Trigonometry. Trigonometry. labelling conventions. Evaluation of areas of non-right-angled triangles using the formulas A = 1 ab sin (C )

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1 8 8 Pythgors theorem 8 Pythgoren trids 8 Three-dimensionl Pythgors theorem 8D Trigonometri rtios 8E The sine rule 8F miguous se of the sine rule 8G The osine rule 8H Speil tringles 8I re of tringles res of study The solution of right-ngled tringles using trigonometri rtios The solution of tringles using the sine nd osine rules Evlution of res of non-right-ngled tringles using the formuls = 1 sin ( nd = s s s s ( ( ( eookplus is rnh of mthemtis tht is used to solve prolems involving the reltionships etween the ngles nd sides of tringles. Often the prolem is desriptive one nd, to solve it onfidently, you need to visulise the sitution nd drw n pproprite digrm or sketh. lelling onventions When we use trigonometry to solve prolems involving tringles, there re severl lelling onventions tht help us remin ler out the reltionships etween the verties, ngles nd lines eing used. These will e eplined s they rise; however, the si onvention used in this ook is shown in the figure elow right. Note the use of itlis. The ngle is t verte, whih is opposite line. The ngle is t verte, whih is opposite line. The ngle is t verte, whih is opposite line. To void luttered digrms, only the verties (,, re usully shown; the ssoited ngles (,, re ssumed. Digitl do 10 Quik Questions Note: Nturlly, we do not need suh lels in ll digrms, nd sometimes we wish to lel verties, ngles nd lines in other wys, ut these will lwys e ler from the digrm nd its ontet. hpter 8 365

2 8 pythgors theorem efore investigting the reltionships etween the ngles nd sides of tringle, we should onsider prolem-solving tehnique tht involves only the sides of tringles: Pythgors theorem. Pythgors theorem is ttriuted to the Greek mthemtiin nd philosopher, Pythgors, round 500. (However, the priniple ws known muh erlier, nd it seems tht even the pyrmid uilders of nient Egypt used the theorem in onstruting the pyrmids. The theorem desries the reltionship etween the lengths of the sides of ll right-ngled tringles. Pythgors theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides, or (hypotenuse = + nd, therefore, to find, = + where is the longest side or hypotenuse nd nd re the two shorter sides. Note: euse the eqution = + hs eome stndrd wy of epressing Pythgors theorem, we often djust the lelling onvention to use for the hypotenuse no mtter how the opposite (right ngle nd verte is lelled. However, this will lwys e ler from the digrm. The longest side is lwys opposite the lrgest ngle (90 for right-ngled tringles nd similrly, the shortest side is opposite the smllest ngle. To find one of the shorter sides (for emple, side, the formul trnsposes to: = nd so = WorkEd EmplE 1 Find the length of the unknown side (to 1 deiml ple in the right-ngled tringle shown. 4 m Think WriTE/displY 7 m Method 1: Using the rule 1 Note tht the tringle is right-ngled nd we need to find the unknown length, given the other two lengths. 4 Lel the sides of the tringle, using the onvention tht is the hypotenuse. 3 Sustitute the vlues into the pproprite formul. 4 Write the nswer using the orret units nd to the pproprite degree of ury. 7 = + lterntively, = = + = = = 65 = = 65 = 65 = 8.06 The unknown side s length is 8.1 m, orret to 1 deiml ple. 366 mths Quest 1 Further mthemtis for the sio lsspd

3 Method : Using S lultor 1 On the NumSolve sreen, use the /V soft keyord to omplete the entry line s: = + Then press E. The list of vriles will pper. Enter the known vlues nd selet the vrile to solve y heking the djent ullet. Then tp: 1 (on the tool r OK Write your nswer. The unknown side s length is 8.1 m, orret to 1 deiml ple. WorkEd EmplE Find the mimum horizontl distne (to the nerest metre ship ould drift from its originl nhored point, if the nhor line is 50 metres long nd it is 4 metres to the ottom of the se from the end of the nhor line on top of the ship s dek. Think WriTE/displY Method 1: Using the rule 1 Sketh suitle digrm of the prolem given. Note tht the tringle is right-ngled nd we need to find the unknown length, given the other two lengths. Simplify the tringle, dding known lengths, nd lel the sides using the onvention tht is the hypotenuse. 50 metres? 4 metres 3 Sustitute the vlues into the pproprite formul. 4 Write the nswer using the orret units nd to the required ury. = + lterntively, 50 = + 4 = = = 50 4 = = = = = = The ship n drift 49 metres, orret to the nerest metre. hpter 8 367

4 Method : Using S lultor 1 On the NumSolve sreen, use the /V soft keyord to omplete the entry line s: = + Then press E. The list of vriles will pper. Enter the known vlues nd selet the vrile to solve y heking the djent ullet. Then tp: 1 (on the tool r OK Write your nswer. The ship n drift 49 metres, orret to the nerest metre. rememer 1. Pythgors theorem is used: ( only on right-ngled tringles ( to find n unknown length or distne, given the other two lengths.. When using Pythgors theorem: ( drw n pproprite digrm or sketh ( ensure the hypotenuse side,, is opposite the right ngle (90 ( = + or = +. EErisE 8 eookplus Digitl do Spredsheet 10 Pythgors theorem pythgors theorem 1 WE 1 Find the length of the unknown side (to 1 deiml ple in eh of the following rightngled tringles d 11.6 e 3 4 f mths Quest 1 Further mthemtis for the sio lsspd

5 n irrft is flying t n ltitude of 5000 metres. If its horizontl distne from the irport is 3 kilometres, wht is the distne (to the nerest metre from the irport diretly to the irrft? 3 Wht is the length (to the nerest millimetre of digonl re on retngulr gte tht is 600 mm wide nd 1800 mm high? 4 WE Find the length of the unknown side (to 1 deiml ple in eh of the following rightngled tringles d 7 5 e f 15 5 lulte the lengths of the sloping sides in the following. (Rememer to onstrut suitle right-ngled tringle mm8 mm 30 mm d e f 6 m 3 m 14 m 305 m 15 m 8 m 1 m 460 m 6 lulte the vlue of the pronumerls mm 4.8 mm 6.3 mm d d.3 Em tip Students must not round n nswer in the middle of lultion nd then ontinue working with the rounded numer, thus ompounding round-off error. This ws most evident in the Geometry nd trigonometry module. [ssessment report 007] hpter 8 369

6 7 One of the smller sides of right-ngled tringle is 16 metres long. The hypotenuse is 8 metres longer thn the other unknown side. Drw suitle tringle to represent this sitution. Write n epression to show the reltionship etween the three sides. Stte the lengths of ll three sides. 8 m The length of side F in the digrm elow is: 3 D 5 E 6 1 m 9 m To the nerest metre, the length of le tht would onnet the roofs of two uildings tht re 40 metres nd 80 metres high respetively nd re 30 metres prt (s shown elow is: 40 metres 45 metres 50 metres D 55 metres E none of these F E D 8 pythgoren trids Pythgoren trid is set of 3 numers whih stisfies Pythgors theorem. n emple is the set of numers 3, 4, 5 where = + So, 5 = = The digrm elow illustrtes this reltionship nother Pythgoren trid is the multiple (sle ftor of of the ove set: 6, 8, 10. Others re 5, 1, 13 nd 0.5, 1., 1.3. Prove these for yourself mths Quest 1 Further mthemtis for the sio lsspd

7 Worked Emple 3 Is the set of numers 4, 6, 7 Pythgoren trid? Think Write 1 Find the sum of the squres of the two smller numers = = 5 Find the squre of the lrgest numer. 7 = 49 3 ompre the two results. The numers form Pythgoren trid if the results re the sme. 4 Write your nswer. The set of numers 4, 6, 7 is not Pythgoren trid. nother wy to generte Pythgoren trids is y using the following rule: Step 1. Squre n odd numer (5 = 5. Step. Find the two onseutive numers tht dd up to the squred vlue ( = 5. Step 3. The trid is the odd numer you strted with together with the two onseutive numers (5, 1, 13. Try to find trid for the odd numer 9. tringle whose sides form Pythgoren trid ontins right ngle, whih is opposite the longest side. This result n e illustrted pproimtely with rope of ny length, y tying 11 eqully sped knots nd forming tringle with sides equl to 3, 4 nd 5 spes, s shown elow. In doing this, right ngle is formed opposite the 5-spe side. Worked Emple 4 tringle hs sides of length 8 m, 15 m nd 17 m. Is the tringle right-ngled? If so, where is the right ngle? Think 1 The tringle is right-ngled if its side lengths form Pythgoren trid. Find the sum of the squres of the two smller sides. Find the squre of the longest side nd ompre to the first result. Write = = = = The tringle is right-ngled. 3 The right ngle is opposite the longest side. The right ngle is opposite the 17 m side. hpter 8 371

8 rememer 1. Pythgoren trid is set of three numers whih stisfies Pythgors theorem.. tringle whose side lengths form Pythgoren trid hs right ngle opposite the longest side. 3. Some ommon trids re: 3, 4, 5 6, 8, 10 9, 1, , 0.4, 0.5 5, 1, 13 10, 4, 6 0.5, 1., 1.3 7, 4, 5 9, 40, 41 EErisE 8 pythgoren trids 1 WE3 re the following sets of numers Pythgoren trids? 9, 1, 15 4, 5, 6 30, 40, 50 d 3, 6, 9 e 0.6, 0.8, 1.0 f 7, 4, 5 g 6, 13, 14 h 14, 0, 30 i 11, 60, 61 j 10, 4, 6 k 1, 16, 0 l, 3, 4 omplete the following Pythgoren trids. Eh set is written from smllest to lrgest. 9,, 15, 4, 5 1.5,.0, d 3,, 5 e 11, 60, f 10,, 6 g, 40, 41 h 0.7,.4, 3 For eh of the sets whih were Pythgoren trids in question 1, stte whih side the right ngle is opposite. 4 WE4 tringle hs sides of length 16 m, 30 m nd 34 m. Is the tringle right-ngled? If so, where is the right ngle? 5 tringle hs sides of length 1 m, 13 m nd 18 m. Is the tringle right-ngled? If so, where is the right ngle? 6 Find the unknown length in eh se elow. Rdius = 3.5 m d 4 m eookplus Digitl do Spredsheet 134 Pythgoren trids 9 41 d d e e f N d 6 km km E 37 mths Quest 1 Further mthemtis for the sio lsspd

9 7 n thlete runs 700 m north nd then.4 km west. How fr wy is the thlete from the strting point? MTHS 8 Find the perimeter of the flg s shown t right. QUEST 9 m Whih of the following is Pythgoren trid? 7, 14, 1 1., 1.5, 3.6 3, 6, 9 D 1, 13, 5 E 15, 0, 5 10 m Whih of the following is not Pythgoren trid? 5, 4, 3 6, 9, 11 13, 84, 85 D 0.9, 4.0, 4.1 E 5, 1, m 180 m 00 m 8 Three-dimensionl pythgors theorem Mny prtil situtions involve 3-dimensionl ojets with perpendiulr plnes nd therefore the pplition of Pythgors eookplus Intertivity int-0189 Three-dimensionl Pythgors theorem theorem. To solve 3-dimensionl prolems, refully drwn nd lelled digrm will help. It is lso of enefit to identify right ngles to see where Pythgors theorem n e pplied. This enles you to progress from the known informtion to the unknown vlue(s. WorkEd EmplE 5 To the nerest entimetre, wht is the longest possile thin rod tht ould fit in the oot of r? The oot n e modelled s simple retngulr prism with the dimensions of 1.5 metres wide, 1 metre deep nd 0.5 metres high. eookplus Tutoril int-0464 Worked emple 5 Think WriTE 1 Drw digrm of the retngulr prism. F G Identify the orienttion of the longest ojet from one orner to the furthest digonlly opposite orner. In this se, it is G. E 0.5 m 1.5 m H D 1.0 m 3 Identify the two right-ngled tringles neessry to solve for the two unknown lengths. 4 Drw the tringles seprtely, identifying the lengths ppropritely. y 1.5 m D 1.0 m y G 0.5 m 5 lulte the length of digonl. = + y = y = y = 3.5 y = 35. y = (to 3 deiml ples The length of is 1.8 metres (to 1 deiml ple. hpter 8 373

10 6 lulte the length of digonl G, using the lulted length for. Note: To void truntion error, use the most urte form, whih is the surd Write the nswer using the orret units nd level of ury. = + (lterntive form.5 + ( = = = 35. = The longest rod tht ould fit in the r oot is 187 entimetres, lulted to the nerest entimetre. WorkEd EmplE 6 To find the height of 100-metre squre-sed pyrmid, with slnt height of 00 metres s shown, lulte the: length of (in surd form length of O (in surd form height of the pyrmid VO (to the nerest metre. Think WriTE/displY D V O 00 m 100 m Method 1: Using the rule lulte the length of digonl in the right-ngled tringle,. Write surds in their simplest form. = = = (lterntive form = = m 100 m The length of is 100 metres. O is hlf the length of. Length of O is 100 or 50 metres. 1 lulte the height of the pyrmid, VO, in the right-ngled tringle, VO. 00 m V = (lterntive form VO = ( VO = VO = VO = m O Write the nswer using the orret units nd level of ury. The height of the pyrmid, VO, is 187 metres, lulted to the nerest metre. 374 mths Quest 1 Further mthemtis for the sio lsspd

11 Method : Using S lultor 1 On the Min sreen, tp: Intertive dvned solve Enter = + into the eqution o nd in the vrile o, then tp OK. omplete the entry line s: solve( = +, = 100 = 100 Then press E. Only the positive solution pplies. djust the entry line to: solve( = +, = 00 = 5 Then press E. Note: Highlighting the nswer nd tpping. will produe n pproimte nswer, ± Only the positive solution pplies. 3 Write the nswer using the orret units nd level of ury. The height of the pyrmid, VO, is 187 metres, lulted to the nerest metre. rememer To solve prolems involving 3-dimensionl Pythgors theorem: 1. Drw nd lel n pproprite digrm.. Identify the right ngles. 3. Identify right-ngled tringles tht enle the informtion given to e used to find the unknown vlue(s. 4. To void truntion error, try to use the surd form (for emple, 37 rther thn if the result is required in further lultions. EErisE 8 Three-dimensionl pythgors theorem 1 WE5 To the nerest entimetre, wht is the longest thin rod tht ould fit inside -metre-ue o (uoid? To the nerest entimetre, wht is the longest drumstik tht ould fit in retngulr toy o whose dimensions re 80 m long y 80 m wide y 60 m high? hpter 8 375

12 3 For eh of the prisms shown, lulte: i the length of ii the length of G. H G E F 10 m E F G H J H I G 5 m D 40 m 400 mm D 300 mm 100 mm F 5 m D 6 m 14 m E 40 m 4 WE6 For eh of the pyrmids shown, lulte: i the length of ii the perpendiulr height. G G 40 m 600 m D 15 m 0 m D 3 4 km 3 km metre long rmp rises to height of 1. metres. How long (to 1 deiml ple is the se of the rmp? 6 M Two guide wires re used to support flgpole s shown. The height of the flgpole would e losest to: 3 m 8 m 1 m D 1 m E 6 m Wire 8.5 m Wire m 4 m 7 Find the vlues of the pronumerls (to 1 deiml ple in the pyrmid t right Find the lengths of nd DH (to deiml ples, where = 7.00 m nd H = m F G E H D 376 Mths Quest 1 Further Mthemtis for the sio lsspd

13 9 mn moves through two-level mze y following the solid lk line, s shown in the digrm. Wht is the diret distne from his strting point,, to his end point, F (to the nerest metre? 40 m D F G 30 m E 30 m 10 m H Not to sle 10 In eh of the following typil uilding strutures find the length of the unknown ross-re shown in red. 5 m 3 m 11 m 3 m.6 m 11 For the offee tle design t right, find the length of the legs (to the nerest millimetre if the offee tle is to e: 500 mm off the ground Offset distne 700 mm off the ground Tle nd the legs re offset from the vertil y distne of: height i 100 mm ii 150 mm. 1 Find the length of the re, G (to the nerest entimetre, tht is needed to reinfore the wedge-shped struture shown. D E G 4.0 m F 1.0 m Em Tip This question involves two steps. Mny students tend to round their nswer to the first step nd then use it in the seond step. Et numers should e retined in the lultor nd used in the seond step, unless the first step is the nswer to speifi question. [ssessment report 005].0 m eookplus 8d Trigonometri rtios Trigonometri rtios inlude the sine rtio, the osine rtio nd the tngent rtio; three rtios of the lengths of sides of right-ngled tringle dependent on given ute ngle. lelling onvention For the trigonometri rtios the following lelling onvention should e pplied: 1. The hypotenuse is opposite the right ngle (90.. The opposite side is diretly opposite the given ngle, θ. 3. The djent side is net to the given ngle, θ. Digitl do WorkSHEET 8.1 hpter 8 377

14 onsider the three tringles drwn elow. We know from the previous hpter on similrity tht, DE nd FG re similr euse the orresponding ngles re the sme. Therefore, the orresponding sides re in the sme rtio (sle ftor. D 30 F 30 E 30 G rtio of lengths of sides opy nd omplete the tle elow y identifying nd mesuring the lengths of the three sides for eh of the three tringles ove. Evlute the rtios of the sides. Length of side Tringle Opposite djent Hypotenuse Rtio of lengths of sides Opposite djent Opposite Hypotenuse Hypotenuse djent DE FG Opposite Notie tht for eh of the rtios, for emple, the vlue is the sme for ll three Hypote nuse tringles. This is the sme for ll right-ngled tringles with the sme ute ngle. Trigonometri rtios re used in right-ngled tringles: 1. to find n unknown length, given n ngle nd side. to find n unknown ngle, given two lengths. sine rtio The sine rtio is defined s follows: Lengthof opposite side The sine of n ngle = Lengthof hypotenuseside. In short, sin (θ = Opposite Hypotenuse Opposite Hypotenuse sin (θ = O H [SOH] 378 mths Quest 1 Further mthemtis for the sio lsspd

15 WorkEd EmplE 7 Find the length (to 1 deiml ple of the line joining the verties nd in the tringle t right. 15 m 50 Think 1 Identify the shpe s right-ngled tringle with given length nd ngle. Lel the sides s per the onvention for trigonometri rtios. WriTE 15 m Hypotenuse = 50 m Opposite Identify the pproprite trigonometri rtio, nmely the sine rtio, from the given informtion. ngle = 50 Opposite side = m Hypotenuse = 15 m [SOH] 3 Sustitute into the formul. Length of opposite side sin (θ = Lengthof hypote nuse side sin (θ = O H sin (50 = 15 4 Isolte nd evlute. 15 sin (50 = = 15 sin (50 = = Write the nswer using the orret units nd level of ury. The length of the line joining verties nd is 11.5 entimetres, orret to 1 deiml ple. osine rtio The osine rtio is defined s follows: Lengtho fdjentside The osine of n ngle = Lengtho f hypote nuse side. dj ent In short, os (θ = Hypot te nuse os (θ = H [H] djent Hypotenuse In Worked emple 7 the sine rtio ws used to find the unknown length. The osine rtio n e used in the sme wy, if it is required. hpter 8 379

16 WorkEd EmplE 8 Find the length of the guy wire (to the nerest entimetre supporting flgpole, if the ngle of the guy wire to the ground is 70 nd it is nhored metres from the se of the flgpole. Think Method 1: Using the rule 1 Drw digrm to represent the sitution nd identify n pproprite tringle. Lel the digrm with the given ngle nd the given side to find n unknown side in right-ngled tringle. WriTE/displY Guy wire 70 m m Hypotenuse 3 hoose the pproprite trigonometri rtio, nmely the osine rtio. 70 m djent ngle = 70 djent side = m Hypotenuse = m [H] 4 Sustitute into the formul. os (θ = H 5 Isolte nd evlute. 6 Write the nswer using the orret units nd level of ury. Method : Using S lultor On the Geometry sreen, drw the shown digrm. Refer to your eookplus for detiled instrutions for onstrution of shpes nd mesurement of unknown vlues. os (70 = 1 os ( 70 = = os ( 70 = The length of the guy wire is 5.85 metres or 585 entimetres, orret to the nerest entimetre. 380 mths Quest 1 Further mthemtis for the sio lsspd

17 Tngent rtio The tngent rtio is defined s follows: Length of opposite side The tngent of n ngle = Length of djent side. In short, tn (θ = Opposite djent tn (θ = O [TO] Opposite djent WorkEd EmplE 9 Find the length of the shdow (to 1 deiml ple st y 3-metre tll pole when the ngle of the sun to the horizontl is 70. Think WriTE/displY Method 1: Using the rule 1 Drw digrm to represent the sitution nd identify n pproprite tringle. 3 m 70 Lel the digrm with the given ngle nd the given side in order to find n unknown side in right-ngled tringle. Opposite 3 m 3 Identify the pproprite trigonometri rtio, nmely the tngent rtio. 70 m djent ngle = 70 Opposite side = 3 m djent side = m [TO] 4 Sustitute into the formul. tn (θ = O tn (70 = 3 5 Isolte nd evlute. 1 tn ( 70 = 3 3 = tn (70 = hpter 8 381

18 6 Write the nswer using the orret units nd level of ury. The length of the shdow is pproimtely 1.1 metres, orret to 1 deiml ple. Method : Using S lultor On the Geometry sreen, drw the shown digrm. Refer to your eookplus for detiled instrutions for onstrution of shpes nd mesurement of unknown vlues. Finding n unknown ngle If the lengths of the sides of tringle re known, unknown ngles within the tringle n e found. WorkEd EmplE 10 Find the smllest ngle (to the nerest degree in 3, 4, 5 Pythgoren tringle. Think Method 1: Using the rule WriTE/displY 1 The smllest ngle is opposite the smllest side. Lel the sides s given y onvention for trigonometri rtios. Opposite Hypotenuse Identify the pproprite rtio from the given informtion. ngle = Opposite side = 3 Hypotenuse = 5 [SOH] 3 Sustitute into the formul. sin (θ = O H sin ( = onvert the rtio to deiml. sin ( = Evlute. = sin 1 (0.6. = Write the nswer using the orret units nd level of ury. The smllest ngle is 37, orret to the nerest degree. 38 mths Quest 1 Further mthemtis for the sio lsspd

19 Method : Using S lultor On the Geometry sreen, drw the shown digrm. Refer to your eookplus for detiled instrutions for onstrution of shpes nd mesurement of unknown vlues. In Worked emple 10 the sine rtio ws used to find the unknown ngle. You would use the sme method if either the osine or tngent rtios were required insted. In the prtiulr se of Worked emple 10, ny of the three rtios ould e used sine ll the sides re known. rememer 1. trigonometri rtio is simply the rtio of one side of right-ngled tringle to nother. sin (θ = O os (θ = tn (θ = O H H. The rtios re used to find n unknown: ( side (given nother side nd n ngle ( ngle (given the lengths of two sides. 3. To solve prolem: ( drw n pproprite right-ngled tringle ( lel the given sides with respet to the given ngle s hypotenuse, opposite or djent ( identify whih trigonometri rtio is involved: SOH H TO helps you to rememer whih omintion of sides re in eh of the three rtios (d use the pproprite formul to solve for the unknown. EErisE 8d Trigonometri rtios 1 WE 7 Find the length of the unknown side (to 1 deiml ple in eh of the following tringles. 0.5 m 1 km 430 mm hpter 8 383

20 eookplus Digitl dos SkillSHEET 8.1 Identifying sides of right-ngled tringle with respet to the given ngle SkillSHEET 8. Finding trigonometri vlues nd ngles d mm e 5 15 m y f 9 mm WE8 ot is moored in lm wters with its depth sounder registering 14.5 m. If the nhor line mkes n ngle of 7 with the vertil, wht is the length of line (to the nerest metre tht is out of the ot? 3 WE9 person is hoping to swim diretly ross stright river from point to point, distne of 15 m. The river rries the swimmer downstrem so tht she tully rehes the other side t point. If the line of her swim,, mkes n ngle of 67 with the river nk, find how fr (to the nerest metre downstrem from point she finished. 4 Find the vlue of the missing side (to 1 deiml ple of the following tringles. 61 Em Tip Lelling nd/or drwing digrms my help students to gin mrks for pplying the orret method. [ssessment report 005] 1 45 m Find the vlue of the unknown sides (to 1 deiml ple of the following shpes. 15 m 6.5 m m 0 m WE 10 Find the size of the unknown ngle (to the nerest degree in eh of the following tringles. d m m 400 mm 500 mm mths Quest 1 Further mthemtis for the sio lsspd

21 7 Find the vlues of the unknown ngle, (to the nerest degree. m 1. m 10 m 11 m 4 m 1 m 8 Find the sizes of the two ute ngles in 6, 8, 10 Pythgoren tringle. 9 m If m is the height rehed y the ldder in the digrm t right, then is equl to: D 0.94 E 1.88 m m The orret epression for the ngle of elevtion, θ, of the rmp is: sin 1 4 os tn D tn 1 4 E os m The orret epression for the vlue of in the figure t right is: tn( 37 os( m 4 4 tn( D E 3 m tn( 37 sin( 37 1 m flgpole metres tll sts 0.6-metre long shdow. The ngle of the sun to the ground is: D 7 E In the digrm t right find θ (to the nerest degree, metres nd y metres (oth to 1 deiml ple m jvelin is thrown so tht 15 m of its pointy end stiks into 60 y the ground. The sun is diretly overhed, sting shdow of 90 m 4 m in length. Determine the ngle (to the nerest degree tht the jvelin mkes with the ground. 15 hot ir lloon is hovering in strong winds, 10 m vertilly ove the ground. It is eing held in ple y tut 1 m length of rope from the lloon to the ground. Find the ngle (to the nerest degree tht the rope mkes with the ground. 16 rmp joins two points, nd. The horizontl distne etween nd is 1. m, nd is 5 m vertilly ove the level of. Find the length of the rmp (in metres to deiml ples. Find the ngle tht the rmp mkes with the horizontl. 17 le r follows diret line from mountin pek (ltitude 150 m to ridge (ltitude 840 m. If the horizontl distne etween the pek nd the ridge is 430 m, find the ngle of desent (to the nerest degree from one to the other hpter 8 385

22 8E introdution sine nd osine rules Often the tringle tht is pprent or identified in given prolem is non-right-ngled. Thus, Pythgors theorem or the trigonometri rtios re not s esily pplied. The two rules tht n e used to solve suh prolems re: 1. the sine rule, nd. the osine rule. For the sine nd osine rules the following lelling onvention should e used. ngle is opposite side (t verte ngle is opposite side (t verte ngle is opposite side (t verte Note: To void luttered digrms, only the verties (, nd re usully shown nd re used to represent the ngles, nd. The sine rule ll tringles n e divided into two right-ngled tringles. Erlier, we sw tht the new side, h, n e evluted in two wys. h h h sin ( = h sin ( = h h = sin ( h = sin ( If we equte the two epressions for h: sin ( = sin ( nd rerrnging the eqution, we otin: sin( = sin( Using similr pproh it n e shown tht: 1. sin( = sin( = sin(. Similrly, if the tringle is lelled using other letters, for emple STU, then: s sin( S = t sin( T = sin( uu The sine rule is used if you re given: 1. two ngles nd one side or. n ngle nd its opposite side length ( omplete rtio nd one other side. For emple, in tringle t 7 m 50 right, = 7 m, = 50 nd = 9 m. ngle ould then e found using the sine rule. 9 m 386 mths Quest 1 Further mthemtis for the sio lsspd

23 WorkEd EmplE 11 Find the unknown length, m, in the tringle t right (to 1 deiml ple m eookplus Tutoril int-0465 Worked emple 11 Think WriTE/displY Method 1: Using the rule 1 Drw the tringle. ssume it is non-rightngled. Lel the tringle ppropritely for the sine rule m 3 onfirm tht it is the sine rule tht n e used s you hve the ngle opposite to the unknown side nd known side ngle rtio. 4 Sustitute known vlues into the two rtios. = = sin( sin ( sin( = = 130 = 7 m = 30 sin( 130 = 7 sin( 30 5 Isolte nd evlute. = 7 sin( 130 sin( 30 = = Write the nswer. The unknown length is 10.7 m, orret to 1 deiml ple. Method : Using S lultor 1 On the NumSolve sreen, use the soft keyord to omplete the entry line s: sin ( β = sin (γγ Then press E. Note tht the vriles for ngles nd will hve to e repled with symols suh s β nd γ. The list of vriles will pper. Enter the known vlues nd selet the vrile to solve y heking the djent ullet. Then tp: 1 (on the tool r OK Write the nswer. The unknown length,, is 10.7 m, orret to 1 deiml ple. Sometimes it is neessry to find the third ngle in tringle in order to pply the sine rule. hpter 8 387

24 WorkEd EmplE 1 Find the unknown length, m (to deiml ples m Think WriTE/displY Method 1: Using the rule 1 Drw the tringle. ssume it is non-rightngled. Lel the tringle ppropritely for the sine rule lulte the third ngle euse it is opposite the unknown side. 4 onfirm tht it is the sine rule tht n e used s you hve the ngle opposite the unknown side nd known side ngle rtio. 5 Sustitute the known vlues into the two rtios. = 180 ( = 15 = = sin( sin ( sin( = = 15 = 7 = 100 sin( 15 = 7 sin( Isolte nd evlute. = 7 sin(1 5 sin( 100 = Write the nswer. The unknown length is 1.84 m, orret to deiml ples. Method : Using S lultor On the Geometry sreen, drw the shown digrm. Refer to your eookplus for detiled instrutions for onstrution of shpes nd mesurement of unknown vlues. 388 mths Quest 1 Further mthemtis for the sio lsspd

25 WorkEd EmplE 13 For tringle PQR, find the unknown ngle (to the nerest degree, P, given p = 5 m, r = 7 m nd R = 48. Think Method 1: Using the rule 1 Drw the tringle nd ssume it is non-rightngled. P WriTE/displY 7 m Q 5 m Lel the tringle ppropritely for the sine Q rule (it is just s esy to use the given lels. p 5 r onfirm tht it is the sine rule tht n e used s you hve the ngle opposite to the unknown ngle nd known side ngle rtio. 4 Sustitute known vlues into the two rtios. 5 Isolte sin (P. 6 Evlute the ngle (inverse sine nd inlude units with the nswer. Method : Using S lultor 1 On the NumSolve sreen, use the soft keyord to omplete the entry line s: 5 7 sin( p = sin ( 48 Then press E. Set lower to 0 nd upper to 180. This instruts the lultor to lulte ll ngles etween 0 nd 180. Selet the vrile p to solve y heking the djent ullet. Then tp: 1 (on the tool r OK 48 P p q r = = sin ( P sin ( Q sin ( R p = 5 P =? r = 7 R = sin( P = sin ( 48 sin( P sin ( 48 = 5 7 sin (P = 5 sin( sin( 48 P = sin 1 7 P = 3.06 = 3 The unknown ngle is 3, orret to the nerest degree. R R eookplus Tutoril int-0466 Worked emple 13 The ngle p must e smller thn 48 s it is opposite to the shorter side. The unknown ngle is 3, orret to the nerest degree. hpter 8 389

26 Sometimes the ngle required for the sine rule is not given. In suh ses simply sutrt the two known ngles from 180, s ws done in method 1, step 3 of Worked emple 1. WorkEd EmplE 14 pir of ompsses (often lled ompss used for drwing irles hs two equl legs joined t the top. The legs re 8 entimetres long. If it is opened to n inluded ngle of 36 degrees etween the two legs, find the rdius of the irle tht would e drwn (to 1 deiml ple. Think WriTE/displY Method 1: Using the rule 1 Drw the sitution nd identify tht the tringle is non-right-ngled m Drw the tringle seprtely from the sitution nd lel it ppropritely for the sine rule. This is n isoseles tringle sine = ; therefore =. Using the ft tht the ngle sum of tringle is 180, find nd. 3 onfirm tht it is the sine rule tht n e used s you hve the ngle opposite to the unknown side nd known side ngle rtio. 4 Sustitute the known vlues into the two rtios. 8 m 36 8 m 180 = = = = 144 = 7 nd therefore = = 7 = = sin ( sin ( sin ( = y = 36 = 8 = 7 y sin ( 36 = 8 sin( 7 5 Trnspose the eqution to get the unknown y itself. 6 Evlute y to 1 deiml ple nd inlude units. y = 8 sin( 36 sin(7 y 4.9 The rdius of the irle is 4.9 m, orret to 1 deiml ple. 390 mths Quest 1 Further mthemtis for the sio lsspd

27 Method : Using S lultor 1 On the NumSolve sreen, use the soft keyord to omplete the entry line s: y 8 = sin( 36 sin ( 7 Then press E. Selet the vrile y to solve y heking the djent ullet. Then tp: 1 (on the tool r OK Write the nswer. The rdius of the irle is 4.9 m, orret to 1 deiml ple. rememer 1. Follow the pproprite lelling onvention when using the sine rule.. For the tringle shown, the sine rule sttes: = = sin( sin ( sin( Note tht only two of the three rtios need e pplied. 3. The sine rule n e used to find n unknown: ( side if its opposite ngle nd side rtio re known ngle ( ngle if its opposite side nd side rtio re known. ngle 4. When two ngles re given, it my e neessry to lulte the third ngle in order to pply the sine rule. Tht is, if nd re the known ngles, then = 180 ( +. EErisE 8E The sine rule 1 WE 11 Find the unknown length,, in eh of the following m 15 m mm d m e 50 km hpter 8 391

28 The reltive positions of the shool, hurh nd post offie in smll town re shown t the verties of the tringle t right. Find the stright-line distne etween the shool nd the post offie (to 1 deiml ple. 3 WE 1 Find the unknown length,, (to 1 deiml ple in eh se elow m Shool 3 km 86 Post Offie 85 7 mm 3 hurh m d 18 m siling epedition followed ourse s shown t right. Find the totl distne overed in the round trip. N 10.5 km 78 5 WE 13 For the following questions give nswers to the nerest degree. In, find the unknown ngle,, given = 6, = 6 nd = 5. In LMN, find the unknown ngle, M, given m = 14.1, n = 7. nd N = 18. In STU, find the unknown ngle, S, given s = 1.7, t = 16.3 nd T = 45. d In PQR, find the unknown ngle, P, given p =, r = 3.5 nd R = 18. e In, find the unknown ngle,, given = 10, = 8 nd = 80. f In PQR, find the unknown ngle, R, given p = 48, q = 1 nd P = onstrut suitle tringle from the following instrutions nd find ll unknown sides nd ngles. One of the sides is 3 m ut the smllest is 15 m. The smllest ngle is 8. 7 WE 14 Steel trusses re used to support the roof of ommeril uilding. The struts in the truss shown re eh mde from 0.8 m steel lengths nd re welded t the ontt points with the upper nd lower setions of the truss m 130 On the lower setion of the truss, wht is the distne (to the nerest entimetre etween eh pir of onseutive welds? Wht is the height (to the nerest entimetre of the truss? 8 M The length of side m is nerest to: D 5.8 E m Em tip Some students inorretly ssume tht ngle X is 90. [ssessment report 006] 30 Em tip Students ommonly found the size for the inorret ngle. [ssessment report 006] X 39 Mths Quest 1 Further Mthemtis for the sio lsspd

29 9 m The orret epression for the vlue of t in the given tringle is: 7 sin n( sin s ( sin s ( 30 sin ( 30 sin( 30 sin( sin s ( sin( 50 D E sin( 50 sin( m The vlue of (to 1 deiml ple in the given figure is: D 3.3 E m m t m In the tringle given, the lrgest ngle (to the nerest degree is: D 67 E 60 8 m 3 7 m 1 m yht sils the three-leg ourse shown. The lrgest ngle etween ny two legs within the ourse, to the nerest degree, is: D 78 E m 15 km km 13 km 13 m The orret epression for ngle S in the given tringle is: sin 1 40 sin( 41 os 1 40 os( sin 1 30 sin( E sin sin( Find the perimeter of the eehive ell shown. 41 sin( 41 D sin logo is in the shpe of n isoseles tringle with the equl sides eing 6.5 m long nd the equl ngles 68. Use the sine rule to find the length (to 1 deiml ple of the unknown side. 16 rope is pegged t one end into the ground, pulled tightly up over rnh nd pegged into the ground t the other end. It is known tht one peg-to-rnh length of rope is 8 m nd it mkes n ngle of 39 with the ground. The other end of the rope mkes n ngle of 48 with the ground. Find (to 1 deiml ple: the length of the rope the distne etween the two pegs mm S 30 hpter 8 393

30 8F 17 plyground swing, whih is.3 m long, mkes n ngle of 74, t its swing point, in one omplete swing. Determine the horizontl distne (in metres to 1 deiml ple etween the etreme positions of the swing set. 18 seni flight leves Geelong nd flies west of north for the 80 km diret journey to llrt. t llrt the plne turns 9 to the right to fly est of north to Kyneton. From here the plne gin turns to the right nd flies the 103 km stright k to Geelong. Determine the ngle (in degrees to 1 deiml ple through whih the plne turned t Kyneton. Find the distne (to the nerest km of the diret flight from llrt to Kyneton. miguous se of the sine rule On your lultor, investigte the vlues for eh of these pirs of sine rtios: sin (30 nd sin (150 sin (110 nd sin (70. You should otin the sme numer for eh vlue in pir. Similrly, sin (60 nd sin (10 give n identil vlue of Now try to find the inverse sine of these vlues; for emple, sin 1 ( is 60. The otuse (greter thn 90 ngle is not given y the lultor. When using the inverse sine funtion on your lultor, the lultor will give only the ute ngle. The sitution is illustrted prtilly in the digrm t right where the sine of the ute ngle equls the sine of the otuse ngle. eookplus elesson eles-0051 miguous se of the sine rule Otuse ute rope tthed to pole n e nhored in two possile positions. Therefore lwys hek your digrm to see if the unknown ngle is the ute or otuse ngle or perhps either. This sitution is illustrted in the two digrms elow. The tringles hve two orresponding sides equl, nd, s well s ngle. The sine of 110 lso equls the sine of 70 ; however, the side is quite different. It is worth noting tht this miguity ours when the smller known side is opposite the known ngle WorkEd EmplE 15 To the nerest degree, find the ngle, U, in tringle, given t = 7, u = 1 nd ngle T is 5. Think Method 1: Using the rule WriTE/displY eookplus Tutoril int-0467 Worked emple 15 1 Drw suitle sketh of the tringle given. s the length of s is not given, side t n e drwn two different wys. Therefore ngle U ould e either n ute or n otuse ngle. Lel the tringles ppropritely for the sine rule. (It is just s esy to use the given lels. T u 1 5 s U S t 7 S u 1 t 7 T 5 s U 394 mths Quest 1 Further mthemtis for the sio lsspd

31 Identify tht it is the sine rule tht n e used s you hve the side opposite to the unknown ngle nd known side ngle rtio. s t u = = sin( S sin ( T sin( U t = 7 T = 5 u = 1 U =? 3 Sustitute the known vlues into the two rtios. 4 Trnspose the eqution to get the unknown y itself. 5 Evlute the ngle (inverse sine. Note tht the vlue is n ute ngle ut it my well e n otuse ngle. 7 1 sin( 5 = sin ( U sin( U sin ( 5 = sin( 5 sin (U = 7 sin (U = U = lulte the otuse ngle. U = = Write the nswer, giving oth the ute nd otuse ngles, s not enough informtion ws given (the informtion ws miguous to preisely position side t. Method : Using S lultor 1 On the Min sreen, use the soft keyord to omplete the entry line s: 1 7 solve sin( u = sin ( 5 Highlight the eqution nd tp: Intertive dvned solve hek Solve numerilly, type in the vrile s u nd input the lower nd upper ngles s 0 nd 180 respetively. This instruts the lultor to lulte ll possile ngles etween 0 nd 180. Then tp OK. The ngle U is either 46 or 134, orret to the nerest degree. The ngle U ould e either the ute or otuse ngle sine the length of the side u is greter thn the length of the side t. The ngle U is either 46 or 134, orret to the nerest degree. WorkEd EmplE 16 In the otuse-ngled tringle PQR t right, find the unknown ngle (to the nerest degree, P. Q 30 m 0 m R 40 P hpter 8 395

32 Think WriTE 1 Lel the tringle ppropritely for the sine rule. (It is just s esy to use the given lels. Q p 30 r 0 R 40 P Identify tht the sine rule is used s you hve the side opposite to the unknown ngle nd known side ngle rtio. 3 Sustitute the known vlues into the two rtios. 4 Trnspose the eqution to get the unknown y itself. 5 Evlute the ngle (inverse sine. Note tht the vlue is n ute ngle while in the digrm given it is n otuse ngle. p q r = = sin( P sin ( Q sin( R p = 30 P =? r = 0 R = sin( P = sin ( 40 sin( P sin ( 40 = sin( 40 sin (P = 0 sin (P = P = lulte the otuse ngle. P = = The ngle P is 105, orret to the nerest degree. rememer If the unknown ngle is n otuse ngle, rememer the following: 1. the inverse sine funtion on lultors evlutes only the ute ngle. for the otuse ngle, evlute s follows: otuse ngle = 180 ute ngle 3. the miguous se of the sine rule ours when the smller known side is opposite the known ngle. EErisE 8F miguous se of the sine rule 1 WE 15 Find oth the ute nd otuse ngles in eh se elow. Epress ll nswers in degrees to 1 deiml ple. In, find the unknown ngle,, given = 10.8, = 6 nd = 6. In STU, find the unknown ngle, S, given t = 1.7, s = 16.3 nd T = 45. In PQR, find the unknown ngle, P, given p = 3.5, r = nd R = 1. d In LMN, find the unknown ngle, M, given n = 0. km, m = 0.5 km nd N = mths Quest 1 Further mthemtis for the sio lsspd

33 WE 16 Find the unknown ngle (to the nerest degree in eh of the following otuse-ngled tringles. d 60 km 3 m m 110 km 5.8 m 11 m m m 5 3 m In the tringle given, ngle is (to the nerest degree: m D E 14 8 m 4 Find the two unknown ngles shown in the digrm elow (to 1 deiml ple. 10 m 7 9 m 9 m y 5 Look t the swinging pendulum shown t right. Drw the two possile positions of the o t the level of the horizontl line. Find the vlue of the ngle, W, t these two etreme positions. Find the smllest nd lrgest distnes etween verte V nd the o. V 8 m 15 W 5 m 8G The osine rule The osine rule is derived from non-right-ngled tringle divided into two right-ngled tringles in similr wy to the derivtion of the sine rule. The differene is tht, in this se, Pythgors theorem nd the osine rtio re used to develop it. The tringle in the figure t right hs een divided into two right-ngled tringles with se sides equl to nd (. In D, h = nd in D, h = ( (Pythgors theorem Equting epressions for h, = ( = + ( = + + = + [1] h D Now, from D, os ( = = os ( Sustitute this vlue of into [1] ove. = + [ os (] hpter 8 397

34 So, the osine rule n e written s: = + os ( In similr wy to tht ove, it n e shown tht: = + os ( = + os ( lso, if the tringle is lelled using other letters, for emple STU, then: s = t + u tu os (S The osine rule is used to find: 1. n unknown length when you hve the lengths of two sides nd the ngle in etween. n unknown ngle when you hve the lengths of ll three sides. The formul my e trnsposed in order to find n unknown ngle. os ( = + or lterntively, os ( = + nd os ( = +. WorkEd EmplE 17 Find the unknown length (to deiml ples,, in the tringle t right. Think 7 m 80 6 m WriTE eookplus Tutoril int-0468 Worked emple 17 1 Identify the tringle s non-right-ngled. Lel the tringle ppropritely for the sine rule or osine rule Identify tht it is the osine rule tht is required s you hve the two sides nd the ngle in etween. 4 Sustitute the known vlues into the osine rule formul. 5 Rememer to get the squre root vlue,. = = = 6 = 80 = 7 = = + os ( = os (80 = os (80 = Evlute the length nd inlude units with the nswer. = 8.39 The unknown length is 8.39 m, orret to deiml ples. 398 mths Quest 1 Further mthemtis for the sio lsspd

35 WorkEd EmplE 18 Find the size of ngle in the tringle elow, to the nerest degree Think Method 1: Using the rule 1 Identify the tringle s non-right-ngled. Lel the tringle ppropritely for the sine rule or osine rule. WriTE/displY Identify tht it is the osine rule tht is used s ll three sides re given. 4 Sustitute the known vlues into the rerrnged form of the osine rule nd simplify. = 4, = 6, = 6, = os ( = + os ( = os ( = os ( = Evlute [ = os 1 (0.3333]. = Evlute the ngle nd inlude units with the nswer. Method : Using S lultor 1 On the NumSolve sreen, use the soft keyord to omplete the entry line s: 6 = os ( Then press E. Selet the vrile to solve y heking the djent ullet nd set the lower nd upper levels to 0 nd 180 respetively. This instruts the lultor to lulte ll possile ngles etween 0 nd 180. Then tp: 1 (on the tool r OK The ngle is 71, orret to the nerest degree. Write the nswer. The ngle is 71, orret to the nerest degree. hpter 8 399

36 rememer 1. Follow the pproprite lelling onvention when using the osine rule.. The osine rule n e used to find n unknown: ( length, if the other two sides nd the ngle in etween them re known. = + os ( ( ngle, if ll three sides re known. os ( = + EErisE 8G The osine rule 1 WE 17 Find the unknown length in eh of the following (to deiml ples m.3 km 1.5 km z d 60 5 m e 30 3 f mm 000 mm km 100 km Em Tip Mny students who use the osine rule forget to tke the squre root t the end. [ssessment report 005] During siling re, the ots followed ourse s shown elow. Find the length,, of its third leg (to 1 deiml ple. 10 km km 400 mths Quest 1 Further mthemtis for the sio lsspd

37 3 Two irles, with rdii 5 m nd 8 m, overlp slightly s shown t right. If the ngle etween the two rdii tht meet t the point of intersetion of the irumferenes is 105, find the distne etween the entres of the irles (to 1 deiml ple. 5 m 8 m WE 18 Find the size of the unknown ngle in eh of the following (to the nerest degree. 1 mm d y 85 km 8 m 5 m 13 mm 0.5 m 19.1 m p 101 km 0 mm 8.6 m 6 m 68 km 5 onsider the siling epedition ourse in question. Find the two unknown ngles (to the nerest degree in the tringulr ourse. 6 onsider the overlpping irles in question 3. Find the two ngles formed etween the line joining the entres of the irles nd eh of the rdii drwn (to the nerest degree. 7 For the tringle shown t right, find ll three unknown ngles (to the 9 nerest degree For the following questions, find nswers to 1 deiml ple. For, find the unknown side,, given = 10 km, = 8 km nd = 30. For, find the unknown ngle,, given = = 10 nd = 6. For, find the unknown side,, given = 7 m, = 3 m nd = 80. d For STU, find the unknown ngle, S, given t = 1.7, s = 16.3 nd u = 4.5. e For PQR, find the unknown ngle, P, given p =, q = 3.5 nd r =.5. f For, find the unknown side,, given = 60, = 10 nd = onstrut suitle tringle from the following instrutions nd find ll unknown sides nd ngles. Two sides re 3 m nd 15 m nd the ngle in etween is M The vlue of (to 1 deiml ple in the digrm t right is: mm mm D 8.5 E none of the ove 11 M The length of side m t right is nerest to: m D E 50 1 M In the tringle given, the lrgest ngle is: D 85 E m 0 m m Not to sle hpter 8 401

38 13 m The orret epression for ngle s is: os os E os os D os m 4 m 6 m s 14 m The orret epression for the vlue of t is: os( D E t eookplus Digitl do WorkSHEET m The 4 surfe ngles t the verte of regulr squre pyrmid re ll the sme. The mgnitude of these ngles for the pyrmid given (to the nerest degree is: D 39 E Find the unknown vlues. 4 m 4 m 1 m 6 m 3 m 8 m 15 m m 100 Regulr squre pyrmid 10 m 8h speil tringles Often, the tringles enountered in prolem solving re either equilterl or right-ngled isoseles tringles. They ehiit some unique fetures tht, when reognised, n e very useful in solving prolems. Equilterl tringles hve three equl sides nd three equl ngles. Therefore, when given the length of one side, ll sides re known. The three ngles re lwys equl to mths Quest 1 Further mthemtis for the sio lsspd

39 Right-ngled isoseles tringles hve one right ngle (90 opposite the longest side (hypotenuse nd two equl sides nd ngles. The two other ngles re lwys lso, the hypotenuse is lwys times the length of the smller sides. hek for yourself using Pythgors theorem WorkEd EmplE 19 Find the vlues of r nd ngle θ in the hegon t right. Regulr hegon 6 m r m Think 1 Tringles in regulr hegon re ll identil. The si ngles t the entre re equl. The mgnitude of eh is one revolution divided y 6. Furthermore, the two sides tht form the tringle re equl. Thus the two equl ngles on the shpe s perimeter re lso 60. ll three ngles re the sme; therefore, ll three sides re equl. Therefore, the tringles in regulr hegon re ll equilterl tringles. WriTE 6 m 60 θ = θ = 60 r = 6 m hpter 8 403

40 WorkEd EmplE 0 Find the vlue of the pronumerl (to 1 deiml ple in the figure. Think WriTE 1 m 45 1 The tringle is right-ngled isoseles tringle. Two ngles re 45 nd the third ngle is m m Two sides re equl nd the longer side opposite the right ngle is times longer thn these equl sides. 3 Write your nswer using the orret ury nd units. = = 1 = The vlue of is 17.0 m, orret to 1 deiml ple. rememer 1. Equilterl tringles hve three equl sides nd three equl ngles. Eh ngle equls Right-ngled isoseles tringles hve one right ngle (90 opposite the longest side (hypotenuse, nd two equl sides nd ngles. The two other ngles re lwys 45. The hypotenuse is lwys times the length of the smller sides EErisE 8h speil tringles 1 WE 19 Find the unknown(s in eh of the following m m 404 mths Quest 1 Further mthemtis for the sio lsspd

41 WE0 Find the unknowns in eh of the following. 45 m 158 m y 7. m 10 mm 3 nswer the following. In, find the unknown ngle,, given = 10, = 10 nd = 90. In STU, find the unknown side, s, given t = 1.7, S = 45 nd T = 45. In PQR, find the unknown ngle, P, given p = 3.5, r = 3.5 nd R = 60. d In LMN, find the unknown side, m, given n = 0., L = 60 nd N = pir of ompsses used for drwing irles hs legs tht re 6 m long. If it is opened s shown in the digrm, wht is the rdius of the irle tht ould e drwn? 60 8i 5 Wht is the height of tree if its shdow, on horizontl ground, is 1 metres long when the sun s rys striking the tree re t 45 to the ground? 6 m In the tringle given, the length of side (in metres is: m D 0 E m squre serviette is prepred for presenttion y ompleting three folds firstly, y tking orner nd pling it on top of the opposite orner; seondly, y tking one of the two orners on the rese tht hs een mde nd pling it on the other one; nd finlly, y pling the two orners t the ends of the longest side on top of eh other. Find the length of the rese mde fter the i first fold ii seond fold iii third fold. With the finl serviette lying flt, wht ngles re produed t the orners? re of tringles Three possile methods n e used to find the re of tringle: Method 1. When the two known lengths re perpendiulr to eh other we would use: re tringle = 1 se Height = 1 h 3 m Height Height 4 m se se hpter 8 405

42 Method. When we re given two lengths nd the ngle in etween we would use: re tringle = 1 sin ( = 1 sin ( 10 m Height sin ( 3 15 m se 1 re se Height 1 sin ( Method 3. When ll three sides re known we would use: re tringle = s (ss (s s (s s where the semi-perimeter, s = ( + +. This formul is known s Heron s formul. It ws developed y Heron (or Hero of lendri, Greek mthemtiin nd engineer who lived round D 6. Let us find the re of the tringle t right to demonstrte tht ll three formuls provide the sme result. For the 3, 4, 5 tringle, the most pproprite method is method 1 euse it is right-ngled tringle. re tringle = 1 5 se Height = = 6 The other two methods my lso e used. re tringle = 1 sin ( = sin (90 = 6 1 = 6 re tringle = s (ss (s s (s s s = ( + + = 66 (6 3 (6 4 (6 5 s = ( = s = 1 = 36 s = 6 = 6 3 WorkEd EmplE 1 Find the re of the tringle t right. 8 mm Think 1 onfirm tht the two given lengths re perpendiulr. WriTE 1 mm 406 mths Quest 1 Further mthemtis for the sio lsspd

43 Sustitute the known vlues into the formul. re tringle = 1 se Height = = 48 3 Write the nswer using orret units. The re of the tringle is 48 mm. WorkEd EmplE Find the re of the tringle t right (to deiml ples. eookplus Tutoril int-0469 Worked emple 9 m 37 6 m Think WriTE/displY Method 1: Using the rule 1 Identify the shpe s tringle with two known sides nd the ngle in etween Identify nd write down the vlues of the two sides, nd, nd the ngle in etween them,. 3 Identify the pproprite formul nd sustitute the known vlues into it. = 6 = 9 = 37 re tringle = 1 sin ( = sin (37 = Write the nswer in orret units. The re of the tringle is 16.5 m, orret to deiml ples. Method : Using S lultor On the Geometry sreen, drw the shown digrm. Refer to your eookplus for detiled instrutions for onstrution of shpes nd mesurement of unknown vlues. hpter 8 407

44 WorkEd EmplE 3 Find the re of tringle PQR (to 1 deiml ple, given p = 6, q = 9 nd r = 4, with mesurements in entimetres. Think WriTE/displY Method 1: Using the rule 1 onfirm tht ll three sides of the tringle hve een given nd therefore Heron s formul is to e used. p 6 Q r 4 R q 9 P Write the vlues of the three sides,, nd, nd lulte the semi-perimeter vlue, s. = p = 6, = q = 9, = r = 4 s = ( + + s = ( s = Sustitute the known vlues into Heron s formul. re tringle = s (ss (s s (s s = 9595.( ( ( = = re = Write the nswer, using the orret units. The re of tringle PQR is 9.6 m, orret to 1 deiml ple. Method : Using S lultor 1 On the Min sreen, use the soft keyord to tp: {N ontinue to press {N until 5 templtes hve een reted. Enter the equtions s shown. t the ottom right of the equtions templte, type the vriles to e solved (i.e. m, s,,, nd press E. Write the nswer, using the orret units. The re of the tringle PQR is 9.6 m, orret to 1 deiml ple. 408 mths Quest 1 Further mthemtis for the sio lsspd

45 rememer 1. Three possile methods re ville for finding the re of tringle: ( When the two known lengths re perpendiulr to eh other we would use: re tringle = 1 se Height ( When we re given two lengths nd the ngle in etween we would use: re tringle = 1 sin ( ( When ll three sides re known we would use: re tringle = s (ss (s s (s s where the semi-perimeter, s = ( + +. lwys use the most effiient method to find the re of tringle. EErisE 8i re of tringles 1 WE1 Find the res of the following tringles (to 1 deiml ple. 4.5 mm 7.0 mm d 3 m 7 m 1 m 10.5 mm 3. mm Em Tip orret epression formultion ontinues to e onern where tehnology is used for lultion. Divisions, suh s , often seemed to e done on lultor without rkets or ny other pproh to sum the numertor efore dividing y. For questions requiring the use of Heron s formul, one mrk would e llowed for the orret nd identified vlue of the semiperimeter, s. [ssessment report 007] 5 m 4 m WE Find the res of the following tringles (to 1 deiml ple m 7 m 4 m d 3 m 100 m m 10. m m hpter 8 409

46 3 WE3 Find the res of the following tringles (to 1 deiml ple. 3 km 8 m 8 m 6 km 6 m 4 km 0 mm d 5. m 6.7 m 3.1 m 4 Find the res of the following tringles (to 1 deiml ple. 4.7 m 4.4 m m km d m m 11. km 10 km 5 Find the re of eh of the following tringles. (Give ll nswers to 1 deiml ple. For, given = 10 km, = 8 km nd = 30 For, given = = 10 m nd = 6 m For, given = 7 m, = 3 m, = 8.4 m nd = 108 d For STU, given t = 1.7 m, s = 16.3 m nd u = 4.5 m e For PQR, given p = units, q = 3.5 units nd r =.5 units f For, given = 60 m, = 10 m nd = 90 6 Find the re of n equilterl tringle with side lengths of 10 m. 7 tringulr rh hs supporting legs of equl length of 1 metres s shown in the digrm elow. Wht is its re? 45 1 m 1 m Mths Quest 1 Further Mthemtis for the sio lsspd

47 8 From the digrm given, find the re of: i one of the tringles ii ll of the tringles. Use nother tehnique to verify your nswer in i. 10 mm 10 mm 9 Find the re of the stte forest s defined y the three fire-spotting towers on the orners of its oundry. 11 km 5. km 10.4 km 10 m If the perimeter of n equilterl tringle is 10 metres, its re is losest to: 100 m 450 m 4800 m D 5500 m E 1700 m 11 m The orret epression for the re of the shpe t right is: sin ( os ( sin (100 6 m 30 4 m 50 D E none of the ove 1 m The orret epression for the re of the otgon shown is: 195 sin ( sin ( sin (60 D 338 sin (60 E sin ( Find the re of the following tringles. 7 km mm eookplus Digitl do Investigtion Prolem solving to find n re hpter 8 411

48 summry right-ngled tringles Pythgors theorem For hypotenuse: = + or = + For shorter sides: = Pythgoren trids Pythgoren trid is set of three numers whih stisfies Pythgors theorem. Some ommon trids re ( 3, 4, 5 ( 6, 8, 10 ( 5, 1, 13 nd (d 7, 4, 5. Three-dimensionl Pythgors theorem To solve prolems involving 3-dimensionl Pythgors theorem: ( Drw nd lel n pproprite digrm. ( Identify the right ngles. ( Identify right-ngled tringles tht enle the informtion given to e used to find the unknown vlue(s. Trigonometri rtios Hypotenuse Opposite djent sin ( θ = O H os (θ = H non-right-ngled tringles The sine rule tn (θ = O or SOH H TO helps you rememer the omintion of sides in eh rtio = = sin( sin ( sin( The sine rule is used when: 1. two ngles nd one side re given. two sides nd non-inluded ngle re given. If two ngles re given, simply lulte the third ngle, if needed, using: = 180 ( + 41 mths Quest 1 Further mthemtis for the sio lsspd

49 miguous se of the sine rule The sine rule is miguous when finding n ngle when the smller known side is opposite the known ngle. The osine rule = + os ( or os ( = + To lulte: ( sides, use the osine rule when two sides nd the inluded ngle re given ( ngles, use the osine rule when ll three sides re given. Speil tringles Equilterl tringles Right-ngled isoseles tringlesr re of tringles To find the re of tringle: ( given perpendiulr dimensions, use re tringle = 1 se Height ( given two sides nd the inluded ngle, use re tringle = 1 sin ( ( given ll three sides only, use re tringle = s (ss (s s (s s where s = hpter 8 413

50 hpter review multiple hoie 1 For the tringle shown, the 3 vlue of is: E D Whih one of the following is not Pythgoren trid? 9, 39, 40 3, 4, 5 0.3, 0.4, 0.5 D 6, 8, 10 E 7, 4, m long strw is the longest tht n fit into ylindril n with rdius of 6 m. The height of the n, in entimetres, is losest to: D 16 E 17 4 retngulr o hs rod positioned s shown in the digrm. The epression tht would enle the ngle the rod mkes with the se of the o to e found, is: tn (θ = 5 1 sin (θ = 4 13 tn (θ = D tn (θ = E os (θ = stepldder is ereted s shown. How fr prt (to deiml ples t the se re the two legs? E 6.97 m 1.15 m 1.41 m 1.50 m D.00 m 3? m m 6 Given ST = 1 m, TU = 16 m nd sin (U = 3 T 3, then sin (S 4 1 m equls: E D Find the vlue of the pronumerl, to the nerest metre. 50 m D E S U 16 m 8 In tringle where = 10, = 0 nd = 6, (to the nerest degree ould e: D 63 or 117 E 61 or To find the distne ross lrge evtion, mesurements were found s shown in the 110 m digrm. The distne, m, ross the evtion is losest to: 75 metres 74 metres 100 metres D 10 metres E none of these 10 regulr hegon is insried in irle of r m rdius m. The perimeter of the hegon, in r entimetres, is: 4π 1 16 D 17 E right-ngled isoseles tringle hs longest side of 141 metres. The other two equl sides hve vlue losest to: 00 m 100 m 50 m D 10 m E none of these 1 The re of tringle XYZ X Y (to the nerest m 40 is: m 17 m 45 D 85 E 65 Z 13 The re of the tringle t right is losest to: 96 m 97 m 98 m D 99 m 40 E 100 m 50 0 m 14 The ross-setion of wter pipe is irulr with rdius, r, of 50 m, s shown elow. The surfe of the wter hs width, w, of 80 m. r = 50 m w = 80 m d m depth of wter in the pipe 414 mths Quest 1 Further mthemtis for the sio lsspd

51 The depth of wter in the pipe, d, ould e: 0 m 5 m 30 m D 40 m E 50 m [V 006] Em Tip Only % of students were suessful in hoosing option. Option, whih orresponded to the distne of the surfe from the entre of the pipe, ws given y 39% of students. further % of students wrongly ssumed tht the ross-setion of the surfe ws semiirulr nd hose option D. possile solution strtegy is s follows. Drw in rdius line tht is perpendiulr isetor of the wter surfe line. Lel points s shown on the digrm elow. From right-ngled tringle D, = 30 m ( = m Then d = = = 0 m. 50 m 30 m 40 m d 0 m D [ssessment report 1 006] 15 If, in tringle elow, sin (α = 0.8, then sin (β is equl to: m 0.75 D 0.8 E m [V 005] short nswer 1.5 m long ldder is pled up ginst wll nd rehes to height of.4 m. Find the distne tht the legs of the ldder re from the se of the wll. Em Tip Mny students provided nswers without suitle working, usully in the Geometry modules. Where the nswer is orret, full mrks re wrded ut, without working, inorret nswers or onsequentil nswers do not reeive ny mrks. [ssessment report 006] 190 mm squre ermi floor tile is to e ut digonlly. Wht is the et length of the ut to e mde? 3 ot sils diretly northwrds for 11 km efore turning towrds the est nd siling 60 km. t this point the ot is 61 km from the strting point. On the seond leg of the trip, did the ot sil diretly estwrds? 4 uoid with 8 m sides is internlly red. Wht is the length of the longest re tht ould e pled inside the uoid? Epress your nswer in surd form. 5 stirse is to rise y 50 mm from the ground floor to the first level of house. The mimum ngle of elevtion llowed for the stirs is 50. Wht is the length of the se of the stirse (to the nerest mm? Wht is the length of the stirse (to the nerest mm? 6 opy nd omplete the following tle using the two tringles given elow. Give eh nswer s oth frtion nd deiml ngle sin os tn 3 = r dge to e fitted on onnet is of n isoseles tringle design. If the height of the dge is not to e more thn 30 mm, wht is the mimum length of the se of the dge (to the nerest mm, if the equl ngles re 5? If the longest side is to e set t 100 mm, wht is the length of the other two equl sides, if the two equl ngles re still 5? 8 hot-ir lloon is nhored to the ground t points nd D s shown in the following digrm. 5-metre length of eess rope is dropped to the ground from the lloon. It is then tied to the ground, t point, s further sfety mesure. hpter 8 415

52 Wht re the smllest nd lrgest ngles tht n e mde t point y the two lengths of rope, nd (to the nerest degree? Using these vlues, determine the furthest nd losest positions (to 1 deiml ple possile for point from point. 9 The hour hnd of lok is 0 mm long nd the minute hnd is 5 mm. The lok fe t right shows the time t 4 o lok. Wht is the distne etween the tips of oth hnds (to the nerest mm? 40 m 35 5 m D 10 n underover ptio is overed with sil s shown in the digrm. Wht re the ngles 5 m 6.3 m mde y the sil t eh of the support 8.1 m poles (to the nerest degree? t whih support pole is the smllest ngle? 5 mm 10 0 mm 11 The infmous ermud Tringle is represented t right. Wht is the distne etween the western nd US northern orners of the tringle 110 km (to the nerest 40 kilometre? 90 km Wht is the lrgest ngle within the tringle (to the nerest degree? 1 DVD storge unit is 1.5 metres tll nd hs se re s shown. Find the front width of the storge unit (to the nerest m. Find the volume of the storge unit (in m 3. 1 m 1 m 13 Wht is the re of Give Wy trffi sign tht is in the shpe of n equilterl tringle with side lengths 45 m (to the nerest m? 14 Wht is the re of the dge desried in question 7 (to the nerest mm? N Etended response 1 sndwih r uses red tht is roughly 10 m squre. The red slies re ut into four equl tringles nd pkged in rdord o with the tringles rrnged s shown. 10 m 10 m 8 m i Wht is the totl length of the two uts required to mke four tringulr piees? ii Wht is the re of the tringulr fe of the pkged sndwih (to 1 deiml ple? The ompleted sndwihes re pled on shelves s shown. iii Wht is the smllest possile gp required etween shelves for the sndwihes to fit? To mintin sndwih freshness, the owner is dvised to prepre sndwihes so tht the surfe re is minimised. i Show tht the surfe re of pkged, tringulr sndwihes is lose to 43 m. ii Would utting the sndwihes into four smll equl squre piees redue the surfe re? If so, y how muh (to the nerest m? Drw suitle digrm(s. iii Find the volume of the sndwih pkge. 416 Mths Quest 1 Further Mthemtis for the sio lsspd

53 n lterntive is to use red whih hs retngulr shpe s shown, nd to prepre it s tringulr piees. i Wht is one disdvntge of using retngulr slies of red for mking four tringulr sndwih piees? ii The four ngles t the entre of the red just fter mking the two uts re no longer right ngles. Find the vlue of the lrgest ngle. iii If the four tringulr piees re lso to e pkged, wht is the smllest possile re of the tringulr fe of the rdord o? Two thin rods re hinged together nd the end of one rod is hinged to the ground, while the end of the other rod is free, s shown in the digrm t right. Luie onduts n investigtion of the tringle formed. She strts y investigting the formtion of right-ngled tringle. i t wht distne from (to deiml ples must Luie ple end so tht right-ngled tringle is formed t? Rememer tht the two rods n move, lthough they re fied t. ii Wht is the ngle mde y the 1.5 m rod with the ground (to the nerest degree? iii Using your nswer from prt ii, wht is the vlue of the other ute ngle? Luie now rings end to position 1 m from end. i Stte the type of tringle formed. ii Wht is the size of the lrgest ngle formed in this tringle (to the nerest degree? iii If the lrgest ngle is now to e 110, wht is the new distne from to (to 3 deiml ples? n lterntive is to move end wy from, s shown t right. How fr is end from, if is to e 110 (in metres to 1 deiml ple? d Luie now investigtes the re of the tringle mde in eh sitution. i Wht is the re of the tringle in prt i (in m to deiml ples? ii Wht is the re of the tringle in prt iii (in m to deiml ples? iii Wht is the re of the tringle in prt (in m to deiml ples? e third rod, 3 metres long, is onneted t point to the right-ngled tringle formed in prt. Its free end rests on the ground. Wht is the horizontl distne etween nd the end of this third rod (to the nerest m? 3 The digrm elow represents pln view (looking down onto of n open-ut mine, whih is roughly in the shpe of prllelogrm. D 1.5 m 1.5 m 1 m 9 m 1 m 1 m E F Mine pit mine surveyor hs een sked to determine the dimensions of this pit. From, she mesures the distnes, nd, to either side of the pit nd lso the ngle in etween. She finds tht: = 86 m, = 97 m nd = 46 o. Find the pit s width, (to the nerest metre. hpter 8 417

54 From D, she ompletes similr eerise to find the length of the pit, EF. This time she finds: DE = 10 m, DF = 111 m nd EDF = 53. The following digrm is ross-setionl view of the pit long its length. E 49º 75º d F R The surveyor needs to find the depth, d. To do this she hs loted lrge oulder, R, t the ottom of the pit nd found tht the ngle etween the horizontl, EF, nd ER is 49, nd the ngle etween FE nd FR is 75. Find (to the nerest metre: the distne ER d the depth, d. e The surveyor s lst tsk is to find the re of the opening of the pit. For this lultion she simply mesures the sides of the pit opening. She finds tht: E = F = 71 m nd F = E = 45 m. Find the re, to the nerest squre metre. Note tht in prllelogrm the digonls iset eh other. 4 Tess s tsk is to rve the right retngulr pyrmid DY shown elow. eookplus Digitl do Investigtion Pper plnes D 4 m Y 8 m 3 m Em Tip This question required the orret use of Heron s formul. One mrk ws llowed for the orret nd identified vlue of s. Ineffetive lultor usge seems to hve led to inorret vlues for s, often due to not rketing the numertor of the frtion efore dividing y. nother ommon error involved using 3 m for one of the sides of tringle Y rther thn reognising it ws n isoseles tringle. [ssessment report 007] lulte the volume, in m 3, of the pyrmid DY. Show tht, orret to the nerest m, length Y is 37 m. Using Y s 37 m, demonstrte the use of Heron s formul to lulte the re, in m, of the tringulr fe Y. [V 007] eookplus Digitl do Test Yourself hpter mths Quest 1 Further mthemtis for the sio lsspd

55 eookplus TiViTiEs hpter opener Digitl do 10 Quik Questions: Wrm up with quik quiz on trigonometry. (pge Pythogors theorem Digitl do Spredsheet 10: Investigte the effet on the hypotenuse of right-ngled tringle s eh perpendiulr side length vries. (pge Pythgoren trids Digitl do Spredsheet 134: Investigte Pythgoren trids. (pge 37 8 Three-dimensionl Pythgors theorem Digitl do WorkSHEET 8.1: Use Pythgors theorem in three dimensions nd trigonometri rtios to lulte unknown lengths. (pge 377 Tutoril WE5 int-0464: Wth tutoril on how to pply Pythgors theorem to solve rel-life prolem. (pge 373 Intertivity int-0189 Three-dimensionl Pythgors theorem: Use the intertivity to identify right-ngled tringles in three dimensions in order to lulte side lengths. (pge 373 8D Trigonometri rtios Digitl dos SkillSHEET 8.1: Prtise identifying sides of right-ngled tringle with respet to the given ngle. (pge 384 SkillSHEET 8.: Prtise finding trigonometri vlues nd ngles. (pge 384 8E The sine rule Tutorils WE 11 int-0465: Wth tutoril on how to use the sine rule to lulte n unknown length in non-right-ngled tringle. (pge 387 WE 13 int-0466: Lern how to use the sine rule to lulte n unknown ngle in non-right-ngled tringle. (pge 389 8F miguous se of the sine rule Tutoril WE 15 int-0467: Wth tutoril on when the miguous se of the sine rule is pplied. (pge 394 elesson eles-0051 miguous se of the sine rule: Disover how one fied ngle nd two defined side lengths n give two different tringles n ute ngle nd n otuse ngle. (pge 394 8G The osine rule Digitl do WorkSHEET 8.: lulte the unknown sides nd ngles in right-ngled nd non-right-ngled tringles. (pge 40 Tutoril WE 17 int-0468: Wth tutoril on how to use the osine rule to lulte n unknown side length of non-right-ngled tringle. (pge 398 8I re of tringles Digitl do Investigtion: Investigte the generl rule for re. (pge 411 Tutoril WE int-0469: Wth worked emple on how to use sine to lulte the re of non-right-ngled tringle. (pge 407 hpter review Digitl dos Investigtion: Investigte the re of plne s wings. (pge 418 Test Yourself: Tke the end-of-hpter test to test your progress. (pge 418 To ess eookplus tivities, log on to hpter 8 419