SAMPLE. Trigonometric ratios and applications

Transcription

1 jetives H P T E R 12 Trigonometri rtios nd pplitions To solve prtil prolems using the trigonometri rtios To use the sine rule nd the osine rule to solve prolems To find the re of tringle given two sides nd n inluded ngle To find the re of setor nd segment of irle To find the length of n r To solve prolems involving ngles of depression nd ngles of elevtion To identify the line of gretest slope of plne To solve prolems in three dimensions inluding determining the ngle etween plnes 12.1 efining sine, osine nd tngent The unit irle is irle of rdius 1 with entre t the origin. SMPLE Sine nd osine my e defined for ny ngle through the unit irle. For the ngle of,point P on the unit irle is defined s illustrted here. The ngle is mesured in n ntilokwise diretion from the positive diretion of the x xis. y (0, 1) ( 1, 0) (0, 0) y θ (0, 0) (0, 1) x (1, 1) P(os(θ ), sin(θ )) x mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird 326

2 hpter 12 Trigonometri rtios nd pplitions 327 os ( )isdefined s the x oordinte of the point P nd sin ( )isdefined s the y oordinte of P. lultor gives pproximte vlues for these oordintes. y y y ( , ) (0.8660, 0.5) ( , ) x x x sin 30 = 0.5 (ext vlue) sin 135 = os 100 = os 30 = os 135 = sin 100 = In this hpter, ngles greter thn 180 or less thn 0 will not e onsidered. Forright-ngled tringle,similr tringle n e onstruted tht lies in the unit irle. From the digrm, = os ( ) nd = sin ( ) The sle ftor is the length. Hene = sin ( ) nd = os ( ) This implies = sin ( ) nd = os ( ) 1 ' θ ' This gives the rtio definition of sine nd osine for right-ngled tringle. The nming of sides with respet to n ngle is s shown. hypotenuse θ djent SMPLE opposite sin = opp ( ) opposite hyp hypotenuse os = dj ( ) djent hyp hypotenuse ) tn = opp dj ( opposite djent mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

3 328 Essentil dvned Generl Mthemtis From the unit irle, note tht sin ( ) = sin (180 ), e.g. sin 45 = sin 135 nd os ( ) = os (180 ), e.g. os (45 ) = os (135 ) This result will e used lter in this hpter. Exmple 1 Find the vlue of x orret to two deiml ples. Exmple 2 (os(180 θ), sin(180 θ) ) x = sin x = 80 sin 29.6 = x = orret to two deiml ples Find the length of the hypotenuse orret to two deiml ples. 10 = os = os 15 = 10 os 15 = m 29.6 y (180 θ) 0 15 θ x m The length of the hypotenuse = m orret to two deiml ples. Exmple 3 Find the mgnitude of. tn x = 11 3 x = tn x = ( ) x = x 3 m 11 m (os(θ ), sin(θ )) SMPLE (to the nerest seond). Rememer tht this is red s 74 degrees, 44 minutes nd 42 seonds. mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird x

4 hpter 12 Trigonometri rtios nd pplitions 329 Exmple 1 Exmple 2 Exmple 3 Exerise 12 1 Find the vlue of x in eh of the following. d 35 5 m 30 15' 7 m x m x m e 5 x 15 m x m f m 40 x m 2 n equilterl tringle hs ltitudes of length 20 m. Find the length of one side. x m 3 The se of n isoseles tringle is 12 m long nd the equl sides re 15 m long. Find the mgnitude of eh of the three ngles of the tringle. 4 pole sts shdow 20 m long when the ltitude of the sun is 49. lulte the height of the pole. SMPLE m 5 This figure represents rmp. Find the mgnitude of ngle. Find the distne. 6 This figure shows vertil mst PQ, whih stnds on horizontl ground. stright wire 20 m long runs from P t the top of the mst to point R on the ground, whih is 10 m from the foot of the mst. 6 m 20 m lulte the ngle of inlintion,,ofthe wire to the ground. lulte the height of the mst. θ R 10 m Q pole 1 m P mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

5 330 Essentil dvned Generl Mthemtis 7 ldder lening ginst vertil wll mkes n ngle of 26 with the wll. If the foot of the ldder is3mfrom the wll, lulte the length of the ldder the height it rehes ove the ground. 8 n engineer is designing stright onrete entry rmp, 60 m long, for r prk 13 m ove street level. lulte the ngle of the rmp to the horizontl. 9 vertil mst is seured from its top y stright les 200 m long fixed t the ground. The les mke ngles of 66 with the ground. Wht is the height of the mst? 10 mountin rilwy rises 400 m t uniform slope of 16 with the horizontl. Wht is the distne trvelled y trin for this rise? 11 The digonls of rhomus iset eh other t right ngles. If = =, find the length of the sides of the rhomus the mgnitude of ngle. 12 pendulum swings from the vertil through n ngle of 15 on eh side of the vertil. If the pendulum is 90 m long, wht is the distne x m etween its highest nd lowest point? 13 piture is hung symmetrilly y mens of string pssing over nil with its ends tthed to two rings on the upper edge of the piture. The distne etween the rings is 30 m nd the ngle etween the two portions is 105.Find the length of the string. 14 The distne = 50 m. If the line of sight of person stnding t to the tree mkes n ngle of 32 with the nk, how wide is the river? 90 m 90 m m SMPLE x m m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

6 hpter 12 Trigonometri rtios nd pplitions ldder 4.7 m long is pled ginst wll. The foot of the ldder must not e pled in flower ed, whih extends distne of 1.7 m from the foot of the wll. How high up the wll n the ldder reh? 16 river is known to e 50 m wide. swimmer sets off from to ross the river nd the pth of the swimmer is s shown. How fr does the person swim? 12.2 The sine rule In Setion 12.1, methods for finding unknown lengths nd ngles for right-ngled tringles were disussed. In this setion nd the next, methods for finding unknown quntities in non-right-ngled tringles re disussed. The sine rule is used to find unknown quntities in tringle when one of the following situtions rises: one side nd two ngles re given two sides nd non-inluded ngle re given. In the first of the two ses, unique tringle is defined, ut for the seond it is possile for two tringles to exist. Lelling onvention The following onvention is followed in the reminder of this module. Interior ngles re denoted y upper se letters nd the length of the side opposite n ngle is denoted y the orresponding lower se letter. Forexmple, the mgnitude of ngle is denoted y, nd the length of side is denoted y. The sine rule sttes tht for tringle sin = sin = sin proof will only e given for the ute-ngled tringle se. The proof for otuse-ngled tringles is similr. Proof In tringle, In tringle, i.e., SMPLE sin = h h = sin sin = h h = sin sin = sin sin = sin 50 m h 60 mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

7 332 Essentil dvned Generl Mthemtis Similrly, strting with perpendiulr from to would give sin = sin Exmple 4 Use the sine rule to find the length of. sin 31 = 10 sin sin 31 = sin 70 = Exmple 5 The length of is 5.48 m orret to two deiml ples. Use the sine rule to find the mgnitude of ngle XZY in the tringle, given tht Y = 25, y = 5 m, nd z = 6 m. 5 sin 25 = 6 sin Z sin Z sin 25 = sin 25 sin Z = 5 = Z = sin 1 ( ) Z = ( ) or ( ) X 70 5 m 6 m Z = or Z = (to the nerest seond) Rememer: sin (180 ) = sin There re two solutions for the eqution sin Z = Note: When using the sine rule in the sitution where two sides nd non-inluded ngle re given, the possiility of two suh tringles existing must e onsidered. Existene n e heked through the sum of the given ngle nd the found ngle not exeeding 180. Z 1 5 m X 30 28' 25" 5 m 6 m 31 Z ' 35" SMPLE Z 2 25 Y Y mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

8 hpter 12 Trigonometri rtios nd pplitions 333 Exmple 4 Exmple 5 Exerise 12 1 Find the vlue of the pronumerl for eh of the following tringles. X Y 50 x m 28 Y 70 Z x m 5.6 m 100 X Z d y m Y X Z x m 2 Find the vlue of for eh of the following tringles. 7 m 8 m m θ θ d 9.4 m θ 38 6 m 12 m 90 Z 9 m m SMPLE Y X 8 m 3 Solve the following tringles (i.e. find ll sides nd ngles). = 12, = 59, = 73 = 75.3, = 5.6, = = 123.2, = 11.5, = 37 d = 23, = 15, = 40 e = 140, = 20, = 10 4 Solve the following tringles (i.e. find ll sides nd ngles). = 17.6, = 48.25, = 15.3 = 129, = 7.89, = 4.56 = 28.35, = 8.5, = lndmrk is oserved from two points nd,whih re 400 m prt. The mgnitude of ngle is found to e 68 nd the mgnitude of ngle is 70.Find the distne of from. θ mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

9 334 Essentil dvned Generl Mthemtis 6 P is point t the top of lighthouse. Mesurements of the length of nd ngles P nd P re tken nd re s shown in the digrm. Find the height of the lighthouse m nd re two points on ostline. They re 1070 m prt. is point t se. The ngles nd hve mgnitudes of 74 nd 69 respetively. Find the distne of from. Y 8 Find X Y X 12.3 The osine rule m 89 The osine rule is used to find unknown quntities in tringle when one of the following situtions rises: two sides nd n inluded ngle re given three sides re given. The osine rule sttes tht for tringle 2 = os or equivlently os = The symmetril results lso hold, i.e. 2 = os 2 = os SMPLE The result will e proved for n ute-ngled tringle. The proof for otuse-ngled tringles is similr. Proof In tringle 2 = x 2 + h 2 (Pythgors theorem) os = x nd therefore x = os h P x mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

10 hpter 12 Trigonometri rtios nd pplitions 335 In tringle Exmple 6 Expnding gives Exmple 6 2 = ( x) 2 + h 2 (Pythgors theorem) 2 = 2 2x + x 2 + h 2 = 2 2x + 2 (s x 2 + h 2 = 2 ) 2 = os (s x = os ) For tringle, find the length of in entimetres orret to two deiml ples. 2 = os 67 = The length of is 9.27 m orret to two deiml ples. Exmple 7 Find the mgnitude of ngle for tringle. os = = = = ( ) The mgnitude of ngle is (to the nerest seond). Exerise 12 1 Find the length of. 6 m 15 m m SMPLE m 5 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

11 336 Essentil dvned Generl Mthemtis Exmple 7 2 Find the mgnitude of ngles nd. 8 m 3 For tringle with = 60 = 16 = 30, find = 14 = 53 = 12, find = 27 = 35 = 46, find the mgnitude of ngle d = 17 = 120 = 63, find e = 31 = 42 = 140, find f = 10 = 12 = 9, find the mgnitude of ngle g = 11 = 9 = 43.2, find h = 8 = 10 = 15, find the mgnitude of ngle 4 setion of n orienteering ourse is s shown. Find the length of leg. 5 Two ships sil from point. t prtiulr time their positions nd re s shown. Find the distne etween the ships t this time. 6 is prllelogrm. Find the length of the digonls: 4 m 48 6 km 4 km 20 5 m N 6 km 4 km 30 5 m SMPLE mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

12 7 weight is hung from two hooks in eiling y strings of length 54 m nd 42 m, whih re inlined t 70 to eh other. Find the distne etween the hooks. 8 Find the length of digonl. Use the sine rule to find the length of. 9 Two irles of rdius 7.5 m nd 6 m hve ommon hord of length 8 m. Find the mgnitude of ngle. Find the mgnitude of ngle. 10 Two stright rods interset t n ngle of 65.point on one rod is 90 m from the intersetion nd point on the other rod is 70 m from the intersetion, s shown on the digrm. Find the distne of from. is the midpoint of.find the distne of from the intersetion re of tringle hpter 12 Trigonometri rtios nd pplitions m m 4 m 5 m m 7.5 m 6 m 8 m ' SMPLE It is known tht the re of tringle is given y the formul re = 1 2 h re = 1 se length height 2 h 90 m m y oserving tht h = sin the following formul n e found. mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

13 338 Essentil dvned Generl Mthemtis re of tringle = 1 sin 2 i.e., the re is given y hlf the produt of the length of two sides nd the sine of the ngle inluded etween them. Exmple 8 Find the re of tringle shown in the digrm. re = sin = m m The re of tringle is m 2 orret to two deiml ples. Exmple 9 Find the re of eh of the following tringles, orret to three deiml ples. 6.4 m 8 m Using the osine rule, F 8.2 m E G 7 m 8 2 = os 64 = os os = = ( ) (the ext vlue n e stored on the grphis lultor s, sy) m SMPLE re of tringle = sin 2 = m 2, orret to three deiml ples. I 12 H mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

14 hpter 12 Trigonometri rtios nd pplitions 339 Exmple 8 E = (180 ( )) = 25 Using the sine rule, F = sin (25 ) 8.2 sin (85 ) = (the ext vlue n e stored on the grphis lultor s E, sy) re of tringle EF = E sin (70 ) Using the sine rule, = m 2, orret to three deiml ples. sin I = 10 sin (12 ) 7 = I = ( ) sine I is n otuse ngle = ( ) (the ext vlue n e stored on the grphis lultor s I, sy) G = (180 (12 + I )) = ( ) (the ext vlue n e stored on the grphis lultor s G, sy) re of tringle GHI = sin (G ) Exerise 12 1 Find the re of eh of the following tringles m 4 m = m 2, orret to three deiml ples. 5.1 m Y M 3.5 m d N m X m 25 5 m SMPLE Z L 5 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

15 340 Essentil dvned Generl Mthemtis Exmple 9 2 Find the re of eh of the following tringles, orret to three deiml ples. 5.9 m 9 m e G 4.1 m 65 5 m 6.3 m 55 F I 3.2 m 12 m 12.5 irle mensurtion Terminology E 24 H d f G 7 m 5.1 m m F 5.9 m 10 4 m 19 In this irle with entre, the intervl is lled hord of the irle. hord is n intervl with endpoints on the irle. If the entre of the irle is on the hord, the intervl is lled dimeter, e.g. intervl in the digrm. ny two points on irle divide the irle into rs. The shorter r is lled the minor r, the longer is the mjor r, e.g. r is minor r nd is mjor r in this digrm. Note tht r nd r re semiirulr rs in this digrm. Every hord divides the interior of irle into two regions lled segments. The smller is lled the minor segment, the lrger is the mjor segment. Inthe ove digrm the minor segment hs een shded. SMPLE Two rdii nd n r define region lled setor. Inthis digrm with irle entre, the shded region is minor setor nd the unshded region is mjor setor. Formuls to find r lengths, hord lengths nd res of regions inside irle will now e developed. I E H mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

16 hpter 12 Trigonometri rtios nd pplitions 341 r length The r nd the orresponding hord re sid to sutend the ngle t the entre of the irle. If the mgnitude of = nd rdius length is r units, then l units, the length of r, will e frtion of the irumferene. Sine irumferene = 2 r l = r = r 180 Now sine 180 = where = l = r where = mg hord length From the digrm, the osine rule gives In tringle P, 2 = r 2 + r 2 2r 2 os = 2r 2 (1 os ) = 2r 2 (1 os ) P = r sin 2 = 2r sin ( 2 Note: 1 os = sin 2 ) = 2 sin re of setor If mg = the re of the setor is frtion of the re of the irle. Now re of irle is given y Exmple 10 re of irle = r 2 re of setor = frtion of r 2 = r gin using = 180 re of setor = 1 2 r 2 where = mg In this irle, entre, rdius length, the ngle sutended t y r hs mgnitude 120.Find SMPLE the ext lengths of i the hord ii the r r θ P θ θ 2 r r r θ 120 r mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

17 342 Essentil dvned Generl Mthemtis the ext re of the minor setor the mgnitude of ngle,indegrees nd minutes, if the minor r hs length 4 m. i Use hord length = 2r sin where r = 10 nd = hord length = 20 sin 60 3 = 20 2 = 10 3 Length of hord is 10 3 m. ii Use l = r where r = 10, = 2 3 (note use of rdins) = = 20 3 Length of r is 20 3 m. (Verify tht length of r is greter thn length of hord s hek.) Use re of setor = 1 2 r 2 where r = 10, = 2 3 = = So re of minor setor = 100 m 2. 3 Use r length = r 4 = 10 = 4 10 ngle = = ( ) re of segment = (to the nerest minute) re of segment shded = re of minor setor re of (note use of rdins) SMPLE So = r r 2 sin Where mg = ut if mg =, = 180 = 1 2 r r 2 sin = 1 2 r 2 ( sin ) θ θ r 4 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

18 Exmple 10 Exmple 11 hpter 12 Trigonometri rtios nd pplitions 343 Generlly speking the formuls re simpler if mg is mesured in rdins. The following formuls ssume is in rdins. Exmple 11 r length = r hord length = 2r sin 2 re of setor = 1 2 r 2 re of segment = 1 2 r 2 ( sin ) irle, entre, with rdius length 20 m hs hord tht is from the entre of the irle. lulte the re of the minor segment formed y this hord. Now re of segment = 1 2 r 2 ( sin ) r = 20 ut needs to e lulted. In, os 2 = = 60 nd = 120 Hene Exerise 12E mg = 2 3 re of segment = 1 ( sin 2 ) m 2 ( ) = m 2 2 ( 4 3 ) 3 = 200 m 2 6 ( ) 3 = m m θ SMPLE 1 Find the r length whih sutends n ngle of mgnitude 105 t the entre of irle of rdius length 25 m. 2 Find the mgnitude, in degrees nd minutes, of the ngle sutended t the entre of irle of rdius length 30 m, y n r of length 50 m hord of length 50 m. 3 hord of length 6 m is drwn in irle of rdius 7 m. Find the length of the minor r ut off y the hord the re of the smller region inside the irle ut off y the hord. mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

19 344 Essentil dvned Generl Mthemtis 4 Sketh, on the sme set of xes, the grphs of ={(x, y) : x 2 + y 2 16} nd = {(x, y) : y 2} Find the re mesure of the region. 5 Use results from hpter 11 to show tht 2r 2 (1 os ) = 2r sin 2 6 Find the re of the region etween n equilterl tringle of side length nd the irumirle of the tringle (the irle tht psses through the three verties of the tringle). 7 person stnds on level ground 60 m from the nerest point of ylindril tnk of rdius length 20 m. lulte the irumferene of the tnk the perentge of the irumferene tht is visile to the person. 8 The minute hnd of lrge lok is 4mlong. How fr does the tip of the minute hnd move etween p.m. nd p.m? Wht is the re overed y the minute hnd etween p.m. nd p.m? 9 Two irles of rdii 3 m nd 4 m hve their entres 5 m prt. lulte the re of the region ommon to oth irles. 10 setor of irle hs perimeter of 32 m nd n re of 63 m 2.Find the rdius length nd the mgnitude of the ngle sutended t the entre of the two possile setors. 11 Twowheels (pulleys) hve rdii of length 15 m nd 25 m nd hve their entres 60 m prt. Wht is the length of the elt required to pss tightly round the pulleys without rossing? 12 frme in the shpe of n equilterl tringle enloses three irulr diss of rdius length 5 m so tht the diss touh eh other. Find the perimeter of the smllest frme whih n enlose the diss the re enlosed etween the diss. SMPLE 12.6 ngles of elevtion nd depression nd erings The ngle of elevtion is the ngle etween the horizontl nd diretion ove the horizontl. The ngle of depression is the ngle etween the horizontl nd diretion elow the horizontl. eye level eye level liff line of sight ngle of elevtion ngle of depression line of sight mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

20 hpter 12 Trigonometri rtios nd pplitions 345 Exmple 12 The pilot of heliopter flying t 400 m oserves smll ot t n ngle of depression of 1.2. lulte the horizontl distne of the ot to the heliopter. H 400 m H = tn = tn 1.2 = 400 tn 1.2 = The horizontl distne is m to the nerest 10 m. Exmple 13 The light on liff-top lighthouse, known to e 75 m ove se level, is oserved from ot t n ngle of elevtion of 7.1. lulte the distne of the ot from the lighthouse. 75 = tn (7.1 ) 75 = tn (7.1 ) = Exmple 14 L 75 m 1.2 (ngle of depression) (digrm not to sle) The distne of the ot from the lighthouse is 602 m to the nerest metre. SMPLE From the point,mn oserves tht the ngle of elevtion of the summit of hill is 10.Hethen wlks towrds the hill for 500 m long flt ground. The summit of the hill is now t n ngle of elevtion of 14. Find the height of the hill ove the level of m The mgnitude of ngle H = (180 14) = 166 The mgnitude of ngle H = [180 ( )] = H mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

21 346 Essentil dvned Generl Mthemtis Using the sine rule in tringle H: 500 sin 4 = H sin sin 10 H = sin 4 = In tringle H: H = sin 14 H H = H sin 14 = erings The height of the hill is 301 m to the nerest metre. The ering (or ompss ering) is the diretion mesured from north lokwise. The ering of from is 030 The ering of from is 120 The ering of from is 210 The ering of from is 330 Exmple 15 The rod from town runs due west for 14 km to town. television mst is loted due south of t distne of 23 km. lulte the distne nd ering of the mst from the entre of town. W km SMPLE tn = = (to two deiml ples) ering = ( ) = y Pythgors theorem T 2 = 2 + T 2 = = 725 T = N S T km θ E N The mst is 27 km from the entre of town (to the nerest kilometre) nd on ering of mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

22 hpter 12 Trigonometri rtios nd pplitions 347 Exmple 12 Exmple 13 Exmple 16 yht strts from point nd sils on ering of 038 for 3000 m. It then lters its ourse to one in diretion with ering of 318 nd fter siling for 3300 m it rehes point. Find the distne the ering of from. The mgnitude of ngle needs to e determined so tht the osine rule n e pplied in tringle. The mgnitude of ngle = (180 ( )) = 100 In tringle N 38 2 = os (100 ) = = m m N 38 N 318 The distne of from is 4830 m (to the nerest metre). To find the ering of from, the mgnitude of ngle must first e found. The sine rule my e used sin = sin sin 100 sin = sin = = ( ) The ering of from = 360 ( ) = SMPLE The ering of from is 356 to the nerest degree. Exerise 12F 1 From the top of vertil liff 130 m high the ngle of depression of uoy t se is 18. Wht is the distne of the uoy from the foot of the liff? 2 The ngle of elevtion of the top of n old himney stk t point 40 m from its se is 41.Find the height of the himney N mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

23 348 Essentil dvned Generl Mthemtis Exmple 14 Exmple 15 Exmple 16 3 mn stnding on top of mountin oserves tht the ngle of depression to the foot of uilding is 41.Ifthe height of the mn ove the foot of the uilding is 500 m, find the horizontl distne from the mn to the uilding. 4 mn lying down on top of liff 40 m high oserves the ngle of depression to uoy in the se elow to e 20.Ifheisinline with the uoy, lulte the distne etween the uoy nd the foot of the liff, whih my e ssumed to e vertil. 5 mn stnding on top of liff 50 m high is in line with two uoys whose ngles of depression re 18 nd 20. lulte the distne etween the uoys. 6 ship sils 10 km north nd then 15 km est. Wht is its ering from the strting point? 7 ship leves port nd stems 15 km due est. It then turns nd goes 22 km due north. Wht is the ering of the ship from? Wht is the ering of port from the ship? 8 yht sils from point on ering of 035 for 2000 m. It then lters ourse to diretion with ering of 320 nd fter siling for 2500 m it rehes point. Find the distne. Find the ering of from. 9 The ering of point from point is 207. Wht is the ering of from? 10 The ering of ship S from lighthouse is 055.seond lighthouse is due est of. The ering of S from is 302.Find the mgnitude of ngle S. 11 yht strts from L nd sils 12 km due est to M. Itthen sils 9 km on ering of 142 to K.Find the mgnitude of ngle MLK. 12 The ering of from is 035. The ering of from is 346. The distne of from is 340 km. The distne of from is 160 km. Find the mgnitude of ngle. Use the osine rule to find the distne from to. 160 km 346 N km SMPLE 13 From ship S two other ships P nd Q re on erings 320 nd 075 respetively. The distne PS = 7.5kmnd the distne QS = 5 km. Find the distne PQ Prolems in three dimensions Prolems in three dimensions re solved y piking out tringles from min figure nd finding lengths nd ngles through these tringles. mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

24 hpter 12 Trigonometri rtios nd pplitions 349 Exmple 17 EFGH is uoid. Find distne distne H the mgnitude of ngle H d the mgnitude of ngle H. 2 = = 164 = 164 = E 7 m The length of is m orret to two deiml ples. H 2 = H = H = m = 213 H = 213 = d 8 m The length of H is m orret to two deiml ples. tn = H H = m = = to the nerest minute. From tringle H H os = = to the nerest minute. H F θ 164 m θ 164 m SMPLE Exmple 18 The figure shows pyrmid with squre se. The se hs sides 6mlong nd the edges V, V, V, V re eh long. Find the length of. Find the length of E. Find the length of VE. d Find the mgnitude of ngle VE. 6 m V E G 8 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

25 350 Essentil dvned Generl Mthemtis 2 = = 72 = 6 2 = The length of is 8.49 m to two deiml ples. E = 1 2 E = = 3 2 = The length of E is 4.24 m orret to two deiml ples. VE 2 = V 2 E 2 d = = = 82 VE = 82 = The length of VE is 9.06 m orret to two deiml ples. sin = VE V 82 = 10 = = The mgnitude of ngle VE is to the nerest minute. SMPLE Exmple 19 ommunitions mst is ereted t the orner,, of retngulr ourtyrd whose sides mesure 60 m nd 45 m. If the ngle of elevtion of the top of the mst from is 12, find 12 the height of the mst the ngle of elevtion of the top of the mst from (where = 45 m). 60 m E E V V E 6 m 45 m H 6 m θ mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

26 hpter 12 Trigonometri rtios nd pplitions 351 Exmple 17 Exmple 18 Exmple 19 2 = = = 5625 = 75 H = tn H = 75 tn 12 = The height of the mst is m, orret to two deiml ples. tn = H 45 = The ngle of elevtion of the top of the mst, H, from is to the nerest minute. Exerise 12G 1 EFGH is uoid with dimensions s shown. Find the length of FH the length of H the mgnitude of ngle HF 8 m d the mgnitude of ngle HG. H E 60 m m θ 12 m 45 m G 45 m H H F 5 m 2 V is right pyrmid with squre se. The sides of the se re 8 m in length. The height, VF,ofthe pyrmid is 12 m. Find V the length of EF the mgnitude of ngle VEF the length of VE d the length of sloping edge e the mgnitude of ngle V E F f the surfe re of the pyrmid. 8 m 3 tree stnds t the orner of squre plying field. Eh side of the squre is 100 m long. t the entre of the field the tree sutends n ngle of 20. Wht ngle does it 20 T sutend t eh of the other three orners of the field? 100 m 100 m SMPLE mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

27 352 Essentil dvned Generl Mthemtis 4 Suppose tht,, nd X re three points in horizontl plne nd is point vertilly ove X. Ifthe length of = 85 m nd the mgnitudes of ngles, nd X re 45, 90 nd 32 respetively, find the distne the height X m 5 Stnding due south of tower 50 m high, the ngle of elevtion of the top is 26. Wht is the ngle of elevtion fter wlking distne 120 m due est? 6 From the top of liff 160 m high two uoys re oserved. Their erings re 337 nd 308. Their respetive ngles of depression re 3 nd 5. lulte the distne etween the uoys. 7 Find the mgnitude of eh of the following ngles for the uoid shown. E HF EH E 6 m H 12 m F 32 X G 5 m 8 From point due north of tower, the ngle of elevtion to the top of the tower is 45. From point, 100 m on ering of 120 from, the ngle of elevtion is 26.Find the height of the tower. 9 nd re two positions on level ground. From n dvertising lloon t vertil height of 750 m, is oserved in n esterly diretion nd t ering of 160. The ngles of depression of nd s viewed from the lloon re 40 nd 20 respetively. Find the distne etween nd. 10 right pyrmid, height 6 m, stnds on squre se of side 5 m. Find the length of sloping edge the re of tringulr fe. 11 light irrft flying t height of 500 m ove the ground is sighted t point due est of n oserver sttioned t point on the ground, mesured horizontlly to e 1 km from the plne. The irrft is flying south west (long ' )t300 km/h. 500 m How fr will it trvel in one minute? Find its ering from ( )tthis time. Wht will e its ngle of elevtion from t this time? ' 1000 m SMPLE 45 ' mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

28 hpter 12 Trigonometri rtios nd pplitions ngles etween plnes nd more diffiult 3- prolems ngles etween plnes onsider ny point P on the ommon line of two plnes 1 nd 2.IfP nd P re drwn t right ngles to the ommon line so tht P is in 1 nd P is in 2, then ngle P is the ngle etween 1 nd 2. Note: If one of the plnes, 2 sy, is horizontl, then P is lled line of gretest slope in the plne 1. Exmple 20 Π 1 Π 2 P Π 1 Π 2 P θ lines of gretest slope ngle of gretest slope ' ' Given the uoid shown in the digrm, find the ngle etween nd the plne the ngle etween the plnes nd '. ' 3 3 To find the ngle etween nd the plne, we need the projetion of ' ' in the plne. So we drop perpendiulr from to the plne, i.e. the ' line, nd join the foot of the perpendiulr to θ ' 3, i.e.. The required ngle lies etween 3 nd. rwing seprte digrms showing the se nd the setion through, nd we hve SMPLE ' ' 3 nd 3 Thus = (3) 2 + (3) 2 = 3 2 nd tn = 3 2 = Hene the required ngle,,is13.3. θ ' ' mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

29 354 Essentil dvned Generl Mthemtis The line ommon to the plnes nd is.ifm is the midpoint of this line, then M is perpendiulr to in plne nd M is perpendiulr to in plne. Thus is the ngle etween the plnes nd. Exmple 21 φ ' ' 3 M 3 ut M = 1 2 = 1 2 (3 2) ( 3 ) 2 2 Hene tn = = 2 3 i.e. the required ngle is 25.2 Three points, nd re on horizontl line suh tht = 70 m, nd = 35 m. The ngles of elevtion of the top of tower re, nd where tn = 1 13, tn = 1 15 nd tn = 1 (s shown in the digrm). 20 The foot of the tower is t the sme level s, nd. Find the height of the tower. tn = 1 13 tn = 1 15 tn = 1 20 Let the height of the tower, PQ, eh m. Then h = Q tn = Q tn = Q tn whih implies Q = 13h, Q = 15h, Q = 20h Now onsider the se tringle Q. Using the osine formul in Q nd Q, ' ' α P Q β φ M 70 m 35 m SMPLE os = (70)2 + (15h) 2 (13h) 2 2(70)(15h) nd os (180 ) = os = (35)2 + (15h) 2 (20h) 2 2(35)(15h) 13h Q 15h 70 m θ γ 20h 35 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

30 hpter 12 Trigonometri rtios nd pplitions 355 (70) 2 + (15h) 2 (13h) 2 Hene = (20h)2 (15h) 2 (35) 2 2(70)(15h) 2(35)(15h) h 2 = 2(175h ) 7350 = 294h 2 Hene h = 5 The height of the tower is 5 m. Exmple 22 sphere rests on the top of vertil ylinder whih is open t the top. The inside dimeter of the ylinder is 8 m. The sphere projets 8 m ove the top of the ylinder. Find the rdius length of the sphere. This 3- prolem n e represented y 2- digrm without loss of informtion. From the digrm, in,ifrdius length of sphere is r m, = (8 r)m, = r m, = 4m Using Pythgors theorem (8 r) = r r + r = r 2 16r + 80 = 0 r = 5 So rdius length of sphere is 5 m. Exmple 23 ox ontins two stndrd golf lls tht fit snugly inside. The ox is 85 mm long. Wht perentge of the spe inside the ox is ir? 2- digrms my e used to represent the 3- sitution. SMPLE 85 mm side view Use r mm = rdius length of ll Now length of ox = 85 mm = 4r mm r = 85 4 i.e. r = end view 8 m 8 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

31 356 Essentil dvned Generl Mthemtis So ox hs dimensions 85 mm y 42.5 mm y 42.5 mm Exmple 20 Exmple 21 Now volume of ox in mm 3 = (using V = h) volume of two golf lls = ( using V = 4 3 r 3 ) = ( ) So perentge ir = = 47.6% (to one deiml ple) Exerise 12H 1 The digrm shows retngulr prism. d = 4 units, = 3 units, G = units. lulte the res of the fes FE, GF,. lulte the mgnitude of the ngle whih plne GF mkes with the se. lulte the mgnitude of the ngle whih plne HG mkes with the se. lulte the mgnitude of the ngle whih G mkes with the se. 2 V is right pyrmid with squre se. V = 2 nd V =. Find the slope of edge V, i.e., the mgnitude of V. Find the slope of the fe V. 5 3 hill hs grdient 12.IfF mkes n ngle of 45 with F E 5 the line of gretest slope, find the grdient of F 12 the mgnitude of F. 4 The ross-setion of right prism is n isoseles tringle. = = 16 m nd the mgnitude of = 58. The equl edges, E nd F re prllel nd eh of length 12 m. lulte the length of the length of E the mgnitude of the ngle etween E nd E. SMPLE 5 vertil tower, T,ofheight 50 m, stnds t point on horizontl plne. The points,, nd lie on the sme horizontl plne, is due west of nd is due south of. The ngles of elevtion of the top, T, ofthe tower from nd re 25 nd 30 respetively. lulte, giving nswers to the nerest metre, the distnes i ii iii lulte the ngle of elevtion of T from the midpoint, M, of. E H F G mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

32 hpter 12 Trigonometri rtios nd pplitions 357 Exmple 23 Exmple 22 6 right squre pyrmid, vertex, stnds on squre se. The height is 15 m nd se side length is. Find the length of the slnt edge the inlintion of slnt edge to the se the inlintion of sloping fe to the se d the mgnitude of the ngle etween two djent sloping fes. 7 post stnds t one orner of retngulr ourtyrd. The elevtions of the top of the post from the nerest orners re 30 nd 45.Find the elevtion from the digonlly opposite orner. 8 V is regulr tetrhedron with se. (ll fes re equilterl tringles.) Find the mgnitude of the ngle etween sloping edge nd the se djent sloping fes. 9 n oserver t point t se level notes n irrft due est t n elevtion of 35.t the sme time n oserver t,2kmdue south of, reports the irrft on ering of 50. lulte the ltitude of the irrft. 10 Four ongruent spheres, rdius length, re pled on horizontl tle so tht eh touhes two others nd their entres form squre. fifth ongruent sphere rests on them. Find the height of the top of this fifth sphere ove the tle. 11 FE represents setion of ski run whih hs uniform inlintion of 30 to the horizontl. E = 100 m, = 100 m. skier trverses the slope from to F. lulte E F the distne tht the skier hs trversed the inlintion of the skier s pth to the horizontl. 12 sphere of rdius length 8 m rests on the top of hollow inverted one of height 15 m whose vertil ngle is 60.Find the height of the entre of the sphere ove the vertex of the one. 13 ue hs edge length m. Wht is the rdius length, in terms of, of the sphere tht just ontins the ue the sphere tht just fits inside the ue? SMPLE 14 In this digrm is vertil nd is horizontl. is right ngle. = 20 m, = 40 m, = 30 m. lulte the inlintion to the horizontl of E where E is the line of gretest slope E where E is the midpoint of. E mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

33 358 Essentil dvned Generl Mthemtis Review hpter summry The sine rule is used to find unknown quntities in tringle when one of the following situtions rises: one side nd two ngles re given two sides nd the non-inluded ngle re given. In the first of the two ses unique tringle is defined ut for the seond it is possile for two tringles to exist. Lelling onvention The following onvention is followed. Interior ngles re denoted y upper se letters nd the length of the side opposite n ngle is denoted y the orresponding lower se letter. e.g. The mgnitude of ngle is denoted y. The length of side is denoted y. The sine rule sttes tht for tringle sin = sin = sin The osine rule is used to find unknown quntities in tringle when one of the following situtions rises: two sides nd n inluded ngle re given three sides re given. The osine rule sttes tht for tringle 2 = os or equivlently os = The symmetril results lso hold, i.e. 2 = os 2 = os SMPLE It is known tht the re of tringle is given y the formul re = 1 2 h re = 1 2 se length height y oserving tht h = sin the following formul n e found: re of tringle = 1 sin 2 i.e. the re is given y hlf the produt of the length of two sides nd the sine of the ngle inluded etween them. h mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

34 hpter 12 Trigonometri rtios nd pplitions 359 The length of the minor r (red line) is given y the formul l = r The re of setor (shded) is given y the formul re = 1 2 r 2 hord length (red line) is given y l = 2r sin 2 The re of segment (shded) is given y re = 1 2 r 2 ( sin ) ngle etween plnes onsider ny point P on the ommon line of two plnes 1 nd 2.IfP nd P re drwn t right ngles to the ommon line so tht P is in 1 nd P is in 2 then ngle P is the ngle etween 1 nd 2. Note: If one of the plnes, 2 sy, is horizontl, then P is lled line of gretest slope in the plne 1. Multiple-hoie questions Π 1 Π 1 Π 2 P Π 2 θ P θ r lines of gretest slope r θ l ngle of gretest slope 1 In tringle XYZ, x = 21 m, y = 18 m nd YXZ = 62. The mgnitude of XYZ, orret to one deiml ple, is E In tringle, = 30, = 21 nd os = 51. The vlue of, tothe nerest whole 53 numer, is E 129 SMPLE 3 In tringle, = 5.2m, = 6.8mnd = 7.3 m. The mgnitude of, orret to the nerest degree, is E 98 4 The re of the tringle,where = 5m, = 3m, = 30 nd = 70,is 2.75 m m m m 2 E 8m 2 Review mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

35 360 Essentil dvned Generl Mthemtis Review 5 The length of the rdius of the irle shown, orret to two deiml ples, is 5.52 m 8.36 m 9.01 m m 130 E m 6 hord of length 5 m is drwn in irle of rdius 6 m. The re of the smller region inside the irle ut off y the hord, orret to one deiml ple, is 1.8 m 2 2.3m m m 2 E 15.5 m 2 7 From point on liff 500 m ove se level, the ngle of depression to ot is 20. The distne from the foot of the liff to the ot, to the nerest metre, is 182 m 193 m 210 m 1374 m E 1834 m 8 tower 80 m high is 1.3 km wy from point on the ground. The ngle of elevtion to the top of the tower from this point, orret to the nerest degree, is E 89 9 mn wlks 5 km due est followed y 7 km due south. The ering he must tke to return to the strt is E ot sils t ering of 215 from to. The ering it must tke from to return to is E 250 Short-nswer questions (tehnology-free) 1 Find x. Find y. SMPLE 2 Find H, where H is the ltitude 30 M, where M is the medin. 40 m 40 m 3 From port P,ship Q is 20 km wy on ering of 112, nd ship R is 12 km wy on ering of 052.Find the distne etween the two ships. 4 In qudrilterl, = 5m, = 5m, = 7m, = 120 nd = 90. Find the length of the digonl the re of tringle the re of tringle d the re of the qudrilterl. 30 x m y 6 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

36 hpter 12 Trigonometri rtios nd pplitions If sin x = sin 37 nd x is otuse, find x. 6 point T is 10 km due north of point S, nd point R, whih is est of stright line joining T nd S,is8kmfrom T nd 7 km from S. lulte the osine of the ering of R from S. 7 In, = 5 m, mgnitude of = 60 nd = 6 m. lulte the sine of. 8 The re of setor of irle with rdius 6 m is 33 m 2. lulte the ngle of the setor. 9 The digrm shows two survey points, nd, whih re on n est west line on level ground. From point, the ering of the foot of tower is 060,while from the ering of the tower is 045.Find i the mgnitude of T ii the mgnitude of T 6 2 Given tht sin 15 =, find 4 T nd T. N 300 m N ot sils 11 km from hrour on ering of 220.Itthen sils 15 km on ering of 340.How fristhe ot from the hrour? 11 heliopter leves heliport nd flies 2.4 km on ering of 150 to hek point. It then flies due est to its se. If the ering of from is 120, find the distnes nd. The heliopter flies t onstnt speed throughout nd tkes five minutes to fly from to. Find its speed. 12 The digrm shows irle of rdius length 13 m nd hord of length 24 m. lulte the length of r the re of the shded region. SMPLE 13 setor of irle hs n r length of 30 m. If the rdius of the irle is 12 m, find the re of the setor. 14 hord PQ of irle, rdius 5 m, sutends n ngle of two rdins t the entre of the irle. Tking to e 3.14, lulte, orret to one deiml ple, the length of the mjor r PQ. 15 From liff top 11 m ove se level, two ots re oserved. ne hs n ngle of depression of 45 nd is due est, the other n ngle of depression of 30 on ering of 120. lulte the distne etween the ots T Review mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird

37 362 Essentil dvned Generl Mthemtis Review Extended-response questions 1 is tower 60 m high on top of hill. The mgnitude of is 49 nd the mgnitude of is 37. Find the mgnitude of ngles, nd. Find the length of. Find the height of the hill, i.e. the length of. 2 The ngle of setor of irle, entre nd rdius length 12 m hs mgnitude 2.5 rdins. The setor is folded so tht nd re joined to form one. lulte the se rdius length of the one the urved surfe re of the one the shortest distne etween two points dimetrilly opposed on the edge of the se. 3 tower 110 m high stnds on the top of hill. From point t the foot of the hill the ngle of elevtion of the ottom of the tower is 7, nd tht of the top is 10. Find the mgnitude of ngles T, T nd T. Use the sine rule to find the length of. Find, the height of the hill. 4 Point S is distne of 120 m from the se of uilding. n the uilding is n eril,. The ngle of elevtion from S to is 57. The ngle of elevtion from S to is 59.Find the distne the distne the distne. 7 5 From the top of ommunitions tower, the ngles of depression of two points nd on horizontl line through the foot of the tower re 30 nd 40. The distne etween the points is 100 m. Find the distne T 100 m the distne T the height of the tower. S T ' 110 m m T top of tower SMPLE 6 ngles V, V nd re right ngles. Find the distne V the distne V the distne d the mgnitude of ngle V. 8 m se of tower 8 m V 6 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & vid Hird