Trigonometry 9 VEcoverge re of study Units & Geometry nd trigonometry In this ch chpter 9 Pythgors theorem 9 Pythgoren trids 9 Three-dimensionl Pythgors theorem 9D Trigonometric rtios 9E The sine rule 9F mbiguous cse of the sine rule 9G The cosine rule 9H Specil tringles 9I re of tringles
0 Further Mthemtics Trigonometry Trigonometry is used to solve problems involving distnces nd ngles from the formtion of tringles. Often the problem is descriptive one nd to confidently solve it, you need to visulise the sitution nd drw n pproprite digrm or sketch. Lbeling conventions Where we re deling with trigonometric figures like tringles, there re severl lbeling conventions tht help us remin cler bout the reltionships between the points, ngles nd lines being used. These will be eplined s they rise; however, the bsic convention used in this book is shown in the figure t right. Note the use of itlics. The ngle is t point, which is opposite line. The ngle is t point, which is opposite line b. c The ngle is t point, which is opposite line c. To void cluttered digrms, only the points b (,, ) re usully shown; the ssocited ngles (,, ) re ssumed. Note: Nturlly, we do not need such lbels in ll digrms, nd sometimes we wish to lbel points, ngles nd lines in other wys, but these will lwys be cler from the digrm nd its contet. Pythgors theorem efore investigting the reltionships between the ngles nd sides of tringle we should consider problem solving technique tht involves only the sides of tringles: Pythgors theorem. Pythgors theorem is ttributed to the Greek mthemticin nd philosopher, Pythgors, round 500. (However, the principle ws known much erlier, nd it seems tht even the pyrmid builders of ncient Egypt used the theorem in constructing the pyrmids.) The theorem describes the reltionship between the lengths of the sides of ll right-ngled tringles. c Hypotenuse Pythgors theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides, or c = + b nd, therefore, to find c, b c = + b where c is the longest side or hypotenuse nd nd b re the two shorter sides. Note: ecuse the eqution c = + b hs become stndrd wy of epressing Pythgors theorem, we often djust the lbeling convention to use c for the hypotenuse no mtter how the opposite (right) ngle point is lbeled. However, this will lwys be cler 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: = c b nd so = c b
hpter 9 Trigonometry Find the length of the unknown side (to deciml plce) in the rightngled tringle shown. Note tht the tringle is right-ngled nd we need to find the unknown length, given the other two lengths. = c = Lbel the sides of the tringle, using the convention tht c is the hypotenuse. Substitute the vlues into the pproprite formul. b = 7 c = + b WORKED Emple Write the nswer using the correct units nd to the pproprite degree of ccurcy. cm lterntively, = + 7 c = + b = 6 + 9 = + 7 = 65 = 6 + 9 = 65 = 65 = 8.06 The unknown side s length is 8. cm. 7 cm WORKED Emple Find the mimum horizontl distnce (to the nerest metre) ship could drift from its originl nchored point, if the nchor line is 50 metres long nd it is metres to the bottom of the se from the end of the nchor line on top of the ship s deck. Sketch suitble digrm of the problem 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 lbel the sides using the convention tht c is the hypotenuse. Substitute the vlues into the pproprite formul. Write the nswer using the correct units nd to the required ccurcy. c = 50 metres b = metres =? c = + b lterntively, 50 = + = c b 6500 = + 576 = 50 = 6500 576 = 6 500 576 = 69 = 6 9 = 6 9 = 8.85 The ship cn drift pproimtely 9 metres.
Further Mthemtics remember remember. Pythgors theorem is used: () only on right-ngled tringles (b) to find n unknown length or distnce, given the other two lengths.. When using Pythgors theorem: () drw n pproprite digrm or sketch (b) ensure the hypotenuse side, c, is opposite the right ngle (90 ) (c) c = + b or c = + b. 9 Pythgors theorem EXEL G bri Spredsheet progrm WORKED Emple Mthcd Pythgors theorem Pythgors visul Geometry Pythgors theorem G progrm Pythgors theorem clcultions Find the length of the unknown side (to deciml plce) in ech of the following right-ngled tringles. b c 5 d.6 e f 7.5 8 9 0.7. n ircrft is flying t n ltitude (distnce bove the ground) of 5000 metres. If its horizontl distnce from the irport is kilometres, wht is the distnce (to the nerest metre) from the irport directly to the ircrft? WORKED Emple Wht is the length (to the nerest millimetre) of digonl brce on rectngulr gte tht is 600 mm wide nd 800 mm high? Find the length of the unknown side (to deciml plce) in ech of the following right-ngled tringles. b c 7 8 0 0 5 9
hpter 9 Trigonometry d e f 5 7 0.6 7. 5 5 lculte the lengths of the sloping sides in the following. (Remember to construct suitble right-ngled tringle.) b c 5 0 0 mm 0.8.6 6. d e f 0 mm 8 mm 6 m m m 05 cm 5 cm 8 m m 60 cm 6 lculte the vlue of the pronumerls. b b c d 7. 0.6 c 6..5 mm 0 5 5. mm 6. mm.6.7 d. 7 One of the smller sides of right-ngled tringle is 6 metres long. The hypotenuse is 8 metres longer thn the other unknown side. Drw suitble tringle to represent this sitution. b Write n epression to show the reltionship between the three sides. c Stte the lengths of ll three sides. 8 multiple choice The length of side F in the digrm t right is: F E D m D 5 E 6 9 multiple choice To the nerest metre, the length of cble tht would connect the roofs of two buildings tht re 0 metres nd 80 metres high respectively nd re 0 metres prt is: 0 metres 5 metres 50 metres D 55 metres E none of the bove
Further Mthemtics Pythgoren trids Pythgoren trid is set of numbers which stisfies Pythgors theorem. n emple is the set of numbers,, 5 where c = + b So, 5 = + 5 = 9 + 6 The digrm below illustrtes this reltionship. 5 nother Pythgoren trid is the multiple (scle fctor) of of the bove set: 6, 8, 0. Others re 5,, nd 0.5,.,.. Prove these for yourself. WORKED Emple Is the set of numbers, 6, 7 Pythgoren trid? 5 0 8 6 Find the sum of the squres of the two + 6 = 6 + 6 smller numbers. = 5 Find the squre of the lrgest number. 7 = 9 ompre the two results. The numbers 7 + 6 form Pythgoren trid if the results re the sme. Write your nswer., 6, 7 is not Pythgoren trid. nother wy to generte Pythgoren trids is by using the following rule: Step. Squre n odd number (5 = 5). Step. Find the two consecutive numbers tht dd up to the squred vlue ( + = 5). Step. The trid is the odd number you strted with together with the two consecutive numbers (5,, ). Try to find trid for the odd number 9.
hpter 9 Trigonometry 5 tringle whose sides form Pythgoren trid contins right ngle, which is opposite the longest side. This result cn be illustrted pproimtely with rope of ny length, by tying eqully spced knots nd forming tringle with sides equl to, nd 5 spces, s shown below. In doing this right ngle is formed opposite the 5-spce side. WORKED Emple tringle hs sides of length 8 cm, 5 cm nd 7 cm. Is the tringle right-ngled? If so, where is the right ngle? 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 compre to the first result. The right ngle is opposite the longest side. 8 + 5 = 6 + 5 8 + 5 = 89 7 = 89 7 = 8 + 5 The tringle is right-ngled. The right ngle is opposite the 7 cm side. remember remember. Pythgoren trid is set of three numbers which stisfies Pythgors theorem.. tringle whose side lengths form Pythgoren trid hs right ngle opposite the longest side.. Some common trids re:,, 5 6, 8, 0 9,, 5 0., 0., 0.5 5,, 0,, 6 0.5,.,. 7,, 5 9, 0,
6 Further Mthemtics 9 Pythgoren trids EXEL Spredsheet Pythgoren trids WORKED Emple WORKED Emple re the following sets of numbers Pythgoren trids? 9,, 5 b, 5, 6 c 0, 0, 50 d, 6, 9 e 0.6, 0.8,.0 f 7,, 5 g 6,, h, 0, 0 i, 60, 6 j 0,, 6 k, 6, 0 l,, omplete the following Pythgoren trids. 9,, 5 b,, 5 c.5,.0, d,, 5 e, 60, f 0,, 6 g, 0, h 0.7,., For ech of the sets which were Pythgoren trids in question stte which side the right ngle is opposite. tringle hs sides of length 6 cm, 0 cm nd cm. Is the tringle right-ngled? If so, where is the right ngle? 5 tringle hs sides of length cm, cm nd 8 cm. Is the tringle right-ngled? If so, where is the right ngle? 6 Find the unknown length in ech cse below. b c 0 Rdius =.5 cm d cm c 0 9 d d. e f 6. e N. 0. d 6 km 0. 0 km E 7 n thlete runs 700 m north nd then. km west. How fr wy is the thlete from the strting point? 8 Find the perimeter of the flg s shown t right. 9 multiple choice Which of the following is Pythgoren trid? 7,,.,.5,.6, 6, 9 D,, 5 E 5, 0, 5 0 multiple choice Which of the following is not Pythgoren trid? 5,, 6, 9,, 8, 85 D 0.9,.0,. E 5,, 00 cm MTHS QUEST 80 cm 00 cm
hpter 9 Trigonometry 7 Three-dimensionl Pythgors theorem Mny prcticl situtions involve -dimensionl objects with perpendiculr plnes nd therefore the ppliction of Pythgors theorem. To solve -dimensionl problems, crefully drwn nd lbelled digrm will help. It is lso of benefit to identify right ngles to see where Pythgors theorem cn be pplied. This enbles you to progress from the known informtion to the unknown vlue(s). WORKED Emple 5 To the nerest centimetre, wht is the longest possible thin rod tht could fit in the boot of cr? The boot cn be modelled s simple rectngulr prism with the dimensions of.5 metres wide, metre deep nd 0.5 metres high. Drw digrm of the rectngulr prism. Identify the orienttion of the longest object from one corner to the furthest digonlly opposite corner. In this cse, it is G. E 0.5 m F.5 m H D G.0 m Identify the two right-ngled tringles necessry to solve for the two unknown lengths. Drw the tringles seprtely, identifying the lengths ppropritely. y.0 m.5 m D y G 0.5 m 5 6 7 lculte the length of digonl. c = + b lculte the length of digonl G, using the clculted length for. Note: To void trunction error use the most ccurte form, which is the surd.5. Write the nswer using the correct units nd level of ccurcy. y =.5 +.0 y =.5 + y =.5 y =.5 y =.80 (to deciml plces) The length of is.8 metres (to deciml plce). c = + b = = =.5 =.8708 0.5 + (.5) 0.5 +.5 (lterntive form) The longest rod tht could to fit in the cr boot is 87 centimetres.
8 Further Mthemtics WORKED Emple 6 To find the height of 00-metre squre-bsed pyrmid, with slnt height of 00 metres s shown, clculte the: length of (in surd form) b length of O (in surd form) c height of the pyrmid VO (to the nerest metre). V 00 m D O 00 m lculte the length of digonl in c = + b (lterntive form) the right-ngled tringle,. Write surds in their simplest form. = = 00 + 00 0 000 = 0 000 = 00 The length of is 00 metres. b O is hlf the length of. 00 b Length of O is --------------- or 50 metres. c lculte the height of the pyrmid, c = c b (lterntive form) VO, in the right-ngled tringle, VO. VO = 00 ( 50 ) VO = VO = 0 000 5000 5 000 VO = 87.088... Write the nswer using the correct units nd level of ccurcy. The height of the pyrmid, VO, is 87 metres. remember remember To solve problems involving -dimensionl Pythgors theorem:. Drw nd lbel n pproprite digrm.. Identify the right ngles.. Identify right-ngled tringles tht enble the informtion given to be used to find the unknown vlue(s).. To void trunction error, try to use the surd form (for emple, 7 rther thn 6....) if the result is required in further clcultions.
hpter 9 Trigonometry 9 9 Three-dimensionl Pythgors theorem WORKED Emple 5 To the nerest centimetre, wht is the longest thin rod tht could fit inside -metrecube bo (cuboid)? To the nerest centimetre, wht is the longest drum stick tht could fit in rectngulr toy bo whose dimensions re 80 cm long by 80 cm wide by 60 cm high? For ech of the prisms shown, clculte: i the length of ii the length of G. G F b F G c H E 0 cm E H J H I G WORKED Emple 6 00 mm D 5 cm 00 mm 0 cm D 00 mm For ech of the pyrmids shown, clculte: i the length of ii the perpendiculr height. G b F 5 m G D 6 m m E 0 m 0 m 600 m D 5 m 0 m D km km 5.5-metre long rmp rises to height of. metres. How long (to deciml plce) is the bse of the rmp? 6 multiple choice Two guide wires re used to support flgpole s shown. The height of the flgpole would be closest to: m 8 m m D m E 6 m Wire 7 Find the vlues of the pronumerls (to deciml plce) in the pyrmid t right. 8.5 m Wire m m c b.0.9 6.
0 Further Mthemtics 8 Find the lengths of nd DH (to deciml plces), where F = 7.00 m nd H = 5.00 m. G 9 For the tent shown in E. m E H the digrm t right, find D.5 m (to the nerest mm): the length of the D cross-brce,. m b the height of the centre pole, EF. F.5 m 0 The feet of cmer tripod, which re on.5-metre legs, form the vertices of n equilterl tringle. The distnce from the centre of the equilterl tringle to the foot on ny of the three legs is 0.75 m. Find the perpendiculr height to the top of the tripod (to deciml plces). mn moves through two-level mze by following F 0 m G the solid blck line, s shown in the digrm. Wht is D the direct distnce from his strting point,, to his end point, F (to the nerest metre)? m E H 0 m 0 m Not to scle In ech of the following typicl building structures find the length of the unknown cross-brce shown in red. 800 mm b WorkSHEET 9. c 000 mm m c.6 m 500 mm For the coffee tble design t right, find the length of the legs (to the nerest millimetre) if the coffee tble is to be: 500 mm off the ground Tble b 700 mm off the ground height nd the legs re offset from the verticl by distnce of: i 00 mm E F ii 50 mm. G.0 m Find the length of the brce, G (to the nerest centimetre), tht is needed to reinforce the wedge-shped structure shown. D.0 m.0 m d 5 m m b ll mesurements re in metres..5 7 m.0 m d Offset distnce
hpter 9 Trigonometry Trigonometric rtios Trigonometric rtios include the sine rtio, the cosine rtio nd the tngent rtio; three rtios of the lengths of sides of right-ngled tringle dependent on given cute ngle. Lbelling convention For the trigonometric rtios the following lbelling convention should be pplied:. The hypotenuse is opposite the right ngle (90 ).. The opposite side is directly opposite the given ngle, θ.. The djcent side is net to the given ngle, θ. onsider the three tringles drwn below. We know from the previous chpter on similrity tht tringles, DE nd FG re similr becuse the corresponding ngles re the sme. Therefore, the corresponding sides re in the sme rtio (scle fctor). D 0 F MQ FurMt fig.9 0 E MQ FurMt fig.9b 0 G Rtio of lengths of sides opy nd complete the tble below by identifying nd mesuring the lengths of the three sides for ech of the three tringles bove. Evlute the rtios of the sides. Length of side Rtio of lengths of sides Opposite djcent Tringle Opposite djcent Hypotenuse ----------------------------- ----------------------------- ---------------------- Opposite Hypotenuse Hypotenuse djcent DE FG Opposite Notice tht for ech of the rtios, for emple ----------------------------, the vlue is the sme Hypotenuse for ll three tringles. This is the sme for ll right-ngled tringles with the sme cute ngle.
Further Mthemtics Trigonometric rtios re used in right-ngled tringles:. 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: Length of opposite side The sine of n ngle = ---------------------------------------------------------------. Length of hypotenuse side Opposite In short, sin θ = ---------------------------- Hypotenuse Opp sin θ = --------- [SOH] Hyp Find the length (to deciml plce) of the line joining the vertices nd in the tringle t right. 5 WORKED Emple 7 Identify the shpe s right-ngled tringle with given length nd ngle. Lbel the sides s per the convention for trigonometric rtios. Identify the pproprite trigonometric rtio, nmely the sine rtio, from the given informtion. 5 cm Hypotenuse = 50 Substitute into the formul. sin θ = Opposite ngle = 50 Opposite side = cm Hypotenuse = 5 cm [SOH] Length of opposite side --------------------------------------------------------------- Length of hypotenuse side Hypotenuse Opp sin θ = --------- Hyp sin 50 = ----- 5 Isolte nd evlute. 5 sin 50 = ----- 5 5 = 5 sin 50 = 5 0.766 =.9 Write the nswer using the correct units The length of the line joining vertices nd nd level of ccurcy. is.5 centimetres. θ cm Opposite 5 cm 50 θ osine rtio The cosine rtio is defined s follows: Length of djcent side The cosine of n ngle = ---------------------------------------------------------------. Length of hypotenuse side djcent In short, cos θ = ---------------------------- Hypotenuse dj cos θ = ----------- [H] Hyp djcent Hypotenuse θ
hpter 9 Trigonometry In worked emple 7 the sine rtio ws required to find the unknown length. The cosine rtio cn be pplied in the sme wy, if it is required. WORKED Emple 8 Find the length of the guy wire (to the nerest centimetre) supporting flgpole, if the ngle of the guy wire to the ground is 70 nd it is nchored metres from the bse of the flgpole. Drw digrm to represent the sitution nd identify n pproprite tringle. Guy Wire Lbel the digrm with the given ngle nd the given side to find n unknown side in right-ngled tringle. 70 m m Hypotenuse hoose the pproprite trigonometric rtio, nmely the cosine rtio. 70 m djcent ngle = 70 djcent side = m Hypotenuse = m [H] Substitute into the formul. dj cos θ = ---------- Hyp cos 70 = -- 5 Isolte nd evlute. ---------------- = -- cos70 = ---------------- cos 70 = 5.876 6 Write the nswer using the correct units The length of the guy wire is 5.85 metres or nd level of ccurcy. 585 centimetres. Tngent rtio The tngent rtio is defined s follows: Length of opposite side The tngent of n ngle = --------------------------------------------------------. Length of djcent side Opposite In short, tn θ = ---------------------- djcent tn θ = Opp ---------- dj [TO] Opposite djcent θ
Further Mthemtics WORKED Emple 9 Find the length of the shdow (to deciml plce) cst by metre tll pole when the ngle of the sun to the horizontl is 70. Drw digrm to represent the sitution nd identify n pproprite tringle. m 70 Lbel the digrm with the given ngle nd the given side in order to find n unknown side in right-ngled tringle. Opposite m 5 6 Identify the pproprite trigonometric rtio, nmely the tngent rtio. ngle = 70 Opposite side = m djcent side = m Opp Substitute into the formul. tn θ = ---------- dj tn 70 = -- Isolte nd evlute. ---------------- = -- tn70 = ---------------- tn70 =.099 Write the nswer using the correct units nd level of ccurcy. 70 m djcent [TO] The length of the shdow is pproimtely. metres. Finding n unknown ngle If the lengths of the sides of tringle re known, unknown ngles within the tringle cn be found.
hpter 9 Trigonometry 5 WORKED Emple 0 Find the smllest ngle (to the nerest degree) in,, 5 Pythgoren tringle. The smllest ngle is opposite the smllest side. Lbel the sides s given by convention for trigonometric rtios. Identify the pproprite rtio from the given informtion. Opposite ngle = Opposite side = Hypotenuse = 5 [SOH] Substitute into the formul. Opp sin θ = ---------- Hyp sin = -- 5 onvert the rtio to deciml. sin = 0.6 5 Evlute : = sin (0.6). = 6.87 6 Write the nswer using the correct units nd level of ccurcy. The smllest ngle is pproimtely 7. 5 Hypotenuse In worked emple 0 the sine rtio ws used to find the unknown ngle. The sme processes would be pplied if either the cosine or tngent rtios were required insted. In the prticulr cse bove, ny of the three rtios could be used since ll the sides re known. remember remember. The trigonometric rtios re simply the rtio of one side of right-ngled tringle to nother.. The rtios re used to find n unknown: () side given nother side nd n ngle (b) ngle given the lengths of two sides.. To solve problem: () drw n pproprite right-ngled tringle (b) lbel the given sides with respect to the given ngle s hypotenuse, opposite or djcent (c) identify which trigonometric rtio is involved: SOH H TO helps to remember which combintion of sides re in ech of the three rtios (d) use the pproprite formul to solve for the unknown.
6 Further Mthemtics 9D Trigonometric rtios bri Geometry SkillSHEET Trig rtios 9. WORKED Emple 7 Find the length of the unknown side (to deciml plce) in ech of the following tringles. b 0 c.5 m km 50 0 mm SkillSHEET 9. d 9 000 mm e 5 f 6 5 cm y 9 mm WORKED Emple 8 WORKED Emple 9 bot is moored in clm wters with its depth sounder registering.5 m. If the nchor line mkes n ngle of 7 with the verticl, wht is the length of line (to the nerest metre) tht is out of the bot? person is hoping to swim directly cross stright river from point to point, distnce of 5 m. The river crries the swimmer downstrem so tht she ctully reches the other side t point. If the line of her swim,, mkes n ngle of 67 with the river bnk, find how fr (to the nerest metre) down strem from point she finished. Find the vlue of the missing side (to deciml plce) of the following tringles. b c 5 m 0 67. 65 d e 6. cm.9 0 m
hpter 9 Trigonometry 7 5 Find the vlue of the unknown sides (to deciml plce) of the following tringles nd shpes. b c 5 cm 0 65 0 cm 70 5 cm 0 d 0 e f 7 m 7 cm km 0 g 6.5 cm 5 5 WORKED Emple 0 6 Find the size of the unknown ngle (to the nerest degree) in ech of the following tringles. b c d 0 6 θ θ m m θ 500 mm 00 mm θ 7 Find the vlues of the unknown ngle, (to the nerest degree). b m. m 0 m m c d m. m m m
8 Further Mthemtics 8 Find the sizes of the two cute ngles in 6, 8, 0 Pythgoren tringle. 9 multiple choice If b m is the height reched by the ldder in the digrm t right, then b is equl to: 5.9.7 m 0.68 b D 0.9 E.88 70 0 multiple choice The correct epression for the ngle of elevtion, θ, of the rmp is: sin -- 5 cos -- 5 5 tn -- 5 D tn -- E cos -- 5 multiple choice The correct epression for the vlue of c in the figure below is: tn 7 ----------------- 5 m cos 7 ----------------- 7 m c 5 ----------------- tn 7 D E ----------------- tn 7 ---------------- sin 7 multiple choice flgpole metres tll csts 0.6-metre long shdow. The ngle of the sun to the ground is: 7 70 7 D 7 E 7 In the digrm below find θ (to the nerest degree), metres nd y metres (both to deciml plce). θ m θ 60 0 y
hpter 9 Trigonometry 9 Introduction Sine nd cosine rules Often the tringle tht is pprent or identified in given problem is non-right-ngled. Thus, Pythgors theorem or the trigonometric rtios re not s esily pplied. The two rules tht cn be used to solve such problems re:. the sine rule, nd. the cosine rule. For the sine nd cosine rules the following lbelling convention should be used. ngle is opposite side (t point ) ngle is opposite side b (t point ) ngle is opposite side c (t point ) Note: To void cluttered digrms, only the points (, nd ) re usully shown. In these instnces, the ngles, nd re ssumed. The sine rule ll tringles cn be divided into two right-ngled tringles. c b b b h c Erlier, we sw tht the new side, h, cn be evluted in two wys. b h h h h sin = -- sin = -- b h = b sin h = sin If we equte the two epressions for h: b sin = sin nd rerrnging the eqution, we obtin: b ----------- = ----------- sin sin Using similr pproch it cn be shown tht:. b c ------------ = ------------ = ------------ sin sin sin. Similrly, if the tringle is lbelled using other letters, for emple STU, then: s t u ----------- = ------------ = ------------ sin S sin T sin U The sine rule is used if you re given:. two ngles nd one side or. n ngle nd its opposite side length ( complete rtio) nd one other side. For emple, in tringle t = 7 cm right, = 7 cm, = 50 nd c = 9 cm. ngle could 50 then be found using the sine rule. c = 9 cm
0 Further Mthemtics Find the unknown length, cm in the tringle t right. 0 Drw the tringle. ssume it is nonright-ngled. 0 Lbel the tringle ppropritely for the 0 c = 7 cm sine rule. 0 b = onfirm tht it is the sine rule tht cn be b c ----------- = ----------- = ------------ used s you hve the ngle opposite to the sin sin sin side b = = 0 unknown side nd known ------------ rtio. ngle c = 7 cm = 0 Substitute known vlues into the two 7 ------------------ = ---------------- rtios. sin0 sin 0 5 Isolte nd evlute. 7 sin0 = ---------------------------- sin 0 = 0.76 = 0.7 Write the nswer ppropritely. The unknown length is 0.7 cm. 6 WORKED Emple 7 cm Sometimes it is necessry to find the third ngle in tringle in order to pply the sine rule. Find the unknown length, cm (to deciml plces). Drw the tringle. ssume it is non-right-ngled. Lbel the tringle ppropritely for the sine rule. c = 5 6 7 WORKED Emple lculte the third ngle becuse it is opposite the unknown side. onfirm tht it is the sine rule tht cn be used s you hve the ngle opposite the unknown side nd side known ------------ rtio. ngle b = 7 = 80 (65 + 00 ) = 5 b c ----------- = ----------- = ------------ sin sin sin c = = 5 b = 7 = 00 7 Substitute the known vlues into the two rtios. --------------- = ------------------ sin5 sin00 7 sin5 Isolte nd evlute. = ------------------------- sin00 =.897 Write the nswer ppropritely. The unknown length is.8 cm. 65 00 00 65 7 cm
hpter 9 Trigonometry WORKED Emple For tringle PQR, find the unknown ngle (to the nerest degree), P, given p = 5 cm, r = 7 cm nd R = 8. Drw the tringle nd ssume it is non-rightngled. Q 5 cm 7 cm 8 R P Lbel the tringle ppropritely for the sine rule Q (it is just s esy to use the given lbels). p = 5 r = 7 8 R P p q r onfirm tht it is the sine rule tht cn be used ----------- = ------------ = ----------- sin P sin Q sin R s you hve the side opposite to the unknown p = 5 P =? side ngle nd known ------------ rtio. r = 7 R = 8 ngle Substitute known vlues into the two rtios. 5 7 ----------- = ---------------- sin P sin 8 5 Isolte sin P. sin P sin 8 ----------- = ---------------- 5 7 sin 8 5 sin P = -------------------------- 7 6 Evlute the ngle (inverse sine) nd include units with the nswer. sin P = 0.508 P =.06 P P = sin sin 8 5 -------------------------- The unknown ngle is bout. 7 Sometimes the ngle clculted using the sine rule does not give the required ngle. In such cses simply subtrct the two known ngles from 80, s ws done in step of worked emple. WORKED Emple pir of compsses (often clled compss) used for drwing circles hs two equl legs joined t the top. The legs re 8 centimetres long. If it is opened to n included ngle of 6 degrees between the two legs, find the rdius of the circle tht would be drwn (to deciml plce). Drw the sitution nd identify tht the tringle is non-right-ngled. 6 8 cm ontinued over pge
Further Mthemtics Drw the tringle seprtely from the sitution nd lbel it ppropritely for the sine rule. This is n isosceles tringle nd since = c, then =. Using the fct tht the ngle sum of tringle is 80, find nd. onfirm tht it is the sine rule tht cn be used s you hve the ngle opposite to the unknown side nd known side ------------ rtio. ngle c = 8 cm 80 = + + 80 = + 6 + = 80 6 = = 7 nd therefore = = 7 Substitute the known vlues into the y --------------- = two rtios. sin6 5 Trnspose the eqution to get the 8 sin6 y = ------------------------- unknown by itself. sin7 6 Evlute y to deciml plce nd include units. 6 b = 8 cm b c ----------- = ---------- = ----------- sin sin sin b = y = 6 c = 8 = 7 8 --------------- sin7 y.9 The rdius of the circle is bout.9 cm. remember remember. Follow the pproprite lbelling convention.. For the tringle shown the sine rule sttes: c b c ----------- = ---------- = ----------- sin sin sin Note tht only two of the three rtios need be pplied.. The sine rule cn be used to find n unknown: side () side if its opposite ngle nd ------------ rtio re known ngle side (b) ngle if its opposite side nd ------------ rtio re known. b ngle. When two ngles re given, it my be necessry to clculte the third ngle in order to pply the sine rule. Tht is, if nd re the known ngles, then = 80 ( + ).
hpter 9 Trigonometry 9E The sine rule WORKED Emple Find the unknown length,, in ech of the following. 9 cm b c 0 0 5 m 85 7 mm Sine rule Mthcd 58 7 d e 50 km 8 55 cm 5 05 The reltive positions of the school, church nd post office in smll town re shown t the vertices of the tringle t right. School km hurch Find the stright-line distnce between the school nd the post office (to deciml plce). 86 Post Office WORKED Emple Find the unknown length,, (to deciml plce) in ech cse below. b 85 7 mm 5 m c 58 8 7 55 cm siling epedition followed course s shown t right. Find the totl distnce covered in the round trip. d 8 cm 9 0.5 km N 78 WORKED Emple 0 5 For the following questions give nswers to the nerest degree. In L, find the unknown ngle,, given b = 6, c = 6 nd = 5. b In LLMN, find the unknown ngle, M, given m =., n = 7. nd N = 8. c In LSTU, find the unknown ngle, S, given s =.7, t = 6. nd T = 5. d In LPQR, find the unknown ngle, P, given p =, r =.5 nd R = 8. e In L, find the unknown ngle,, given b = 0, c = 8 nd = 80. f In LPQR, find the unknown ngle, R, given p = 8, q = nd P = 0. 6 onstruct suitble tringle from the following instructions nd find ll unknown sides nd ngles. One of the sides is cm but the smllest is 5 cm. The smllest ngle is 8.
Further Mthemtics WORKED Emple 7 Steel trusses re used to support the roof of commercil building. The struts in the truss shown re ech mde from 0.8 m steel lengths nd re welded t the contct points with the upper nd lower sections of the truss. 0 0 0.8 m 0 On the lower section of the truss, wht is the distnce (to the nerest centimetre) between ech pir of consecutive welds? b Wht is the height (to the nerest centimetre) of the truss? 8 multiple choice The length of side m is nerest to:.. 8.5 D 5.8 E.0 5 5. 70 m 9 multiple choice The correct epression for the vlue of t in the given tringle is: 7sin00 ---------------------- sin 0 7 m 00 5.5 m 5.5sin00 -------------------------- sin0 0 50 t 5.5sin0 ----------------------- sin00 5.5sin00 D -------------------------- sin50 7sin50 E ------------------- sin00 0 multiple choice The vlue of (to deciml plce) in the given tringle is:..6 5. D. E.6 multiple choice In the tringle given, the lrgest ngle (to the nerest degree) is: 80 o 8 o 8 o D 67 o E 60 o 60 8 cm 70 60 6 cm 7 cm
hpter 9 Trigonometry 5 multiple choice ycht sils the three-leg course shown. The lrgest ngle between ny two legs within the course, to the nerest degree, is: o 55 o 5 o D 78 o E 90 o multiple choice The correct epression for ngle S in the given tringle is: sin 0sin ---------------------- 0 cos 0cos ----------------------- 0 S 0 0 sin 0sin ---------------------- 0 D sin sin0 ---------------------- 0 E sin 0 ---------------------- 0sin 5 km 5 km 8 km Find the perimeter of beehive comprtment shown. 0 mm
6 Further Mthemtics mbiguous cse of the sine rule Investigte, on your clcultor, the vlues for ech of the given pirs of sine rtios:. sin 0 nd sin 50. sin 0 nd sin 70. You should obtin the sme number for ech vlue in pir. Similrly, sin 60 nd sin 0 give n identicl vlue of 0.8660. Now try to find the inverse sine of these vlues; for emple, sin (0.8660) is 60. The obtuse (greter thn 90 ) ngle is not given by the clcultor. When using the inverse sine function on your clcultor, the clcultor will give only the cute ngle. The sitution is illustrted prcticlly in the digrm t right where the sine of the cute ngle equls the sine of the obtuse ngle. Obtuse cute The rope rope ttched cn be to pole, t left, cn be nchored in two nchored in two possible positions. Therefore lwys check your digrm to see if the unknown ngle to be found is the cute or obtuse ngle or perhps either. This sitution is illustrted in the two digrms below. The tringles hve two corresponding sides equl, nd b, s well s ngle. The sine of 0 lso equls the sine of 70 ; however, the side c is quite different. It is worth noting tht this mbiguity occurs when the smller known side is opposite the known ngle. WORKED Emple b b 0 70 c c 5 To the nerest degree find the ngle, U, in tringle, given t = 7, u = nd ngle T is 5. Drw suitble sketch of the tringle given. s the length of s is not given, side t cn be drwn two different wys. Therefore ngle U could be either n cute or n obtuse ngle. Lbel the tringles ppropritely for the sine rule. (It is just s esy to use the given lbels.) Identify tht it is the sine rule tht cn be used s you hve the side opposite to the side unknown ngle nd known ------------ rtio. ngle Substitute the known vlues into the two rtios. Trnspose the eqution to get the unknown by itself. 7 T u = U 5 s U s t u ---------- = ---------- = ----------- sins sint sinu t = 7 T = 5 u = U =? --------------- = ----------- sin5 sinu sin sin5 ----------- = --------------- 7 sin5 sin U = ---------------------------- 7 S t = 7 S u = t = 7 T 5 s U
hpter 9 Trigonometry 7 5 Evlute the ngle (inverse sine). Note tht the vlue is n cute ngle but it sin U = 0.7 88 U = 6. my well be n obtuse ngle. 6 lculte the obtuse ngle. U = 80 6. =.57 7 Write the nswer, giving both the cute nd obtuse ngles, s not enough informtion ws given (the informtion ws mbiguous) to precisely position side t. The ngle U is either 6 or. WORKED Emple 6 In the obtuse-ngled tringle PQR, find the unknown ngle (to the nerest degree), P. Lbel the tringle ppropritely for the Q sine rule. (It is just s esy to use the given lbels.) p = 0 r = 0 Q 0 cm 0 cm 0 R P R 0 P Identify tht the sine rule is used s you p q r ---------- = ----------- = ---------- hve the side opposite to the unknown sinp sinq sinr side p = 0 P =? ngle nd known ------------ rtio. ngle r = 0 R = 0 Substitute the known vlues into the 0 0 ---------- = --------------- two rtios. sinp sin0 Trnspose the eqution to get the sinp sin0 ---------- = --------------- unknown by itself. 0 0 sin0 0 sin P = ---------------------------- 0 5 Evlute the ngle (inverse sine). Note sin P = 0.968 tht the vlue is n cute ngle while in P = 7.6 the digrm given it is n obtuse ngle. 6 lculte the obtuse ngle. P = 80 7.6 = 05.8 P 05 remember remember If the unknown ngle is n obtuse ngle, remember the following:. the inverse sine function on clcultors evlutes only the cute ngle. for the obtuse ngle, evlute s follows: obtuse ngle = 80 cute ngle. the mbiguous cse of the sine rule occurs when the smller known side is opposite the known ngle.
8 Further Mthemtics 9F mbiguous cse of the sine rule WORKED Emple 5 WORKED Emple 6 Find both the cute nd obtuse ngles in ech cse below. Epress ll nswers in degrees to deciml plce. In L, find the unknown ngle,, given b = 0.8, c = 6 nd = 6. b In LSTU, find the unknown ngle, S, given t =.7, s = 6. nd T = 5. c In LPQR, find the unknown ngle, P, given p =.5, r = nd R =. d In LLMN, find the unknown ngle, M, given n = 0. km, m = 0.5 km nd N = 8. Find the unknown ngle (to the nerest degree) in ech of the following obtuse-ngled tringles. b c d 60 km m 0.5 m 0 km 5.8 m m 7 m 0 0 7 m 5 multiple choice In the tringle given, ngle is (to the nerest degree): 8 9.5 cm 78 9 D 8 cm E Find the two unknown ngles shown in the digrm below. 0 cm 9 cm 9 cm 7 y 5 Look t the swinging pendulum shown t right. Drw the two possible positions of the bob t the level of the horizontl line. b Find the vlue of the ngle, W, t these two etreme positions. c Find the smllest nd lrgest distnces between verte V nd the bob. V 8 cm 5 W 5 cm
hpter 9 Trigonometry 9 The cosine rule The cosine rule is derived from non-right-ngled tringle divided into two rightngled tringles in similr wy to the derivtion of the sine rule. The difference is tht, in this cse, Pythgors theorem nd the cosine rtio re used to develop it. The tringle in the figure below hs been divided into two right-ngled tringles with bse sides equl to nd (c ). b h c c D In LD, h = b nd in LD, h = (c ) (Pythgors theorem) Equting epressions for h, b = (c ) = b + (c ) = b + c c + = b + c c [] Now, from LD, cos = -- b = b cos Substitute this vlue of into [] bove. = b + c c(b cos ) So, the cosine rule cn be written s: = b + c bc cos b c In similr wy to tht bove, it cn be shown tht: b = + c c cos c = + b b cos lso, if the tringle is lbelled using other letters, for emple STU, then: s = t + u tu cos S The cosine rule is used to find:. n unknown length when you hve the lengths of two sides nd the ngle in between. n unknown ngle when you hve the lengths of ll three sides. The formul my be trnsposed in order to find n unknown ngle. b + c cos = --------------------------- bc + c b + b c or lterntively, cos = --------------------------- nd cos = ---------------------------. c b
0 Further Mthemtics Find the unknown length (to deciml plces),, in the tringle t right. Identify the tringle s non-right-ngled. Lbel the tringle ppropritely for the sine rule or cosine rule. c = 7 = 80 b = 6 WORKED Emple 7 Identify tht it is the cosine rule tht is required s you hve the two sides nd the ngle in between. Substitute the known vlues into the cosine rule formul. b = 6 = 80 c = 7 = = b + c bc cos = 6 + 7 6 7 cos 80 = 6 + 9 8 cos 80 = 70.6 5 Remember to get the squre root vlue,. = 70.6 = 8.9 6 Evlute the length, nd include units with the nswer. = 8.9 The unknown length is 8.9 cm. 7 cm 80 6 cm Find the size of ngle in the tringle t right, to the nerest degree. Identify the tringle s non-right-ngled. Lbel the tringle ppropritely for the sine rule or cosine rule. c = 6 b = 6 5 6 WORKED Emple 8 Identify tht it is the cosine rule tht is used s ll three sides re given. Substitute the known vlues into the rerrnged form of the cosine rule nd simplify. = =, b = 6, c = 6, = + c b cos = --------------------------- c + 6 6 cos = --------------------------- 6 6 cos = ----- 8 cos = 0. Evlute ( = cos (0.)). = 70.5 7 Evlute the ngle nd include units ngle is pproimtely 7. with the nswer. 6 6
hpter 9 Trigonometry remember remember. Follow the pproprite lbelling convention.. The cosine rule cn be used to find n unknown: () length, if the other two sides nd the ngle in between them re known. = b + c bc cos (b) ngle, if ll three sides re known. b + c cos = --------------------------- bc c b 9G The cosine rule WORKED Emple 7 Find the unknown length in ech of the following (to deciml plces). b c 5 0 m 55. km.5 km z osine rule Mthcd 60 5 m d 0 6 e f 0 00 00 km 7 000 mm 000 mm 00 km During siling rce, the bots followed course s shown below. Find the length,, of its third leg (to deciml plce). 0 km 07 7 km
Further Mthemtics Two circles, with rdii 5 cm nd 8 cm, overlp slightly s shown t right. If the ngle between the two rdii tht meet t the point of intersection of the circumferences is 05, find the distnce between the centres of the circles (to deciml plce). 5 cm 8 cm 05 WORKED Emple 8 Find the size of the unknown ngle in ech of the following (to the nerest degree). b mm c d y 85 km 8 m 5 m mm 0.5 cm 9. cm p 0 km 0 mm 8.6 cm 6 m 68 km 5 onsider the siling epedition course in question. Find the two unknown ngles (to the nerest degree) in the tringulr course. 6 onsider the overlpping circles in question. Find the two ngles formed between the line joining the centres of the circles nd ech of the rdii drwn (to the nerest degree). 7 For the tringle shown, find ll three unknown ngles (to the nerest degree). 8 For the following questions, find nswers to deciml plce. For L, find the unknown side, b, given = 0 km, c = 8 km nd = 0. b For L, find the unknown ngle,, given = b = 0 nd c = 6. c For L, find the unknown side, c, given = 7 m, b = m nd = 80. d For LSTU, find the unknown ngle, S, given t =.7, s = 6. nd u =.5. e For LPQR, find the unknown ngle, P, given p =, q =.5 nd r =.5. f For L, find the unknown side,, given b = 60, c = 0 nd = 5. 9 onstruct suitble tringle from the following instructions nd find ll unknown sides nd ngles. Two sides re cm nd 5 cm nd the ngle in between is 8. 9 0 multiple choice The vlue of (to deciml plce) in the digrm t right is:.5.6 8. D 8.5 E none of the bove 0 mm 60 50 mm multiple choice The length of side m t right is nerest to: 0 6. 6.5 D.6 E 50 0 m 60 0
hpter 9 Trigonometry multiple choice In the tringle given, the lrgest ngle is: 9 5 56 D 85 E cm 0 cm 5 cm Not to scle multiple choice The correct epression for ngle s is: cos 6 + 5 --------------------------- 6 cos + 5 6 --------------------------- 5 5 cm cos 6 + 5 --------------------------- 6 cm 6 cm s D cos + 6 5 --------------------------- 5 E cos 5 + 6 --------------------------- 5 6 multiple choice The correct epression for the vlue of t is: 80 + cos 0 80 0 0 6 D 80 0.5 80 7 t E 80 + 7 5 multiple choice The surfce 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: 8 D 9 E 7 5 cm Regulr squre pyrmid 0 m 6 Find the unknown vlues. c cm cm 8 m 6 cm b m m m 00 WorkSHEET 9.
Further Mthemtics Specil tringles Often, the tringles encountered in problem solving re either equilterl or rightngled isosceles tringles. They ehibit some unique fetures tht, when recognised, cn be very useful in solving problems. 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. = b = c = = = = 60 60 60 Right-ngled isosceles tringles hve one right ngle (90 ) opposite the longest side (hypotenuse) nd two equl sides nd ngles. The two other ngles re lwys 5. 5 = b = c = 5 = 60 60 b = = c = = = = 60 0 0 5 5 = c = b = = = 5 = 90 5 = c = 0 b = 0 = = 5 = 90 = c = 5 b = 5 = = 5 = 90 lso, the hypotenuse is lwys times the length of the smller sides. heck for yourself using Pythgors theorem. 0 WORKED Emple 9 Find the vlues of r nd ngle θ in the hegon t right. Tringles in regulr hegon re ll identicl. The si ngles t the centre re equl. The mgnitude of ech is 6 cm 60 one revolution divided by 6. θ = 60 6 = 60 θ = 60 5 b = 0 0 = c = = = 5 = 90 6 cm Regulr hegon r cm θ
hpter 9 Trigonometry 5 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. r = 6 cm WORKED Emple 0 Find the vlue of the pronumerl (to deciml plce) in the figure. The tringle is right-ngled isosceles tringle. Two ngles re 5 nd the third ngle is 90. Two sides re equl nd the longer side opposite the right ngle is times bigger thn these equl sides. Write your nswer using the correct ccurcy nd units. cm 5 5 cm c = = = 6.970 56 The vlue of is 7.0 cm. cm 5 remember remember. Equilterl tringles hve three equl sides nd three equl ngles. Ech ngle equls 60.. Right-ngled isosceles tringles hve one right ngle (90 ), opposite the longest side (hypotenuse), nd two equl sides nd ngles. The two other ngles re lwys 5. The hypotenuse is lwys times the length of the smller sides. = b = c = = = = 60 = c = b = = = 5 = 90
6 Further Mthemtics 9H Specil tringles WORKED Emple 9 Find the unknown(s) in ech of the following. b c 00 cm 60 5 b 60 60 5. cm WORKED Emple 0 Find the unknowns in ech of the following. b c 5 m 58 cm y 7. m 0 mm nswer the following. In L, find the unknown ngle,, given b = 0, c =0 nd = 90. b In LSTU, find the unknown side, s, given t =.7, S = 5 nd T = 5. c In LPQR, find the unknown ngle, P, given p =.5, r =.5 nd R = 60. d In LLMN, find the unknown side, m, given n = 0., L = 60 nd N = 60. compss used for drwing circles hs legs tht re 6 cm long. If it is opened s shown in the digrm, wht is the rdius of the circle tht could be drwn? 60 5 Wht is the height of tree if its shdow, on horizontl ground, is metres long when the sun s rys striking the tree re t 5 to the ground? 6 multiple choice In the tringle given, side in metres is: 0 0 0 0 m D E 0 0 7 0 cm squre serviette is prepred for presenttion by completing three folds firstly, by tking corner nd plcing it on top of the opposite corner; secondly, by tking one of the two corners on the crese tht hs been mde nd plcing it on the other one; nd finlly, by plcing the two corners t the ends of the longest side on top of ech other. Find the length of the crese mde fter the i first fold ii second fold iii third fold. b With the finl serviette lying flt, wht ngles re produced t the corners?
hpter 9 Trigonometry 7 re of tringles Three possible methods might be used to find the re of tringle: Method. When the two known lengths re perpendiculr to ech other we would use: re tringle = -- se Height = -- bh cm Height Height cm se se Method. When we re given two lengths nd the ngle in between we would use: re tringle = -- b sin b = 0 m = -- = 5 m b sin = b sin Method. When ll three sides re known we would use: + b + c re tringle = ss ( ) ( s b) ( s c) where the semi-perimeter, s = --------------------. This formul is known s Heron s formul. It ws developed by Heron (or Hero) of lendri, Greek mthemticin nd engineer who lived round D 6. Let us find the re of the tringle below to demonstrte tht ll three formuls provide the sme result. 5 For the,, 5 tringle, the most pproprite method is method bove becuse it is right-ngled tringle. re tringle = -- se Height = -- = 6 The other two methods my lso be used. re tringle = -- b sin = se se Height = -- sin 90 = 6 = 6 re tringle = ss ( ) ( s b) ( s c) + b+ c s = -------------------- = 66 ( ) ( 6 ) ( 6 5) + + 5 s = -------------------- = 6 s = ----- = 6 s = 6 = 6 re = b Height = b sin
8 Further Mthemtics Find the re of the tringle t right (to deciml plces). Identify the shpe s tringle with two known sides nd the ngle in between. b = 9 7 = 6 WORKED Emple Identify nd write down the vlues of the two sides, nd b, nd the ngle in between them,. Identify the pproprite formul nd substitute the known vlues into it. = 6 b = 9 = 7 re tringle = -- b sin = -- 6 9 sin 7 = 6.9 Write the nswer in correct units. The re of the tringle is 6.5 m. 9 m 7 6 m Find the re of tringle PQR (to deciml plce), given p = 6, q = 9 nd r =, with mesurements in centimetres. onfirm tht ll three sides of the Q tringle hve been given nd therefore Heron s formul is to be used. p = 6 r = R q = 9 P Write the vlues of the three sides,, b = p = 6, b = q = 9, c = r = nd c, nd clculte the semi-perimeter + b+ c vlue, s. s = -------------------- 6+ 9+ s = -------------------- s = 9.5 Substitute the known vlues into Heron s formul. re tringle = = ss ( ) ( s b) ( s c) 9.5( 9.5 6) ( 9.5 9) ( 9.5 ) WORKED Emple = 9.75 re = 9.56 Write the nswer, using the correct units. The re of tringle PQR is 9.6 cm. e = 9.5.5 0.5 5.5
hpter 9 Trigonometry 9 Find the re of the tringle t right. onfirm tht the two given lengths re perpendiculr. Substitute the known vlues into the formul. WORKED Emple re tringle = -- se Height = -- 8 = 8 Write the nswer using correct units. The re of the tringle is 8 mm. remember remember 8 mm mm. Three possible methods re vilble for finding the re of tringle: () When the two known lengths re perpendiculr to ech other we would use: re tringle = -- se Height (b) When we re given two lengths nd the ngle in between we would use: re tringle = -- b sin (c) When ll three sides re known we would use: re tringle = ss ( ) ( s b) ( s c) where the semi-perimeter, + b+ c s = --------------------. lwys use the most efficient method to find the re of tringle. 9I re of tringles WORKED Emple Find the res of the following tringles (to deciml plce). 0 b m c d 00 m 7 cm 7 cm 80 0 0. m 80 m m 05 7.5 m re of tringle Mthcd WORKED Emple Find the res of the following tringles (to deciml plce). b c d km 0 mm 8 m 8 m 6 km 5. cm 6 m km. cm 6.7 cm
50 Further Mthemtics WORKED Emple Find the res of the following tringles (to deciml plce). b c d. mm.5 mm 7 cm 0.5 mm 7.0 mm cm m m 5 m Find the res of the following tringles (to deciml plce). b c.5 km d.7 m. m 0 cm 60 50 m 0 75 m 70 70. km 0 km 5 Find the re of ech of the following tringles. (Give ll nswers to deciml plce.) For L, given = 0 km, c = 8 km nd = 0 b For L, given = b = 0 cm nd c = 6 cm c For L, given = 7 m, b = m, c = 8. m nd = 08 d For LSTU, given t =.7 m, s = 6. m nd u =.5 m e For LPQR, given p = units, q =.5 units nd r =.5 units f For L, given b = 60 cm, c = 0 cm nd = 90 6 Find the re of n equilterl tringle with side lengths of 0 cm. 7 tringulr rch hs supporting legs of equl length of metres s shown in the digrm. Wht is its re? 5 5 m m 8 From the digrm given, find the re of: i one of the tringles ii ll of the tringles b use nother technique to verify your nswer in i. 0 mm 0 mm 9 Find the re of the stte forest s defined by the three fire-spotting towers on the corners of its boundry. km 5. km 0. km 0 multiple choice If the perimeter of n equilterl tringle is 0 metres, its re is closest to: 00 m 50 m 800 m D 5500 m E 700 m
hpter 9 Trigonometry 5 multiple choice The correct epression for the re of the shpe t right is: -- 6 sin 80 -- 6 cos 00 -- 6 sin 00 D -- 6 E none of the bove 6 m 0 m 50 multiple choice The correct epression for the re of the octgon shown is: 95 sin 5 69 sin 5 95 sin 60 D 8 sin 60 E 5 6.5 sin 67.5 Find the re of the following tringles. 7 km 5 b 0 5 mm 5 6.5 Problem solving to find n re is the most common size for sheet of pper used in photocopy mchines nd computer printers. nd 5 sheets of pper re geometriclly relted to the sheet s shown below (see previous chpter). 5 One common property is tht when the sheet of pper is folded by joining the two digonlly opposite corners common shpe is obtined. s with mny other common shpes, such s rectngles ( = L W), generl epression for its re cn be formulted. Folded sheet Your tsk is to find generl epression for the re of the unique shpe bove, in terms of length, L, nd width, W, of the sheet. Like ll good problem-solving tsks there re mny different pproches (t lest si known to dte) to this problem.
5 Further Mthemtics summry Right-ngled tringles Pythgors theorem c = + b or c = Pythgoren trids Pythgoren trid is set of three numbers, which stisfies Pythgors theorem. Some common trids re (),, 5 (b) 6, 8, 0 (c) 5,, nd (d) 7,, 5. Three-dimensionl Pythgors theorem To solve problems involving three-dimensionl Pythgors theorem: () Drw nd lbel n pproprite digrm. (b) Identify the right ngles. (c) Identify right-ngled tringles tht enble the informtion given to be used to find the unknown vlue(s). Trigonometric rtios + b Hypotenuse Opposite θ djcent Opp dj Opp sin θ = --------- cos θ = --------- tn θ = --------- or SOH H TO Hyp Hyp dj Non-right-ngled tringles The sine rule b c ----------- = ----------- = ------------ sin sin sin c b The sine rule is used when:. two ngles nd one side re given. two sides nd non-included ngle re given. If two ngles re given, simply clculte the third ngle, if needed, using: = 80 ( + )
hpter 9 Trigonometry 5 mbiguous cse of the sine rule The sine rule is mbiguous when finding n ngle when the smller known side is opposite the known ngle. The cosine rule = b + c b + c bc cos or cos = --------------------------- bc To clculte: () sides, use the cosine rule when two sides nd the included ngle re given (b) ngles, use the cosine rule when ll three sides re given. Specil tringles 60 5 c = 60 60 Equilterl tringles 5 Right-ngled isosceles tringles re of tringles To find the re of tringle: () given perpendiculr dimensions, use re tringle = -- se Height (b) given two sides nd the included ngle, use re tringle = -- b sin (c) given ll three sides only, use re tringle = + b+ c where s = -------------------- ss ( ) ( s b) ( s c)
5 Further Mthemtics HPTER review 9 9 9 D 9D 9D 9E Multiple choice For the tringle shown, the vlue of is: 0 0 80 D 6.7 5 E 0 Which one of the following is not Pythgoren trid? 9, 9, 0,, 5 0., 0., 0.5 D 6, 8, 0 E 7,, 5 5-cm-long strw is the longest tht cn fit into cylindricl cn with rdius of 6 cm. The height of the cn, in centimetres, is closest to: 8 9 5 D 6 E 7 rectngulr bo hs rod positioned s shown in the digrm. The epression tht would enble the ngle the rod mkes with the bse of the bo to be found, is: tn θ = sin θ = tn θ = D tn θ = E cos θ = 5 stepldder is erected s shown. How fr prt (to deciml plces) t the bse re the two legs?.5 m. m m.50 m D.00 m E 6.97 m? m 6 Given ST = cm, TU = 6 cm nd sin U = --, then sin S equls: ----- 6 6 ----- 6 ----- 9 -- D E 5 ----- ----- ----- ----- 5 ----- 5 cm S T U 6 cm
hpter 9 Trigonometry 55 7 Find the vlue of the pronumerl below, to the nerest metre. 5 50 m 50 0 D 6 5 E 68 8 In tringle where b = 0, c = 0 nd = 6, (to the nerest degree) could be: 6 6 6 D 6 or 7 E 6 or 9 9 To find the distnce cross lrge ecvtion, mesurements were found s shown in the digrm. The distnce,, cross the ecvtion is closest to: 75 metres 7 metres 00 metres 0 m D 0 metres 0 m 5 E none of the bove 0 regulr hegon is inscribed in circle of rdius cm. The perimeter of the hegon, in centimetres, is: π r = cm 6 r D 7 E 8 right-ngled isosceles tringle hs longest side of metres. The other two equl sides hve vlue closest to: 00 m 00 m 50 m D 0 m E none of the bove The re of tringle XYZ (to the nerest m ) is: 55 X 0 70 5 7 m D 85 E 65 The re of the tringle below is closest to: 96 cm 97 cm 98 cm D 99 cm E 00 cm 50 0 0 cm Y 0 m Z 9E 9F 9G 9H 9H 9I 9I
56 Further Mthemtics Short nswer 9 9 9 9 9D 9D.5-m-long ldder is plced up ginst wll nd reches to height of. m. Find the distnce tht the legs of the ldder re from the bse of the wll. 90-mm-squre cermic floor tile is to be cut digonlly. Wht is the ect length of the cut to be mde? bot sils directly northwrds for km before turning towrds the est nd siling 60 km. t this point the bot is 6 km from the strting point. On the second leg of the trip, did the bot sil directly estwrds? cuboid with 8-cm sides is internlly brced. Wht is the length of the longest brce tht could be plced inside the cuboid? (epress in surd form.) 5 stircse is to rise by 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 bse of the stircse (to the nerest mm)? b Wht is the length of the stircse (to the nerest mm)? 6 opy nd complete the following tble using the two tringles given. Give ech nswer s both frction nd deciml. ngle 0 5 60 sin cos ------ = 0.8660 0 60 5 5 tn 9D 9E 9F 7 cr bdge to be fitted on bonnet is of n isosceles tringle design. If the height of the bdge is not to be more thn 0 mm, wht is the mimum length of the bse of the bdge (to the nerest mm), if the equl ngles re 5? b If the longest side is to be set t 00 mm, wht is the length of the other two equl sides, if the two equl ngles re still 5? 8 hot-ir blloon is nchored to the ground t points nd D s shown in the digrm. 5-metre length of ecess rope is dropped to the ground from the blloon. It is then tied to the ground, t point, s further sfety mesure. b 0 m 5 5 m D Wht re the smllest nd lrgest ngles tht cn be mde t point by the two lengths of rope, nd (to the nerest degree)? Using these vlues, determine the furthest nd closest positions (to deciml plce) possible for point from point.
hpter 9 Trigonometry 57 9 The hour hnd of clock is 0 mm long nd the minute hnd is 5 mm. The clock fce t right shows the time t o clock. Wht is the distnce between the tips of both hnds (to the nerest mm)? 5 mm 0 0 mm 9G 0 n undercover ptio is covered with sil s shown in the digrm. Wht re the ngles mde by the sil t ech of the support poles (to the nerest degree)? b t which support pole is the smllest ngle? 5 m 8. m 6. m 9G The infmous ermud Tringle is represented t right. Wht is the distnce between the western nd northern corners of the tringle (to the nerest kilometre)? b Wht is the lrgest ngle within the tringle (to the nerest degree)? US 0 km 0 90 km N 9G D storge unit is.5 metres tll nd hs bse re s shown. Find the front width of the storge unit (to the nerest cm). b Find the volume of the storge unit (in cm ). cm cm 9H 9I Wht is the re of Give Wy trffic sign (to the nerest cm ), which is in the shpe of n equilterl tringle whose side lengths re 5 cm? Wht is the re of the bdge described in question 7 b (to the nerest mm )? 9I 9I nlysis sndwich br uses bred tht is roughly 0 cm squre. The bred slices re cut into four equl tringles nd pckged in crdbord bo with the tringles rrnged s shown. 0 cm 0 cm 8 cm