ONLINE PAGE PROOFS. Trigonometry Kick off with CAS 12.2 Trigonometry 12.3 Pythagorean triads

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1 Kik off with S 12.2 Trigonometry 12.3 Pythgoren trids Trigonometry 12.4 Three-dimensionl Pythgors theorem 12.5 Trigonometri rtios 12.6 The sine rule 12.7 miguous se of the sine rule 12.8 The osine rule 12.9 Speil tringles re of tringles Review

2 12.1 Kik off with S Eploring the sine rule with S The sine rule n e used to find unknown side lengths nd ngles in tringles. For ny tringle, the sine rule sttes tht sin = sin = sin efore ompleting ny questions involving ngles on your S, ensure tht your ngle setting is in degrees mode. 1 Use S to determine the vlue of the following: i sin(40 ) ii sin(140 ) i sin(25 ) ii sin(155 ) 2 Wht do you notie out your nswers to questions 1 nd 1? n you spot reltionship etween the given ngles? Use S to eplore if this works for other pirs of ngles with the sme reltionship. 3 Use the sine rule to determine the size of ngle in the domin 0 < 90 given: = 16.2, = 25.1, = 33 4 Whih ngle etween 90 nd 180 gives the sme sine vlue s your nswer to question 3? 5 Is your nswer to question 4 potentil seond solution to your tringle (from question 3)? Try to drw tringle with these given vlues. Plese refer to the Resoures t in the Prelims setion of your eookplus for omprehensive step-y-step guide on how to use your S tehnology.

3 12.2 Unit 4 OS M3 Topi 1 onept 1 Right-ngled tringles onept summry Prtie questions Trigonometry Trigonometry 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. 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. 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 2 = nd, therefore, to find, = where is the longest side or hypotenuse nd nd re the two shorter sides. (hypotenuse) 612 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

4 Intertivity Pythgors theorem int-6473 WoRKED EXMPLE 1 Note: euse the eqution 2 = 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: 2 = 2 2 nd so = 2 2. Find the length of the unknown side (orret to 1 deiml ple) in the right-ngled tringle shown. THINK 1 Note tht the tringle is right-ngled nd we need to find the unknown length, given the other two lengths. 2 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. WoRKED EXMPLE 2 WRITE/DRW = 4 = = 7 2 = lterntively, 2 = = = = 65 = = 65 = = = 65 The unknown side s length is 8.1 m, orret to 1 deiml ple. Find the mimum horizontl distne (orret to the nerest metre) ship ould drift from its originl nhored point, if the nhor line is 250 metres long nd it is 24 metres to the ottom of the se from the end of the nhor line on top of the ship s dek. 4 m 7 m THINK 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. WRITE/DRW Topi 12 TRIgonoMETRy 613

5 2 Simplify the tringle, dding known lengths, = 250 metres nd lel the sides using the onvention tht is the hypotenuse. =? = 24 metres 3 Sustitute the vlues into the pproprite formul. 4 Write the nswer using the orret units nd to the required ury. EXERISE 12.2 PRTISE ONSOLIDTE Trigonometry 2 = = = = = = = lterntively, = 2 2 = = = The ship n drift 249 metres, orret to the nerest metre. 1 WE1 Find the length of the unknown side (orret to 1 deiml ple) in the right-ngled tringle shown. 7 m 12 m 2 Find the length of the unknown side (orret to 1 deiml ple) in the right-ngled tringle shown. 18 m 16 m 3 WE2 Find the mimum horizontl distne (orret to the nerest metre) ship ould drift from its originl nhored point, if the nhor line is 200 metres long nd it is 50 metres to the ottom of the se from the end of the nhor line on top of the ship s dek. 4 n etension ldder is used to pint windows on multi-level house. The ldder etends fully to 6 metres nd to e sfe the se of the ldder is kept 1.5 metres from the house. t full etension how fr vertilly n the ldder reh to pint the windows (orret to 1 deiml ple)? 5 Find the length of the unknown side (orret to 1 deiml ple) in eh of the following right-ngled tringles MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

6 d 11.6 e 3 f n irrft is flying t n ltitude of 5000 metres. If its horizontl distne from the irport is 3 kilometres, wht is the distne (orret to the nerest metre) from the irport diretly to the irrft? 7 Wht is the length (orret to the nerest millimetre) of digonl re on retngulr gte tht is 2600 mm wide nd 1800 mm high? 8 Find the length of the unknown side (orret to 1 deiml ple) in eh of the following right-ngled tringles d e f lulte the lengths of the sloping sides in the following. (Rememer to onstrut suitle right-ngled tringle.) d 15 6 m m e 8 mm 10 mm 14 m 30 mm 8 m 12 m 10 lulte the vlue of the pronumerls f m m 4.8 mm 5.3 mm 6.3 mm d d m Topi 12 Trigonometry 615

7 MSTER 11 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. 12 The length of side F in the digrm elow is: F 2 3 E 2 D 5 E 6 1 m D 13 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 is: 40 metres 45 metres 50 metres D 55 metres E none of these 14 Find the length of the unknown side in eh of the following right-ngled tringles. Where neessry, give your nswer orret to 3 deiml ples The sun shines ove Guy s hed suh tht the length of his shdow on the ground is 1.6 m. If the distne from the top of his hed to the shdow of the top of his hed on the ground is 2.0 m, how tll is Guy (in metres)? 16 lulte the length of the pronumerl in eh of the following. (Where pplile, onstrut suitle right-ngled tringle.) Give your nswer orret to 2 deiml ples. d m 215 m 300 m 2.6 m 230 m 616 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

8 12.3 WoRKED EXMPLE 3 THINK Pythgoren trids Pythgoren trid is set of 3 numers whih stisfies Pythgors theorem. n emple is the set of numers 3, 4, 5 where 2 = So, 5 2 = = The digrm t right illustrtes this reltionship. nother Pythgoren trid is the multiple (sle ftor of 2) of the ove set: 6, 8, 10. Others re 5, 12, 13 nd 0.5, 1.2, 1.3. Prove these for yourself. 5 4 Is the set of numers 4, 6, 7 Pythgoren trid? 1 Find the sum of the squres of the two smller numers. WRITE = = 52 2 Find the squre of the lrgest numer. 7 2 = 49 3 ompre the two results. The numers form Pythgoren trid if the results re the sme 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 2 = 25). Step 2. Find the two onseutive numers tht dd up to the squred vlue ( = 25). Step 3. The trid is the odd numer you strted with together with the two onseutive numers (5, 12, 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 t right. In doing this, right ngle is formed opposite the 5-spe side WoRKED EXMPLE 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. WRITE = = 289 Topi 12 TRIgonoMETRy 617

9 2 Find the squre of the longest side nd ompre to the first result = = The tringle is right-ngled. 3 The right ngle is opposite the longest side. The right ngle is opposite the 17 m side. EXERISE 12.3 PRTISE ONSOLIDTE Pythgoren trids 1 WE3 Is the set of numers 5, 6, 8 Pythgoren trid? 2 Is the set of numers 5, 12, 13 Pythgoren trid? 3 WE4 tringle hs side lengths 3 m, 4 m nd 5 m. Is the tringle right-ngled? If so, where is the right ngle? 4 tringle hs side lengths 5 m, 8 m nd 10 m. Is the tringle right-ngled? If so, where is the right ngle? 5 re the following sets of numers Pythgoren trids? 9, 12, 15 4, 5, 6 30, 40, 50 d 3, 6, 9 e 0.6, 0.8, 1.0 f 7, 24, 25 g 6, 13, 14 h 14, 20, 30 i 11, 60, 61 j 10, 24, 26 k 12, 16, 20 l 2, 3, 4 6 omplete the following Pythgoren trids. Eh set is written from smllest to lrgest. 9,, 15, 24, , 2.0, d 3,, 5 e 11, 60, f 10,, 26 g, 40, 41 h 0.7, 2.4, 7 For eh of the sets whih were Pythgoren trids in question 5, stte whih side the right ngle is opposite. 8 tringle hs sides of length 16 m, 30 m nd 34 m. Is the tringle right-ngled? If so, where is the right ngle? 9 tringle hs sides of length 12 m, 13 m nd 18 m. Is the tringle right-ngled? If so, where is the right ngle? 10 Find the unknown length in eh se elow. Rdius = 3.5 m d 24 m d d e e f N d 26 km km E 618 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

10 11 n thlete runs 700 m north nd then 2.4 km west. How fr wy is the thlete from the strting point? 12 Find the perimeter of the flg (eluding the pole) s shown elow. MTHS QUEST 200 m 300 m 180 m MSTER 13 Whih of the following is Pythgoren trid? 7, 14, , 1.5, 3.6 3, 6, 9 D 12, 13, 25 E 15, 20, 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, 12, Find the perimeter of the flg, eluding the pole, shown in the figure elow. 11 m lulte the vlue of the pronumerl in eh of the following mm d 4.0 mm 2.4 mm y 6 m 4 m 9 m Topi 12 Trigonometry 619

11 12.4 Three-dimensionl Pythgors theorem Mny prtil situtions involve three-dimensionl ojets with perpendiulr plnes nd therefore the pplition of Pythgors theorem. To solve three-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 EXMPLE 5 THINK orret 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. 1 Drw digrm of the retngulr prism. 2 Identify the orienttion of the longest ojet from one orner to the furthest digonlly opposite orner. In this se, it is G. 3 Identify the two right-ngled tringles neessry to solve for the two unknown lengths. 4 Drw the tringles seprtely, identifying the lengths ppropritely. WRITE/DRW E 0.5 m F y 1.5 m 1.5 m D H D 1.0 m 5 lulte the length of digonl. 2 = y 2 = = = 3.25 y = 3.25 = (orret to 3 deiml ples) The length of is 1.8 metres (orret to 1 deiml ple). 6 lulte the length of digonl G, using the lulted length for. Note: To void rounding error, use the most urte form, whih is the surd Write the nswer using the orret units nd level of ury. = G 1.0 m (lterntive form) = = = 3.5 = (m) y G 0.5 m The longest rod tht ould fit in the r oot is 187 entimetres, orret to the nerest entimetre. 620 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

12 WoRKED EXMPLE 6 To find the height of 100-metre squre-sed pyrmid, with slnt height of 200 metres s shown, lulte the: V 200 m length of (in surd form) length of O (in surd form) height of the pyrmid VO (orret to the nerest metre). D O 100 m THINK lulte the length of digonl in the right-ngled tringle,. Write surds in their simplest form. EXERISE 12.4 PRTISE Three-dimensionl Pythgors theorem 1 WE5 wooden plnk of the gretest possile length is pled inside grden shed. Use the digrm to lulte the length of the plnk of wood orret to 1 deiml ple. 2 lulte the length of: E WRITE = (lterntive form) = = = = = The length of is m G. G 2 m 2 metres. O is hlf the length of. Length of O is or 50 2 metres. 2 1 lulte the height of the pyrmid, = 2 2 (lterntive form) VO, in the right-ngled tringle, VO. VO = (50 2) 2 2 Write the nswer using the orret units nd level of ury. = = = The height of the pyrmid, VO, is 187 metres, orret to the nerest metre. 3.2 m 100 m 200 m 50 2 m 100 m R V O 95 m D E 4 m F 2 m Topi 12 TRIgonoMETRy 621

13 3 WE6 Find: the length (in surd form) the length G (in surd form) the height of the pyrmid EG (orret to the nerest m). E ONSOLIDTE 18 m D G 5 m 16 m 4 Use the digrm of the pyrmid to nswer this question. Wht is the length of the digonl? Wht is the length of E? E O D OE = 300 m = 500 m 5 orret to the nerest entimetre, wht is the longest thin rod tht ould fit inside ue with side length 2 m? 6 orret 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? 7 For eh of the prisms shown, lulte: i the length of G F F H E 120 m E ii the length of G. G H J H I G 25 m D 40 m 400 mm D 300 mm 1200 mm F 5 m 6 m D 14 m E 40 m 622 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

14 8 For eh of the pyrmids shown, lulte: i the length of G ii the perpendiulr height. G 40 m 600 m D 15 m 20 m D 3 4 km km metre long rmp rises to height of 1.2 metres. How long (orret to 1 deiml ple) is the se of the rmp? 10 Two guide wires re used to support flgpole s shown. The height of the flgpole would e losest to: 3 m 8 m Wire 12 m D 21 m E 62 m 11 Find the vlues of the pronumerls (orret to 1 deiml ple) in the pyrmid t right. 12 Find the lengths of nd DH (orret to 2 deiml ples), where = 7.00 m nd H = m. 13 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)? 30 m 14 In eh of the following typil uilding strutures find the length of the unknown ross-re shown in pink. D m Wire 2 m 4 m 3.0 F 4.9 F E 30 m 10 m 40 m D E 6.1 G H G H Not to sle 5 m 3 m 11 m 3 m 2.6 m 8 m O 12 m Topi 12 Trigonometry 623

15 MSTER 15 For the offee tle design t right, find the length of the legs (orret to the nerest millimetre) if the offee tle is to e: 500 mm off the ground 700 mm off the ground Tle height nd the legs re offset from the vertil y distne of: i 100 mm ii 150 mm. 16 lulte the length of the pronumerl in the following sphere. Offset distne 12.5 Trigonometri rtios O 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 ). 2. The opposite side is diretly opposite the given ngle, θ. 3. The djent side is net to the given ngle, θ. onsider the three tringles shown here. We know from the previous topi on similrity tht Δ, ΔDE nd ΔFG re similr euse the orresponding ngles re the sme. Therefore, the orresponding sides re in the sme rtio (sle ftor) m D F 30 E 30 G 624 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

16 Intertivity Trigonometri rtios int-2577 WoRKED EXMPLE 7 Rtio of lengths of sides opy nd omplete the tle y identifying nd mesuring the lengths of the three sides for eh of the three tringles shown. Evlute the rtios of the sides. Length of side Tringle Opposite djent DE FG Hypotenuse Rtio of lengths of sides Opposite djent Opposite Hypotenuse Hypotenuse djent opposite Notie tht for eh of the rtios, for emple, the vlue is the sme for hypotenuse ll three 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 2. to find n unknown ngle, given two lengths. Sine rtio The sine rtio is defined s follows: length of opposite side The sine of n ngle = length of hypotenuse side. In short, sin(θ) = opposite hypotenuse sin(θ) = O H [SOH] Opposite Hypotenuse Find the length (orret to 1 deiml ple) of the line joining the verties nd in the tringle. 15 m 50 θ 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 θ = 50 m Opposite Topi 12 TRIgonoMETRy 625

17 2 Identify the pproprite trigonometri rtio, nmely the sine rtio, from the given informtion. osine rtio The osine rtio is defined s follows: length of djent side The osine of n ngle = length of hypotenuse side. In short, os(θ) = djent hypotenuse os(θ) = H ngle = 50 Opposite side = m Hypotenuse = 15 m 3 Sustitute into the formul. sin(θ) = sin(θ) = O H sin(50 ) = 15 4 Isolte nd evlute. 15 sin(50 ) = Write the nswer using the orret units nd level of ury. WoRKED EXMPLE 8 [H] [SOH] length of opposite side length of hypotenuse side = 15 sin(50 ) = = The length of the line joining verties nd is 11.5 entimetres, orret to 1 deiml ple. Hypotenuse djent 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. Find the length of the guy wire (orret to the nerest entimetre) supporting flgpole, if the ngle of the guy wire to the ground is 70 nd it is nhored 2 metres from the se of the flgpole. THINK 1 Drw digrm to represent the sitution nd identify n pproprite tringle. WRITE/DRW Guy wire 2 m 70 θ 626 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

18 2 Lel the digrm with the given ngle nd the given side to find n unknown side in right-ngled tringle. m Hypotenuse 3 hoose the pproprite trigonometri rtio, nmely the osine rtio. WoRKED EXMPLE 9 Tngent rtio The tngent rtio is defined s follows: The tngent of n ngle = In short, tn(θ) = opposite djent tn(θ) = O length of opposite side length of djent side. [TO] Opposite Find the length of the shdow (orret to 1 deiml ple) st y 3-metre tll pole when the ngle of the sun to the horizontl is 70. THINK 1 Drw digrm to represent the sitution nd identify n pproprite tringle m djent ngle = 70 djent side = 2 m Hypotenuse = m 4 Sustitute into the formul. os(θ) = H 5 Isolte nd evlute. 6 Write the nswer using the orret units nd level of ury. WRITE/DRW [H] os(70 ) = 2 1 os(70 ) = 2 2 = os(70 ) = The length of the guy wire is 5.85 metres or 585 entimetres, orret to the nerest entimetre. djent θ 3 m 70 Topi 12 TRIgonoMETRy 627

19 2 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 4 Sustitute into the formul. tn(θ) = O [TO] tn(70 ) = 3 5 Isolte nd evlute. 1 tn(70 ) = 3 3 = tn(70 ) = Write the nswer using the orret units nd The length of the shdow is level of ury. pproimtely 1.1 metres, orret to 1 deiml ple. WoRKED EXMPLE 10 Finding n unknown ngle If the lengths of the sides of tringle re known, unknown ngles within the tringle n e found. Find the smllest ngle (orret to the nerest degree) in 3, 4, 5 Pythgoren tringle. THINK 1 The smllest ngle is opposite the smllest side. Lel the sides s given y onvention for trigonometri rtios. 2 ll side lengths re known, therefore, ny one of the 3 rtios n e used. hoose one rtio, for emple, sine rtio. WRITE/DRW Opposite 3 4 ngle = Opposite side = 3 Hypotenuse = 5 3 Sustitute into the formul. sin(θ) = O H 5 Hypotenuse [SOH] sin() = onvert the rtio to deiml. = MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

20 5 Evlute. = sin 1 (0.6). = Write the nswer using the orret units nd level of ury. EXERISE 12.5 PRTISE Trigonometri rtios The smllest ngle is 37, orret to the nerest degree. 1 WE7 Find the length of the unknown side (orret to 1 deiml ple) in the following tringle m 2 Find the length of the unknown side (orret to 1 deiml ple) in the following tringle mm 3 WE8 ot is moored in lm wters with its depth sounder registering 14.5 m. If the nhor line mkes n ngle of 72 with the vertil, wht is the length of line (orret to the nerest metre) tht is out of the ot? 4 Find the length of the rmp (orret to the nerest entimetre), if the ngle the rmp mkes to the ground is 32 nd the rmp overs 3.6 m horizontlly. 5 WE9 Find the length of shdow (orret to 1 deiml ple) st y 4.5 m tll flg pole when the ngle of the sun to the horizontl is person is hoping to swim diretly ross stright river from point to point, distne of 215 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 (orret to the nerest metre) downstrem from point she finished. 7 WE10 Find the size of the unknown ngle (orret to the nerest degree) in the tringle elow. θ 2 m 2 m 8 Find the size of the unknown ngle (orret to the nerest degree) in the tringle elow. θ 500 mm 400 mm Topi 12 Trigonometry 629

21 ONSOLIDTE 9 Find the length of the unknown side (orret to 1 deiml ple) in eh of the following tringles km 430 mm 43 d y 15 m 92 mm 10 Find the vlue of the missing side (orret to 1 deiml ple) of the following tringles m Find the vlue of the unknown sides (orret to 1 deiml ple) of the following shpes m 20 m m m 12 Find the size of the unknown ngle (orret to the nerest degree) in eh of the tringles θ 3 θ 6 d 3 m 5.2 m θ O 1.2 m θ O 630 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

22 MSTER 13 Find the vlues of the unknown ngle, (orret to the nerest degree). 2 m 1.2 m 10 m 11 m 14 Find the sizes of the two ute ngles in 6, 8, 10 Pythgoren tringle. 15 The orret epression for the vlue of in the figure t right is: tn(37 ) 4 5 tn(37 ) os(37 ) 4 4 D tn(37 ) E 4 m 1 m 4 sin(37 ) 16 In the digrm elow find θ (orret to the nerest degree), metres nd y metres (oth orret to 1 deiml ple). 20 θ 60 4 m m jvelin is thrown so tht 15 m of its pointy end stiks into the ground. The sun is diretly overhed, sting shdow of 90 m in length. Determine the ngle (orret to the nerest degree) tht the jvelin mkes with the ground. 18 hot ir lloon is hovering in strong winds, 10 m vertilly ove the ground. It is eing held in ple y tut 12 m length of rope from the lloon to the ground. Find the ngle (orret to the nerest degree) tht the rope mkes with the ground. 19 rmp joins two points, nd. The horizontl distne etween nd is 1.2 m, nd is 25 m vertilly ove the level of. Find the length of the rmp (in metres orret to 2 deiml ples). Find the ngle tht the rmp mkes with the horizontl. 20 le r follows diret line from mountin pek (ltitude 1250 m) to ridge (ltitude 840 m). If the horizontl distne etween the pek nd the ridge is 430 m, find the ngle of desent (orret to the nerest degree) from one to the other. y 5 m 3 m 37 Topi 12 Trigonometry 631

23 12.6 Unit 4 OS M3 Topi 1 onept 2 The sine rule onept summry Prtie questions Intertivity The sine rule int-6275 The sine rule Introdution sine nd osine rules Often the tringle tht is pprent or identified in given prolem is non-rightngled. 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 2. 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. ll tringles n e divided into two right-ngled tringles. h Erlier, we sw tht the new side, h, n e evluted in two wys. sin() = h h = sin() h sin() = h 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: sin() = sin() = Similrly, if the tringle is lelled using other letters, for emple STU, then: s sin(s) = t sin(t) = u sin(u) This n e generlised s follows: in ny tringle, the rtio of side length to the sine of the opposite ngle is onstnt. h sin() 632 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

24 The sine rule is used if you re given: 1. two ngles nd one opposite side or 2. n ngle nd its opposite side length ( omplete rtio) nd one other side. WoRKED EXMPLE 11 THINK For emple, in tringle t right, = 7 m, = 50 nd = 9 m. ngle ould then e found using the sine rule. Find the unknown length, m, in the tringle (orret to 1 deiml ple). 1 Lel the tringle ppropritely for the sine rule. WRITE/DRW = = 7 m 2 We hve the ngle opposite to the unknown side nd known side sin() = sin() rtio, therefore, the ngle = = 130 sine rule n e used. = 7 m = 30 3 Sustitute known vlues into the two rtios. sin(130 ) = 7 sin(30 ) 4 Isolte nd evlute. = 7 sin(130 ) sin(30 ) = Write the nswer. The unknown length is 10.7 m, orret to 1 deiml ple. WoRKED EXMPLE 12 Sometimes it is neessry to find the third ngle in tringle in order to pply the sine rule. Find the unknown length, m (orret to 2 deiml ples). THINK 1 Lel the tringle ppropritely for the sine rule. WRITE/DRW = = = 7 m 7 m 50 = 9 m 7 m Topi 12 TRIgonoMETRy 633

25 2 lulte the third ngle euse it is opposite the unknown side. 3 Write the sine rule nd identify the vlues of the pronumerls. WoRKED EXMPLE 13 For tringle PQR, find the unknown ngle (orret to the nerest degree), P, given p = 5 m, r = 7 m nd R = 48. THINK 1 Drw the tringle nd ssume it is non-right-ngled. WRITE/DRW Q P 7 m 5 m 48 R 2 Lel the tringle ppropritely for the sine Q rule (it is just s esy to use the given lels). p = 5 r = 7 48 R P 3 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. sin() = = 180 ( ) = 15 sin() = = 15 = 7 = Sustitute the known vlues into the rule. sin(15 ) = 7 sin(100 ) 5 Isolte nd evlute. = 7 sin(15 ) sin(100 ) = Write the nswer. The unknown length is 1.84 m, orret to 2 deiml ples. p sin(p) = r sin(r) p = 5 P =? r = 7 R = 48 4 Sustitute known vlues into the two rtios. 5 sin(p) = 7 sin(48 ) 5 Isolte sin(p). sin(p) = sin(48 ) 5 7 sin(p) = 5 sin(48 ) 7 6 Evlute the ngle (inverse sine) nd inlude units with the nswer. P = sin 1 5 sin(48 ) 7 = The unknown ngle is 32, orret to the nerest degree. 634 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

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 step 2 of Worked emple 12. WoRKED EXMPLE 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 (orret to 1 deiml ple). THINK 1 Drw the sitution nd identify tht the tringle is non-right-ngled. 2 Drw the tringle seprtely from the sitution nd lel it ppropritely. The sine rule nnot e used stright wy s we do not hve oth known ngle nd known length opposite to the known ngle. Therefore, we need to find either or first. This is n isoseles tringle sine = ; therefore =. Using the ft tht the ngle sum of tringle is 180, find nd. 3 Write the formul for the sine rule nd identify the vlues of the pronumerls. 4 Sustitute the known vlues into the formul. 5 Trnspose the eqution to get the unknown y itself. 6 Evlute y orret to 1 deiml ple nd inlude the units. WRITE/DRW = 8 m m = 8 m 180 = + + = = = 144 = 72 nd, therefore, = = 72 sin() = sin() = y = 36 = 8 = 72 y sin(36 ) = 8 sin(72 ) y = 8 sin(36 ) sin(72 ) y 4.9 The rdius of the irle is 4.9 m, orret to 1 deiml ple. EXERISE 12.6 PRTISE The sine rule 1 WE11 Find the unknown length, mm Topi 12 TRIgonoMETRy 635

27 2 The length of side m is nerest to: D 5.8 E WE12 Find the unknown length,, orret to 1 deiml ple m X ONSOLIDTE 18 m siling epedition followed tringulr ourse s shown. Find the totl distne overed in the round trip. N 10.5 km 30 5 WE13 In ΔPQR, find the unknown ngle, R, given p = 48, q = 21 nd P = 110, orret to the nerest degree. 6 onstrut suitle tringle from the following instrutions nd find ll unknown sides nd ngles. One of the sides is 23 m; the smllest side is 15 m; the smllest ngle is WE14 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. 0.8 m On the lower setion of the truss, wht is the distne (orret to the nerest entimetre) etween eh pir of onseutive welds? Wht is the height (orret to the nerest entimetre) of the truss? 8 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 (orret to 1 deiml ple) of the unknown side. 9 Find the unknown length,, in eh of the following m m m d 250 km MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

28 10 The reltive positions of the shool, hurh nd post offie in smll town re shown t the verties of the tringle. Find the stright-line distne etween the shool nd the post offie (orret to 1 deiml ple). 11 Find the unknown length, (orret to 1 deiml ple) in eh se m 7 mm 55 m For the following questions give nswers orret to the nerest degree. In Δ, find the unknown ngle,, given = 6, = 6 nd = 52. In ΔLMN, find the unknown ngle, M, given m = 14.1, n = 27.2 nd N = 128. In ΔSTU, find the unknown ngle, S, given s = 12.7, t = 16.3 nd T = 45. d In ΔPQR, find the unknown ngle, P, given p = 2, r = 3.5 nd R = 128. e In Δ, find the unknown ngle,, given = 10, = 8 nd = The orret epression for the vlue of t in the given tringle is: 7 sin(100 ) sin(30 ) 5.5 sin(100 ) D sin(50 ) 7 m m t 5.5 sin(100 ) sin(30 ) E 7 sin(50 ) sin(100 ) 14 The vlue of (orret to 1 deiml ple) in the given figure is: D 3.3 E 3.6 Shool 15 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 The orret epression for ngle S in the given tringle is: sin 1 40 sin(41 ) 41 S os 1 40 os(41 ) 30 Post Offie 3 km sin(30 ) sin(100 ) km km 45 hurh 3 18 km 4 Topi 12 Trigonometry 637

29 MSTER 12.7 sin 1 30 sin(41 ) 40 D sin 1 41 sin(41 ) 30 E sin sin(41 ) 17 Find the perimeter of the eehive ell shown. 18 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 (orret to 1 deiml ple): the length of the rope the distne etween the two pegs. 19 plyground swing, whih is 2.3 m long, mkes n ngle of 74, t its swing point, in one omplete swing. Determine the horizontl distne (in metres orret to 1 deiml ple) etween the etreme positions of the swing set. 20 seni flight leves Geelong nd flies west of north for the 80 km diret journey to llrt. t llrt the plne turns 92 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 orret to 1 deiml ple) through whih the plne turned t Kyneton. Find the distne (orret to the nerest km) of the diret flight from llrt to Kyneton. miguous se of the sine rule Using S, 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. Otuse ute Similrly, sin(60 ) nd sin(120 ) give n identil vlue of Now try to find the inverse sine of these vlues; for emple, sin 1 rope tthed to (0.8660) is 60. The otuse (greter thn 90 ) ngle pole n e nhored is not given y the lultor. When using the inverse sine in two possile positions. funtion on your lultor, the lultor will give only the ute ngle. The sitution is illustrted prtilly in the digrm ove where the sine of the ute ngle equls the sine of the otuse ngle. 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 on the net pge. The tringles hve two orresponding sides equl, nd, s well s 10 mm 638 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

30 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 ute ngle. Tht is, n miguous se ours if < 90 nd sin < : WoRKED EXMPLE orret to the nerest degree, find the ngle, U, in tringle, given t = 7, u = 12 nd ngle T is 25. THINK WRITE/DRW 1 Drw suitle sketh of the tringle given. s the S S length of s is not given, side t n e drwn two u = 12 u = 12 t = 7 different wys. Therefore ngle U ould e either 25 t = 7 25 U T s U T s n ute or n otuse ngle. Lel the tringles ppropritely for the sine rule. (It is just s esy to use the given lels.) 2 Identify tht it is the sine rule tht n e used s you t hve the side opposite to the unknown ngle nd sin(t ) = u sin(u ) known side ngle rtio. t = 7 T = 25 u = 12 U =? 3 Sustitute the known vlues into the two rtios. 7 sin(25 ) = 12 sin(u ) 4 Trnspose the eqution to get the unknown y itself. sin(u ) = sin(25 ) sin(25 ) sin(u ) = 7 5 Evlute the ngle (inverse sine). Note tht the vlue is sin(u ) = n ute ngle ut it my lso e n otuse ngle. 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. 70 The ngle U is either 46 or 134, orret to the nerest degree. WoRKED EXMPLE 16 In the otuse-ngled tringle PQR, find the unknown ngle (orret to the nerest degree), P. Q 30 m 20 m R 40 P Topi 12 TRIgonoMETRy 639

31 THINK 1 Lel the tringle ppropritely for the sine rule. (It is just s esy to use the given lels.) WRITE/DRW Q p = 30 r = 20 2 Identify tht the sine rule is used s you hve the side opposite to the unknown ngle nd known side ngle rtio. EXERISE 12.7 PRTISE miguous se of the sine rule 1 WE15 To the nerest degree, find the ngle, U, in tringle, given t = 12, u = 16 nd ngle T is To the nerest degree, find the ngle, U, in tringle, given t = 7, u = 8 nd ngle T is WE16 In the otuse-ngled tringle PQR shown, find the unknown ngle (orret to the nerest degree), P. Q R 23 m R P p sin(p) = r sin(r) p = 30 P =? P 18 m r = 20 R = 40 3 Sustitute the known vlues into the two rtios. 30 sin(p) = 20 sin(40 ) 4 Trnspose the eqution to get the unknown y itself. sin(p) = sin(40 ) sin(40 ) sin(p) = 20 5 Evlute the ngle (inverse sine). Note tht the vlue sin(p) = is n ute ngle while in the digrm given it is n P = otuse ngle. 6 lulte the otuse ngle. P = = The ngle P is 105, orret to the nerest degree. 4 In the otuse-ngled tringle PQR shown, find the unknown ngle (orret to the nerest degree), Q. R 4.4 m Q 7.2 m P MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

32 ONSOLIDTE 5 Find oth the ute nd otuse ngles orret to one deiml ple. In Δ, find the unknown ngle,, given = 10.8, = 6 nd = Find oth the ute nd otuse ngles orret to one deiml ple. In ΔSTU, find the unknown ngle, S, given t = 12.7, s = 16.3 nd T = Find oth the ute nd otuse ngles orret to one deiml ple. In ΔPQR, find the unknown ngle, P, given p = 3.5, r = 2 nd R = Find oth the ute nd otuse ngles orret to one deiml ple. In ΔLMN, find the unknown ngle, M, given n = 0.22 km, m = 0.5 km nd N = Find the unknown ngle (orret to the nerest degree) in the following otuse-ngled tringle. 60 km 10 Find the unknown ngle (orret to the nerest degree) in the following otuse-ngled tringle. 3 m m 11 Find the unknown ngle (orret to the nerest degree) in the following otuse-ngled tringle m 110 km 11 m 12 Find the unknown ngle (orret to the nerest degree) in the following otuse-ngled tringle m 7 m 25 Topi 12 Trigonometry 641

33 13 In the tringle given, ngle is (orret to the nerest degree): D 141 E m 14 Find the two unknown ngles shown in the digrm (orret to 1 deiml ple) m MSTER 12.8 Unit 4 OS M3 Topi 1 onept 3 The osine rule onept summry Prtie questions Intertivity The osine rule int m 9 m 9 m 27 y 15 Look t the swinging pendulum shown. W 8 m Drw the two possile positions of the o 5 m 15 t the level of the horizontl line. V Find the vlue of the ngle, W, t these two etreme positions. Find the smllest nd lrgest distnes etween verte V nd the o. 16 If ot trvelled the pth shown: wht is the otuse ngle etween the PQ nd QR legs of the trip? wht is the distne trvelled from P to Q? The osine rule 800 m Q 32 R P 1.4 km 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. h The tringle in the figure hs een divided into two right-ngled tringles with se sides equl to nd ( ). D In ΔD, h 2 = 2 2 nd in ΔD, h 2 = 2 ( ) 2 (Pythgors theorem) Equting epressions for h 2, 2 2 = 2 ( ) 2 2 = ( ) 2 = = [1] 642 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

34 WoRKED EXMPLE 17 THINK Now, from ΔD, Find the unknown length (orret to 2 deiml ples),, in the tringle. 1 Identify the tringle s non-right-ngled. os() = = os() Sustitute this vlue of into [1] ove. 2 = [ os()] So, the osine rule n e written s: In similr wy to tht ove, it n e shown tht: 2 = os() 2 = os(). lso, if the tringle is lelled using other letters, for emple STU, then: s 2 = t 2 + u 2 2tu os(s). The formul my e trnsposed in order to find n unknown ngle. 2 Lel the tringle ppropritely for the sine rule or osine rule. 2 = os(). os() = or lterntively, os() = nd os( ) = The osine rule is used to find: 1. n unknown length when you hve the lengths of two sides nd the ngle in etween 2. n unknown ngle when you hve the lengths of ll three sides. 7 m 80 6 m WRITE/DRW = 7 = 3 Identify tht it is the osine rule tht is required s you hve the two sides nd the ngle in etween. 80 = 6 = 6 = 80 = 7 = Topi 12 TRIgonoMETRy 643

35 4 Sustitute the known vlues into the osine rule formul nd evlute the right-hnd side. 5 Rememer to get the squre root vlue,. = = Write the nswer, rounding off to the required numer of deiml ples nd inluding the units. WoRKED EXMPLE 18 THINK EXERISE 12.8 PRTISE Find the size of ngle in the tringle, orret to the nerest degree. 1 Identify the tringle s non-right-ngled. 2 Lel the tringle ppropritely for the sine rule or osine rule. 3 s ll three sides re given, the osine rule should e used. Write the rule nd identify the vlues of the pronumerls. 4 Sustitute the known vlues into the formul nd simplify. The osine rule WE17 Find the unknown length orret to 2 deiml ples Find the unknown length orret to 2 deiml ples mm 2000 mm WRITE/DRW = 6 = 6 = 4 os() = = 4, = 6, = 6, = os() = os() = os() = Evlute [ = os 1 (0.3333)]. = Round to the nerest degree nd stte your nswer. The ngle is 71, orret to the nerest degree. 4 2 = os() 2 = os(80 ) = os(80 ) = = 8.39 The unknown length is 8.39 m, orret to 2 deiml ples. 644 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

36 3 WE18 Find the size of the unknown ngle (orret to the nerest degree). ONSOLIDTE 8 m 5 m 4 onstrut suitle tringle from the following instrutions nd find ll unknown sides nd ngles. Two sides re 23 m nd 15 m nd the ngle in etween is Find the unknown length in eh of the following (orret to 2 deiml ples). 10 m 60 5 m z 6 m d 2.3 km 1.5 km km During siling re, the ots followed tringulr ourse s shown. Find the length,, of the third leg (orret to 1 deiml ple). 10 km km km 7 Two irles, with rdii 5 m nd 8 m, overlp s shown. 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 (orret to 1 deiml ple). 5 m 8 m Find the size of the unknown ngle in eh of the following (orret to the nerest degree). 12 mm y 85 km 13 mm 20.5 m 19.1 m p 101 km 20 mm 28.6 m 68 km 9 onsider the siling epedition ourse in question 6. Find the two unknown ngles (orret to the nerest degree) in the tringulr ourse. 10 For the tringle shown, find ll three unknown ngles (orret to the nerest degree) Topi 12 Trigonometry 645

37 MSTER For the following questions, give nswers orret to 1 deiml ple. For Δ, find the unknown side,, given = 10 km, = 8 km nd = 30. For Δ, find the unknown ngle,, given = = 10 nd = 6. For Δ, find the unknown side,, given = 7 m, = 3 m nd = 80. d For ΔSTU, find the unknown ngle, S, given t = 12.7, s = 16.3 nd u = e For ΔPQR, find the unknown ngle, P, given p = 2, q = 3.5 nd r = 2.5. f For Δ, find the unknown side,, given = 260, = 120 nd = In the tringle given, the lrgest ngle is: D 85 E The orret epression for ngle s is: os os E os The orret epression for the vlue of t is: os D os os(120 ) D E m 15 m 20 m 15 The 4 surfe ngles t the verte of regulr squre-sed pyrmid re ll the sme. The 15 m mgnitude of these ngles for the pyrmid shown t right (orret to the nerest degree) is: D 39 E Find the unknown vlues. 4 m 4 m 2 m 12 m m 6 m 8 m Speil tringles 5 m 4 m 6 m s 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 60. t 10 m 646 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

38 WoRKED EXMPLE 19 THINK 60 = = = 3 = = = = = = 60 = 60 = = = 14 = = = 60 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 = = 13 = 13 2 = = 45 = 90 = = 10 = 10 2 = = 45 = 90 lso, the hypotenuse is lwys 2 times the length of the smller sides. hek for yourself using Pythgors theorem. Find the vlues of r nd ngle θ in the hegon shown. 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 WRITE/DRW 6 m = = 5 = 5 2 = = 45 = = 20 = = 20 2 = = 45 = 90 Regulr hegon 6 m r m θ 2 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. θ = = 60 r = 6 m Topi 12 TRIgonoMETRy 647

39 WoRKED EXMPLE 20 Find the vlue of the pronumerl (orret to 1 deiml ple) in the figure m THINK 1 The tringle is right-ngled isoseles tringle. Two ngles re 45 nd the third ngle is Two sides re equl nd the longer side opposite the right ngle is 2 times longer thn these equl sides. 3 Write your nswer using the required ury nd inlude units. EXERISE 12.9 PRTISE Speil tringles 1 WE19 Find the vlue of θ in the regulr otgon shown. 5 m 2 Find the vlue of the unknown length. θ 9 m 3 WE20 Find the vlue of the pronumerl (orret to 1 deiml ple) in the figure shown. 4 Find the vlue of the unknown length. m m 158 m WRITE/DRW 12 m m = 2 = 12 2 = The vlue of is 17.0 m, orret to 1 deiml ple m ONSOLIDTE 5 Find the vlue of the unknowns Find the vlue of the unknown length. 100 m MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

40 7 Find the vlue of the unknown length Find the vlue of the unknowns. y 7.2 m MSTER Intertivity re of tringles int mm 9 In Δ, find the unknown ngle,, given = 10, = 10 2 nd = In ΔSTU, find the unknown side, s, given t = 12.7, S = 45 nd T = In ΔPQR, find the unknown ngle, P, given p = 3.5, r = 3.5 nd R = In ΔLMN, find the unknown side, m, given n = 0.22, 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? 14 Wht is the height of tree if its shdow, on horizontl ground, is 12 metres long when the sun s rys striking the tree re t 45 to the ground? 15 In the tringle given, the length of side (in metres) is: d 20 e m 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 2 = 1 2 h 60 Topi 12 Trigonometry 649

41 Unit 4 OS M3 Topi 1 onept 4 re of tringle onept summry Prtie questions Intertivity Using Heron s formul to find the re of tringle int m Height 4 m se se Height Method 2. When we re given two lengths nd the ngle in etween we would use: = 10 m 32 = 15 m re tringle = 1 2 sin() = 1 sin() 2 = se 1 re = 2 se Height 1 = sin() Method 3. When ll three sides re known we would use: re tringle = semi-perimeter, s = 2 s(s )(s )(s ) where the ( + + ). 2 Height = sin () This formul is known s Heron s formul. It ws developed y Heron (or Hero) of lendri, Greek mthemtiin nd engineer who lived round e 62. Let us find the re of the tringle elow to demonstrte tht ll three formuls provide the sme result. 3 For the 3, 4, 5 tringle, the most pproprite method is method 1 euse it is right-ngled tringle. re tringle = 1 se Height 2 = 6 The other two methods my lso e used. 5 4 = re tringle = 1 2 sin() = sin(90 ) = 6 1 = MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

42 WoRKED EXMPLE 21 THINK ( + + ) re tringle = s(s )(s )(s ) s = 2 ( ) = 6(6 3)(6 4)(6 5) = = = 12 2 = 36 = 6 = 6 Find the re of the tringle shown. 1 The two given lengths re perpendiulr. Write the most pproprite formul for finding the re. WRITE re tringle = 1 se Height 2 2 Sustitute the known vlues into the formul. = = 48 3 Write the nswer using orret units. The re of the tringle is 48 mm 2. WoRKED EXMPLE 22 8 mm 12 mm Find the re of the tringle (orret to 2 deiml ples). THINK 1 Identify the shpe s tringle with two known sides nd the ngle in etween. 2 Identify nd write down the vlues of the two sides, nd, nd the ngle in etween them,. 3 Identify the pproprite formul nd sustitute the known vlues into it. WRITE/DRW = 9 37 = 6 = 6 = 9 = 37 re tringle = 1 sin( ) 2 2 = sin(37 ) = Write the nswer using orret units. The re of the tringle is m 2, orret to 2 deiml ples. 9 m 37 6 m Topi 12 TRIgonoMETRy 651

43 WoRKED EXMPLE 23 Find the re of tringle PQR (orret to 1 deiml ple), given p = 6, q = 9 nd r = 4, with mesurements in entimetres. THINK 1 ll three sides of the tringle hve een given; therefore, Heron s formul n e used to find the re. WRITE/DRW Q p = 6 r = 4 2 Write the vlues of the three sides,, nd, nd lulte the semiperimeter vlue, s. 3 Sustitute the known vlues into Heron s formul nd evlute. EXERISE PRTISE re of tringles 1 WE21 Find the re of the tringle shown m 2 Find the re of the tringle shown. 10 m 32 m R 3 WE22 Find the re of the tringle shown (orret to 2 deiml ples). 5 m 42 q = 9 4 Find the re of the tringle shown (orret to 2 deiml ples). 7 m P = p = 6, = q = 9, = r = 4 ( + + ) s = 2 ( ) = 2 = 9.5 re tringle = s(s )(s )(s ) = 9.5(9.5 6)(9.5 9)(9.5 4) = = re = Write the nswer, using the orret units. The re of tringle PQR is 9.6 m 2, orret to 1 deiml ple. 7.8 m m 5 WE23 Find the re of Δ (orret to 1 deiml ple) given = = 10 m nd = 6 m. 652 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

44 ONSOLIDTE 6 Find the re of n equilterl tringle with side lengths of 10 m. 7 Find the res of the following tringles (orret to 1 deiml ple). 4.5 mm 7 m 7.0 mm 12 m d 3.2 mm 3 m 10.5 mm 4 m 8 Find the res of the following tringles (orret to 1 deiml ple). 7 m 7 m 100 m m d 10.2 m 5 m 4 m 80 3 m m 9 Find the res of the following tringles (orret to 1 deiml ple). 8 m 8 m 20 mm 6 m d 3 km 5.2 m 3.1 m 4 km 6 km 6.7 m 10 Find the res of the following tringles (orret to 1 deiml ple). 2.5 km d 4.6 m 4.4 m 112 m m m km 10 km Topi 12 Trigonometry 653

45 MSTER 11 Find the re of eh of the following tringles. (Give ll nswers orret to 1 deiml ple.) Δ, given = 10 km, = 8 km nd = 30 Δ, given = 7 m, = 3 m, = 8.42 m nd = 108 ΔSTU, given t = 12.7 m, s = 16.3 m nd u = 24.5 m d ΔPQR, given p = 2 units, q = 3.5 units nd r = 2.5 units e Δ, given = 260 m, = 120 m nd = tringulr rh hs supporting legs of equl length of 12 metres s shown in the digrm. Wht is its re? m 12 m 13 From the digrm given t right, find the re of: i one of the tringles ii ll of the tringles. Use nother tehnique to verify your nswer in i. 14 Find the re of the stte forest s defined y the three fire-spotting towers on the orners of its oundry. 11 km 5.2 km 10.4 km 15 If the perimeter of n equilterl tringle is 210 metres, its re is losest to: 2100 m m m 2 D 5500 m 2 E 1700 m 2 16 The orret epression for the re of the shpe shown is: sin(80 ) m os(100 ) 6.13 m sin(100 ) 2 30 D E none of the ove 17 The orret epression for the re of the otgon t right is: 195 sin(45 ) 169 sin(45 ) 195 sin(60 ) d 338 sin(60 ) e sin(67.5 ) 18 Find the re of the following tringles km 5 mm 10 mm mm 654 MTHS QUEST 12 FURTHER MTHEMTIS VE Units 3 nd 4

46 ONLINE ONLY Review The Mths Quest Review is ville in ustomisle formt for you to demonstrte your knowledge of this topi. The Review ontins: Multiple-hoie questions providing you with the opportunity to prtise nswering questions using S tehnology Short-nswer questions providing you with the opportunity to demonstrte the skills you hve developed to effiiently nswer questions using the most pproprite methods ONLINE ONLY tivities To ess eookplus tivities, log on to Intertivities omprehensive set of relevnt intertivities to ring diffiult mthemtil onepts to life n e found in the Resoures setion of your eookplus. Etended-response questions providing you with the opportunity to prtise em-style questions. summry of the key points overed in this topi is lso ville s digitl doument. REVIEW QUESTIONS Downlod the Review questions doument from the links found in the Resoures setion of your eookplus. studyon is n intertive nd highly visul online tool tht helps you to lerly identify strengths nd weknesses prior to your ems. You n then onfidently trget res of gretest need, enling you to hieve your est results. Topi 12 TRIgonoMETRy 655

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