3.9 41 9.3 2.5 36 G H
4G The osine rule 179 LEVEL 2 12 DEF is tringle for whih DF = 37 m, EF = 46 m nd DFE = 44. Use the osine rule to find the length of DE, to the nerest millimetre. 13 tringle RST hs RST = 51, STR = 63, RT = 40 nd RS = 48. Wht is the size of TRS? Find the length of using the osine rule. nswer orret to three signifint figures. 14 Ruy drives four-wheel drive long trk from point due west to point, distne of 14 km. She then turns nd trvels 19 km to point. Use the osine rule to lulte the distne Ruy is from her strting point. nswer orret to one deiml ple. 15 Pssengers in r trvelling est, long rod tht runs west est, see stle 10 km wy in the diretion 65 E. When they hve trvelled further 4 km est long the rod, wht will e the distne to the stle? nswer orret to two deiml ples. 16 stepldder hs legs of length 120 m nd the ngle etween them is 15. lulte the distne (to the nerest entimetre) etween the legs on the ground. 17 tringle hs sides mesuring 4 m, 5 m nd 7 m. Wht is the size of the smllest ngle in this tringle? nswer in minutes. Wht is the size of the lrgest ngle in this tringle? nswer in minutes. LEVEL 3 18 running iruit is in the shpe of tringle with lengths of 6 km, 6.5 km nd 7 km. Wht re the sizes of the ngles (in minutes) etween eh of the sides? 19 The lengths of the sides of tringle re in the rtio 7 : 8 : 9. Find the size of eh ngle, orret to the nerest minute. R S 48 51 40 63 SMPLE PGES 4 m 103 7 m T 5 m
180 hpter 4 on-right-ngled trigonometry 4H 4H Misellneous prolems Trigonometry is used to solve mny prtil prolems. Use the steps elow to solve them. SOLVIG TRIGOOMETRI WORD PROLEM 1 Red the question nd underline the key terms. 2 Drw digrm nd lel the informtion from the question. 3 If the tringle is right-ngled, use SOH H TO. 4 If the tringle does not hve right ngle: use the sine rule if given two sides nd two ngles. use the osine rule if given three sides nd one ngle. 5 hek tht the nswer is resonle nd units re orret. Emple 16: Solving prolems involving non-right-ngled tringles The dimensions of lok of lnd re shown opposite. Wht is the length of, orret to the nerest metre? Wht is the length of y, orret to the nerest metre? SOLUTIO: 1 Looking t ΔPQS, three sides (inluding ) nd one ngle re given nd the side we wnt to find () is opposite the known ngle. Determine whih rule fits this sitution. 2 Write the osine rule to find side, hnging to PQS to mth the tringle in question. 3 Sustitute the vlues for p, q, s nd P. 4 Evlute to the nerest metre. 5 Looking t ΔQRS, two sides (inluding y, nd is known from prt ) nd two ngles re given, nd the unknown side is opposite known ngle. Write out the rule tht fits this sitution. 6 Sustitute the known vlues. 7 Multiply oth sides y sin 38. Use the osine rule in ΔPQS to find. p 2 = q 2 + s 2 2qs os P 4H 2 = 210 2 + 320 2 2 210 320 os 70 2 = 100532.4927 so = 317.068 317m Use the sine rule to find side in ΔQRS to find y. y sin Y = sin X y sin 38 = 317.068 sin 82 317.068 sin 38 y = sin 82 = 197.125328 197 m 8 Evlute to the nerest metre. (ote: Do not use the pproimte vlue from prt s it my result in n error.) P 320 m 70 210 m SMPLE PGES S Q 38 82 y R
4H Misellneous prolems 181 Emple 17: Solving prolems involving tringle without right ngle 4H Mdison mesures the ngle of elevtion to the top of wll s 32. She wlks 10 m horizontlly towrds the wll nd mesures the ngle of elevtion s 51. Find the height of the wll. nswer to the nerest metre. 32 51 10 m h SOLUTIO: 1 Drw digrm onsisting of two tringles. 2 Lel eh digrm. h 3 To find the height of the wll h we need to 129 lulte the vlue of. 32 51 10 m D 4 Looking t Δ there re two sides (inluding ) nd two ngles. Determine whih rule to use. Use the sine rule in Δ to find. 5 To lulte, use the ngle sum of = 180 (32 + 129 ) = 19 tringle is 180. 6 Write the sine rule. sin = sin 7 Sustitute the known vlues for, nd. sin 32 = 10 sin 19 10 sin 32 8 Multiply oth sides of the eqution y sin 32. = sin 19 9 Evlute. = 16.276753 16m 10 Determine whether to use trigonometri rtio or In ΔD, D is right ngle. Use rule to find h, nd whih one. trigonometri rtio SOH to find h. 11 Write the rtio for SOH. sin = h 12 Sustitute the known vlue of. sin 51 = h 13 Multiply oth sides of the eqution y. sin 51 = h h = sin 51 14 Evlute using the et vlue of from step 9. = 12.64941286 15 Write the nswer orret to the nerest metre. 12 m 16 Write the nswer in words. The wll is 12 m high. SMPLE PGES
182 hpter 4 on-right-ngled trigonometry 4H Eerise 4H LEVEL 1 Emple 16 1 The ering of Y from X is 240 nd the distne of Y from X is 20 km. Wht is the vlue of? If Z is 18 km due north of X, lulte the distne of Y from Z, orret to the nerest kilometre. Wht is the vlue of to the nerest degree? 2 ndrew plns to uild tringulr flower ed in the lower prt of irulr grden. The length of TS is 24 m, TSR is 45 nd ΔRTS is 40. Wht is the size of SRT? Use the sine rule to find the length of TR. nswer orret to the nerest metre. 3 The digrm shows informtion out the lotions of towns P, Q nd R. mer tkes 2 hours nd 30 minutes to wlk diretly from Town P to Town Q. Wht is mer s wlking speed, orret to the nerest km/h? Wht is the distne from P to R? nswer orret to the nerest kilometre. How long would it tke mer to wlk from P to R? nswer to the nerest minute. 4 M is 50 km north of O. The ering of from M is 108 nd from O it is 061. nswer the following questions to the nerest kilometre. Wht is the distne etween M nd? Wht is the distne etween nd O? 5 XYZ represents tringulr re of lnd. The ering of X from Y is 041 nd the ering of Z from Y is 305. The distne XY is 27 m nd the distne YZ is 37 m. Wht is the size of ngle XYZ? nswer orret to the nerest degree. lulte the re of the lnd to the nerest squre metre. Wht is the length of the oundry XZ, to the nerest metre? T orth P 40 19 km 24 m 45 M O Y 108 61 R 111 Q S orth R 17 km SMPLE PGES Z 37 m Y X Z X 27 m
4H Misellneous prolems 183 Emple 17 6 ship is loted t nd two rdr sttions 75 km prt re loted t nd ( is est of ). The ering of the ship from one rdr sttion is 038 nd from the other rdr sttion it is 306. orth 38 orth 306 How fr is the ship from the rdr sttion t, to the nerest kilometre? How fr is the ship from the rdr sttion t, to the nerest kilometre? 7 tringle DEG hs right ngle t E nd DG is 8 m in length. line is drwn from D to F D with DFG = 144 nd GDF = 7. 8 m Find the length of DF. nswer orret to two deiml ples. 7 Find the size of DFE. Find the length of FE. nswer orret to two deiml ples. G 144 F E 8 Two ldders re the sme distne up the wll. The shorter ldder is 4.9 m long nd mkes n ngle of 48 with the ground. The longer ldder is 6.7 m long. How fr up the wll do the ldders reh? nswer orret to one deiml ple. Wht is the ngle, to the nerest degree, tht the longer ldder mkes with the ground? Find the distne etween the ldders on the ground. nswer orret to one deiml ple. 9 The ngle of depression from to D is 65. The length of is 18 m nd the length of D is 16 m. Wht is the size of D? nswer in minutes. lulte the ngle of elevtion from D to. Give your nswer to the nerest minute. Find the length of, orret to the nerest metre. 4.9 m 48 6.7 m SMPLE PGES D 16 m 65 18 m
184 hpter 4 on-right-ngled trigonometry 4H 10 The digrm shows five rods: DE, DG, EF, GE nd GF. The ering of FG is 304, ED is 292 nd DG is 040. The distne from E to F is 8 km nd E is due west of F. Find the size of GFE. 40 Wht is the distne GE? nswer orret to one deiml D ple. Wht is the size of DGE, GED nd GDE? nswer to the nerest degree. d Wht is the distne DE? nswer orret to one deiml ple. e Wht is the distne DG? nswer orret to one deiml ple. 11 survey of prk is shown in the digrm. pth is proposed from to D nd fene is required from to to D. Wht is the length of the pth? nswer orret to the nerest metre. lulte the required length of fening. nswer orret to the nerest metre. Wht is the re of the prk? nswer orret to the nerest squre metre. 12 hris nd Dyln re 55 m prt. They oth oserve ird in the sky t. Prove tht the distne etween Dyln nd the ird (D) is given y: 55 sin 35 D = sin 16 Wht is the distne etween hris nd the ird? nswer orret to three signifint figures. How high is the ird ove the ground? nswer orret to three signifint figures. LEVEL 2 LEVEL 3 13 ot trvels 90 km from point X to point Y on ering of 065. It then turns nd trvels on ering of 140 from point Y to point Z. Point Z is on horizontl line from its strting point (X). Wht re the sizes of XYZ nd YXZ? How fr is the ot from its strting point (X to Z)? nswer orret to two deiml ples. 14 Investigte nvigtionl methods used y originl nd Torres Strit Islnder peoples, nd one other ulture tht uses different methods thn those used overed in this hpter. You n use the welinks provided in the Intertive Tetook. Write report outlining the methods used nd show how they would e used to nvigte from strting point to destintion. 74 D 37 35 51 D hris Dyln G 292 E 41 m 99 F 304 35 m ird SMPLE PGES
4I Rdil survey 185 4I Rdil survey rdil survey is used to rete digrm of piee of lnd tht is in the shpe of polygon. It involves drwing lines from entrl point rditing out to the orners of the polygon. The length of the lines nd the ngles etween them t the entrl point (usully lelled O for origin) re mesured. ODUTIG OMPSS RDIL SURVEY Pddok 8 m Pper 95 11 m 9 m 52 55 O 12 m 158 1 plne tle ( surveyor s tle) nd lrge sheet of pper re pled in the entre of field. 2 point is mrked ner the middle of the pper nd lelled O. 3 line is drwn on the pper from O to reflet the line of sight to eh orner (the rdil line). 4 The distne from the O on the pper to the first orner on the lnd oundry is mesured. 5 The length of the line is mrked off on the line drwn on the pper using suitle sle, e.g. 1 m on the ground is 1 m on the pper. 6 Steps 3 5 re repeted to drw lines from O to every orner of the piee of lnd. 7 oundry lines re drwn onneting the ends of the rdil lines, representing the lnd oundry. 8 The ngles t O etween the rdil lines re mesured using ompss. Emple 18: lulting re using rdil survey The digrm shows rdil survey of tringulr piee of lnd. Wht is the re of Δ? nswer in squre metres orret to one deiml ple. SOLUTIO: 1 The three smller tringles eh hve known ngles etween known side lengths. Use the sine rule for re of tringle. 2 For eh tringle, sustitute the two sides nd the inluded ngle into the re formul, nd evlute. 3 dd up the res orret to one deiml ple. In ΔO: In ΔO: 4 m 142 5 m 82 7 m Tle = 1 2 sin 4I = 1 5 7 sin 82 = 17.330 m2 2 SMPLE PGES 136 = 1 4 7 sin 142 = 8.619 m2 2 In ΔO: = 1 4 5 sin 136 = 6.947 m2 2 Totl re = 17.330 + 8.619 + 6.947 = 32.9 m 2 O
186 hpter 4 on-right-ngled trigonometry 4I Emple 19: lulting distne using ompss rdil survey 4I ompss rdil survey for field is shown opposite. Wht is the size of POQ? Wht is the size of POR? lulte the length of PQ, orret to two deiml ples. d lulte the length of PR, orret to two deiml ples. e Wht is the perimeter of the field? nswer orret to three signifint figures. SOLUTIO: 1 Sutrt the ering of P from the ering of Q. 2 lulte the ngle etween R nd the north diretion. POQ = 145 72 = 73 290 TR 4.7 m 290 T R orth 70 72 O 4.8 m P 072 T 3 Identify the ngle etween P nd the 72 north diretion from the ering of P. 4 dd the two ngles formed with the north POR = 70 + 72 diretion (70 nd 72 ). = 142 5 In ΔPOQ, POQ nd two sides re known, so use the osine formul to find side (PQ is ). 6 Sustitute the vlues for,, nd. 7 lulte the vlue of 2. 8 Tke the squre root of oth sides. 9 Write the nswer to two deiml ples. 2 = 2 + 2 2 os 2 = 4.8 2 + 4.5 2 2 4.8 4.5 os 73 2 = 30.65954236 = 5.54 m Length PQ is 5.54 metres 10 Write the osine formul to find side. d 2 = 2 + 2 2 os 11 Sustitute the vlues for,, nd. 12 lulte the vlue of 2. 13 Tke the squre root of oth sides. 2 = 4.7 2 + 4.8 2 2 4.7 4.8 os 142 2 = 80.6850452 = 8.98 m 14 Write the nswer to two deiml ples. Length PR is 8.98 metres 15 The field hs four sides. Perimeter is the sum of the lengths of eh side. e Perimeter = 5.54 + 8.98 + 4.5 + 4.7 = 23.72 16 Write the nswer to three sig.figs. = 23.7 m 4.7 m O 4.5 m 4.8 m 360 290 = 70 P 072 T Q 145 T SMPLE PGES ote: Point O, the origin, my e on the oundry of the piee of lnd, or even outside it.
4I Rdil survey 187 Eerise 4I LEVEL 1 Emple 18 Emple 19 1 The digrm elow shows plne tle rdil survey of piee of lnd. ll length mesurements re in metres. 15 14 13 56 43 O 47 12 D Wht is the re of ΔO? nswer orret to two deiml ples. Wht is the re of ΔO? nswer orret to two deiml ples. Wht is the re of ΔOD? nswer orret to two deiml ples. d Wht is the totl re of the piee of lnd? nswer to the nerest squre metre. e Find the length of, orret to two deiml ples. f Find the length of, orret to two deiml ples. g Find the length of D, orret to two deiml ples. h lulte the perimeter of the piee of lnd. nswer to the nerest metre. 2 rdil survey of tringulr field is shown. 316 Wht is the size of O? Wht is the size of O? 58 m Wht is the size of O? 82 m d Wht is the re of ΔO? nswer orret to two deiml O ples. 65 m e Wht is the re of ΔO? nswer orret to two deiml ples. f Wht is the re of ΔO? nswer orret to two deiml 214 ples. g Wht is the totl re of the tringulr field? nswer to the nerest squre metre. h Wht is the length of? nswer orret to two deiml ples. i Wht is the length of? nswer orret to two deiml ples. j Wht is the length of? nswer orret to two deiml ples. k lulte the perimeter of the piee of lnd. nswer to the nerest metre. 075 SMPLE PGES
188 hpter 4 on-right-ngled trigonometry 4I 3 rdil survey of lnd DEFG is shown opposite. Wht is the size of FOG? Find the re of tringle FOG to the nerest squre metre. Wht is the length of FG, orret to the nerest metre? 4 The digrm opposite is ompss rdil survey of field VWXYZ. ll distnes re in metres. nswer the following questions, orret to one deiml ple. Wht is the length of XY? Wht is the length of ZY? Wht is the length of VZ? d Wht is the length of VW? e Wht is the length of XW? f lulte the perimeter of the field. 5 rdil survey of lnd D is shown opposite. nswer the following questions orret to the nerest metre. Wht is the re of ΔO? Wht is the re of ΔO? Wht is the re of ΔOD? d Wht is the re of ΔOD? e Wht is the re of the lnd? W 300 40 V 270 324 224 D D 344 G 260 LEVEL 2 LEVEL 3 6 le, lke nd onnor re stnding in field. onnor () is 15 metres wy from lke () on ering of 032. le () is 20 metres wy from lke on ering of 315. Drw digrm to represent the positions, nd. Mrk the informtion from the question on the digrm. Wht is the size of the to the nerest degree? Wht is the re of tringle? nswer orret to the nerest squre metre. d How fr is le from onnor? nswer orret to the nerest metre. 50 m 35 X O 33 m O 18 m 55 40 40 027 66 m 30 m O 24 m 52 m 45 m Z 135 E 036 F 157 Y 090 080 SMPLE PGES
hpter 4 Summry 189 Key ides nd hpter summry Right-ngled trigonometry Degrees nd minutes ngles of elevtion nd depression erings Trigonometry with otuse ngles re of tringle The sine rule The osine rule Rdil surveys vigtion y different ultures Use the mnemoni SOH H TO. SOH: Sine-Opposite-Hypotenuse H: osine-djent-hypotenuse TO: Tngent-Opposite-djent One degree is equl to 60 minutes (1 = 60 ). One minute is equl to 60 seonds (1 = 60 ). Horizontl ngle of elevtion ompss ering diretion given y stting the ngle either side of north or south, suh s S60 E ute ngle (0 to 90 ) sin positive os positive tn positive Horizontl ngle of depression True ering diretion given y mesuring the ngle lokwise from north, suh s 120 T Otuse ngle (90 to 180 ) sin positive os negtive tn negtive re of tringle is hlf the produt = 1 sin of two sides multiplied y the sine of 2 the ngle etween the two sides (inluded ngle). Sine rule is used in non-right-ngled tringle given informtion out two sides nd two ngles. To find side, use sin = sin = sin. To find n ngle, use sin = sin = sin. osine rule is used in non-right-ngled tringle given informtion out three sides nd one ngle. To find side, use 2 = 2 + 2 2 os. To find n ngle, use os = 2 + 2 2. 2 SMPLE PGES Rdil survey involves mesuring the ngles etween rdil lines drwn from entrl point in the diretion of the orners of the lnd, nd mesuring the lengths of the rdil lines Other ultures use different nvigtion methods relying on the strs, previling winds or urrents, or lndmrks tht re reorded in vrious wys. Summry
190 hpter 4 on-right-ngled trigonometry Review Multiple hoie 1 Wht is the length of? 14 os 63 14 sin 63 14 os 63 D 14 sin 63 2 Wht is the orret epression for? 5 sin 29 5 sin 39 sin 39 sin 29 5 sin 112 sin 39 3 How would sin e lulted? 7 sin 64 8 5 8 sin 64 D D 5 sin 112 sin 29 5 sin 64 8 7 8 sin 64 4 Wht is the re of the tringle to the nerest squre metre? 115 m 2 185 m 2 205 m 2 D 220 m 2 5 Wht is the orret formul for the osine rule to find ngle? os = 2 + 2 2 os = 2 + 2 2 2 2 os = 2 + 2 2 2 D os = 2 + 2 2 2 6 Wht is the length of? nswer orret to two deiml ples. 6.93 6.94 48.14 D 48.15 7 The lrgest ngle in the tringle is. Wht is the vlue of os? os = 62 + 7 2 5 2 os = 52 + 7 2 6 2 2 6 7 2 5 7 os = 52 + 7 2 6 2 2 5 7 D os = 52 + 6 2 7 2 2 5 6 39 5 20 m 63 5 50 6 64 5 14 7 29 32 m 9 8 40 18 m SMPLE PGES 7 5
hpter 4 Review 191 Short-nswer 1 Find the vlue of, orret to two deiml ples. 56 35 m 25 m 34 17 mm 2 Find the unknown ngle in eh tringle. nswer orret to the nerest minute. 17 8 3 Susn looked from the top of liff, 62 m high, nd notied ship t n ngle of depression of 31. How fr ws the ship from the se of the liff? nswer orret to one deiml ple. 4 Emm rode for 8.5 km on ering of 43 W from her home. How fr north is Emm from home? nswer orret to one deiml ple. How fr west is Emm from home? nswer orret to one deiml ple. Wht is the ompss ering of her home from her urrent position? 16 34 SMPLE PGES 5 Find the vlue of the following trigonometri rtios, orret to two deiml ples. tn 123 41 os 93 26 sin 130 59 d os 167 35 e 5 sin 134 f 7 tn 128 g 11 os 149 h 4 tn 137 32 62 m 31 41 56 Review