4Measurement and geometry. Trigonometry

Transcription

1 4Mesurement nd geometry In the seond entury BCE, the Greek stronomer Hipprhus ould lulte distnes to the Moon nd the Sun nd he ws the first sientist to hrt the positions of over 850 strs. How ws he le to hieve this over 2100 yers go? Hipprhus strted new rnh of mthemtis lled trigonometry, mening tringle mesure, whih uses ngles, tringles nd irles to lulte lengths nd distnes tht nnot e physilly mesured. is used widely tody in engineering, surveying, nvigtion, stronomy, eletronis nd onstrution.

2 NEW CENTURY MATHS for the Austrlin Curriulum9 Shutterstok.om/trvellight n Chpter outline Profiieny strnds 4-01 The sides of right-ngled tringle U C 4-02 The trigonometri rtios U C 4-03 Similr right-ngled tringles U R C 4-04 on lultor U F 4-05 Finding n unknown side U F PS 4-06 Finding more unknown sides U F PS 4-07 Finding n unknown ngle U F PS n Wordnk djent side In right-ngled tringle, the side tht is next to given ngle nd pointing to the right ngle hypotenuse The longest side of right-ngled tringle, the side opposite the right ngle minute ( 0 1 ) A unit for mesuring ngle size, of degree 60 opposite side In right-ngled tringle, the side tht is fing given ngle thet (y) A letter of the Greek lphet used s pronumerl for ngles trigonometri rtio The rtio of two sides in right ngled tringle, for exmple, sine is the rtio of the opposite side to the hypotenuse

3 Chpter n In this hpter you will: use similrity to investigte the onstny of the sine, osine nd tngent rtios for given ngle in right-ngled tringles pply trigonometry to solve right-ngled tringle prolems find unknown sides nd ngles in right-ngled tringles where the ngle is mesured in degrees (STAGE 5.2) find unknown sides nd ngles in right-ngled tringles where the ngle is mesured in degrees nd minutes SkillChek Worksheet StrtUp ssignment 4 MAT09MGWK Simplify eh frtion Convert eh frtion to deiml, orret to three deiml ples For eh tringle, nme the hypotenuse. A B C x w y T 6 P 8 10 Q 4 Solve eh eqution. x 5 ¼ 7 h 4 ¼ 8:3 45 y ¼ 9 5 For eh tringle, find the vlue of n n 65 n 5.2 n Round eh time to the nerest hour. 8 h 18 min 3 h 45 min 1 h 30 min 7 Convert eh time to hours nd minutes. 4.7 h 2.25 h 6.85 h

4 NEW CENTURY MATHS for the Austrlin Curriulum The sides of right-ngled tringle The three sides of right-ngled tringle hve speil nmes. These nmes depend on the positions of the sides reltive to given ngle. The hypotenuse is the longest side nd is lwys opposite the right ngle The opposite side diretly fes the given ngle The djent side runs from the given ngle to the right ngle We hve lredy lernt out the hypotenuse in Pythgors theorem. Adjent mens next to. In the digrm elow, the mrked \O hs lso een lelled with the Greek letter, y, ( thet ). The hypotenuse is OP. The opposite side is XP. The djent side is OX. O hypotenuse djent P opposite X If the mrked ngle is \P, lso lelled with the Greek letter ( lph ), then the opposite nd djent sides re swpped, ut the hypotenuse stys the sme. The hypotenuse is OP. The opposite side is OX. The djent side is XP. O hypotenuse opposite α P djent X Exmple 1 For eh tringle, nme the hypotenuse, opposite nd djent sides for ngle y. r G 17 8 q p E 15 F Solution Hypotenuse is 17. Opposite side is 8. Adjent side is 15. Hypotenuse is p Opposite side is r. Adjent side is q. Hypotenuse is EF. Opposite side is EG. Adjent side is FG

5 Chpter Exmple 2 For ngles nd, nme the djent side Solution For, the djent side is 7. For, the djent side is 24. α 25 β Exerise 4-01 The sides of right-ngled tringle See Exmple 1 1 For the mrked ngle in eh tringle, nme the hypotenuse, opposite nd djent sides. C u 13 5 w v 12 A B d T e y x z f S R 9 2 For nlkm, nme the ngle: opposite the hypotenuse opposite side KM opposite side LK d djent to side KM e djent to side LK. M L K 3 Whih one of these sttements is flse out ntsu? A The djent side to \S is US. B The djent side to \T is TU. C The hypotenuse is UT. D The opposite side to \T is US. U T S

6 NEW CENTURY MATHS for the Austrlin Curriulum9 4 For eh tringle, find the opposite sides for ngles y ( thet ) nd f ( phi ). D 12 d φ 9 e 15 φ E F f φ See Exmple 2 5 For eh tringle, find the djent sides for ngles y ( thet ) nd f ( phi ). H φ 36 φ I φ J 6 Whih side of right-ngled tringle is fixed nd does not depend on given ngle? Selet the orret nswer A, B, C or D. A djent B hypotenuse C opposite D shortest 7 Given eh desription of right-ngled tringle, sketh the tringle with the orretly-lelled verties nd ngle. nabc hs hypotenuse AB nd side AC opposite ngle y nxyz hs hypotenuse YZ nd side XZ djent to ngle nprq hs side RQ opposite \P nd djent to \R d ndef is right-ngled t E, with the opposite nd djent sides of \D equl Just for the reord The Greek lph-et Here re eight letters (in lower-se nd pitls) from the Greek lphet:, Alph, Bet g, C gmm d, D delt u, H thet p, P pi r, R sigm x, X omeg The nient Greeks gretly influened the development of mthemtis. It is trditionl to use Greek letters s vriles, prtiulrly in geometry nd trigonometry. 1 Find out how mny letters there re in the Greek lphet, nd nme eh one. 2 Compre the Greek lphet with our Romn lphet. 3 Cn you see where the word lphet omes from? Explin how it originted

7 Chpter Puzzle sheet mth-up MAT09MGPS10040 Tehnology GeoGer: rtios MAT09MGTC The trigonometri rtios There re three speil frtions lled trigonometri rtios tht relte the lengths of two sides of right-ngled tringle: sine, osine nd tngent. Summry Tehnology worksheet Exel worksheet: vlues MAT09MGCT00026 Tehnology worksheet Exel spredsheet: vlues MAT09MGCT00011 The trigonometri rtios Rtio Arevition Mening sine sin sin u ¼ opposite hypotenuse osine os os u ¼ djent hypotenuse tngent tn tn u ¼ opposite djent Exmple 3 In naxp, find sin y, os y nd tn y. Solution A 5 13 For ngle y, opposite ¼ 12, djent ¼ 5, hypotenuse ¼ 13. sin u ¼ opposite hypotenuse ¼ 12 os u ¼ djent 13 hypotenuse ¼ 5 13 X 12 P tn u ¼ opposite djent ¼ 12 5 Mnemonis for sin, os nd tn A useful mnemoni (memory id) for rememering the three rtios is to look t the initils of the words in the rtios: sin ¼ opposite hypotenuse ¼ O djent S.O.H. os ¼ H hypotenuse ¼ A H C.A.H. tn ¼ opposite djent ¼ O A T.O.A. If you rememer SOH-CAH-TOA (pronouned so-r-toe-h ), then you n rememer the rtios for sin, os nd tn. Some students lso lern phrse where the first letter of eh word follows the SOH-CAH-TOA sequene, for exmple, Sun Over Hed Cused AHuge Tn On Arms. Find your own mnemoni for the three rtios

8 NEW CENTURY MATHS for the Austrlin Curriulum9 Exmple 4 For the tringle elow, find: sin A os B tn A d sin B 21 C 20 Video tutoril The trigonometri rtios MAT09MGVT10007 Solution For ngle A, opposite ¼ 20, djent ¼ 21, hypotenuse ¼ 29 For ngle B, opposite ¼ 21, djent ¼ 20, hypotenuse ¼ 29 sin A ¼ opposite hypotenuse ¼ tn A ¼ opposite djent ¼ A os B ¼ d sin B ¼ 29 djent hypotenuse ¼ opposite hypotenuse ¼ B Given one rtio, finding nother rtio Exmple 5 If tn R ¼ 8, find the vlue of sin R nd os R. 15 Solution tn R ¼ opposite djent ¼ 8, so drw rightngled tringle tht hs n ngle R with 15 opposite side 8 nd djent side 15. Let x e the length of the hypotenuse. Find x using Pythgors theorem. R x hypotenuse 15 djent 8 opposite x 2 ¼ 8 2 þ 15 2 ¼ 289 p x ¼ ffiffiffiffiffiffiffi 289 ¼ 17 ) sin R ¼ opposite hypotenuse ¼ 8 17 os R ¼ djent hypotenuse ¼

9 Chpter Exerise 4-02 The trigonometri rtios See Exmple 3 1 For eh mrked ngle, find the sine, osine nd tngent rtios. e R g f S 55 d e 7.7 f m M α n k W Y T X See Exmple 4 2 For the tringle elow, find: os X tn Y sin X d sin Y 3 Y 5 4 X 3 Complete eh sttement elow with the orret ngle ( or ). H sin ¼ IJ HJ d os ¼ HI HJ sin ¼ HI HJ e tn ¼ IJ IH f os ¼ IJ HJ tn ¼ IH IJ J β α I 4 For eh tringle elow, find: i tn y ii os y iii os f iv tn f 7 F u v φ φ w H φ f is the Greek letter phi G d 84 φ e φ f Q φ S R

10 NEW CENTURY MATHS for the Austrlin Curriulum9 5 For eh frtion, write orret trigonometri rtio involving ngle X or Y in the tringle d X Y Z 11 6 Whih rtio is equl to p? Selet the orret m nswer A, B, C or D. A os y B os f C tn y D tn f p φ m n 7 Whih sttement is true for this tringle? Selet the orret nswer A, B, C or D. A sin U ¼ os W B tn U ¼ sin W C os U ¼ tn W D tn U ¼ tn W U V 8 Sketh right-ngled tringle for eh trigonometri rtio, then use Pythgors theorem to find the length of the unknown side nd the other two trigonometri rtios for the sme ngle. tn A ¼ 5 sin B ¼ 3 os X ¼ 9 d sin Y ¼ W See Exmple Similr right-ngled tringles In eh right-ngled tringle elow, \A ¼ 32. C C 2 32 A B 1 C Worksheet Investigting the tngent rtio MAT09MGWK10039 Tehnology GeoGer: Trigonometri rtios MAT09MGTC00006 C B 32 A B 3 32 A A 32 Furthermore, euse \B ¼ 90, \C ¼ ¼ 58 euse of the ngle sum of tringle. These four tringles re lled similr tringles euse their orresponding ngles re equl. They hve the sme shpe ut re not the sme size. In ft, they re enlrgements or redutions of one nother B 141

11 Chpter Exmple 6 For eh tringle on the previous pge, mesure the length of eh side (orret to the nerest mm) nd then lulte sin A, os A nd tn A s deimls (orret to two deiml ples). Write your results in the tle elow BC (opp) Side length (mm) AB (dj) AC (hyp) sin A ¼ BC AC Trigonometri rtio os A ¼ AB AC tn A ¼ BC AB Solution Side length (mm) Trigonometri rtio BC AB AC (opp) (dj) (hyp) sin A ¼ BC AC os A ¼ AB AC tn A ¼ BC AB Note: See Tehnology: Similr right-ngled tringles on pge 144 for GeoGer tivity sed on this exmple. Exerise 4-03 Similr right-ngled tringles See Exmple 6 1 For eh similr right-ngled tringle elow, mesure the length of eh side (orret to the nerest mm) nd then lulte sin Y, os Y nd tn Y s deimls (orret to two deiml ples). Copy nd omplete the tle opposite. X X X Z 1 60 Y 2 Y 60 3 Z Y 60 Z

12 NEW CENTURY MATHS for the Austrlin Curriulum9 X Z 4 60 Y Side length (mm) Trigonometri rtio XZ (opp) ZY (dj) XY (hyp) sin Y ¼ XZ XY os Y ¼ ZY XY tn Y ¼ XZ ZY Wht do you notie out the vlue of sin Y for ll four similr right-ngled tringles? Use your lultor to evlute sin 60 y pressing sin 60 =. Wht do you notie out your nswer? d Wht do you notie out the vlue of os Y for ll four similr tringles? e Use your lultor to evlute os 60. Wht do you notie out your nswer? f Wht do you notie out the vlue of tn Y for ll four similr tringles? g Use your lultor to evlute tn 60. Wht do you notie out your nswer? 2 Drw four similr right-ngled tringles tht hve n ngle of 48, mesure the length of eh side (orret to the nerest mm) nd then lulte sin 48, os 48 nd tn 48 s deimls (orret to two deiml ples). Copy nd omplete the tle elow Side length (mm) Opp Adj Hyp sin 48 ¼ opp hyp Trigonometri rtio os 48 ¼ dj hyp tn 48 ¼ opp dj Exmine the vlue of sin 48 for ll four similr tringles, then evlute sin 48 on lultor. Wht do you notie? Exmine the vlues of os 48, then evlute os 48 on lultor. d Exmine the vlues of tn 48, then evlute tn 48 on lultor. 3 For eh trigonometri rtio, drw lrge right-ngled tringle with the given ngle, then y mesurement nd lultion, find the vlue of the rtio orret to three deiml ples. Compre your nswer to the lultor s nswer. tn 55 os 39 sin 67 d os

13 Chpter Tehnology Similr right-ngled tringles In this tivity you will use GeoGer to mesure nd lulte trigonometri rtios. 1 Before you strt, set ngles to mesure in degrees, then lik Options, Rounding nd 1 Deiml Ple. Note: If the ngle mesure is set to rdins, under Options, Advned selet Degree. In the Grphis window, right-lik nd mke sure Axes nd Grid re enled. Alterntively, lose the Alger window nd lik on the the top left-hnd side. ions ner Use Intervl etween two points nd onstrut right-ngled tringle

14 NEW CENTURY MATHS for the Austrlin Curriulum9 Use Angle to mesure the right ngle nd nother ngle (s shown elow). Now keeping the tringle right-ngled, use the Move Tool to djust ny verties so tht one ngle is Drw two more similr right-ngled tringles with 32 ngle. 3 Lel the verties of eh tringle. Right-lik on eh vertex nd lik Show Lel s shown elow. If neessry, relel the verties s A, B nd C y right-liking on eh vertex nd seleting Renme. Mke sure \A ¼ 32 nd \B ¼ Copy this tle Side length (mm) BC (opp) AB (dj) AC (hyp) sin A ¼ BC AC Trigonometri rtio os A ¼ AB AC tn A ¼ BC AB Clik Options, Rounding nd 5 Deiml Ples. To mesure the sides of eh tringle, selet nd lik on side of the tringle. You will see the mesurements pper in entimetres. Convert to millimetres for your tle. d For eh tringle, lulte eh trigonometri rtio orret to 2 deiml ples. e Wht do you notie out the vlues of eh rtio for ll of the tringles?

15 Chpter Homework sheet 1 MAT09MGHS on lultor In the previous setion, we disovered tht for ny prtiulr ngle, the sine, osine nd tngent rtios sty onstnt (the sme) for ll right-ngled tringles with tht ngle. For exmple, sin lwys, no mtter wht size the similr right-ngled tringle. Summry For ny given ngle, the vlues of the sine, osine nd tngent rtios re onstnt. This mens tht the vlue of trigonometri rtio n e esily found on lultor rther thn through onstruting nd mesuring tringles. Stge 5.2 Worksheet lultions MAT09MGWK10041 Puzzle sheet squresw MAT09MGPS10042 Degrees, minutes nd seonds Angles re mesured in degrees, ut one degree n e sudivided into 60 minutes. One minute n e further sudivided into 60 seonds. The revitions for minutes nd seonds re shown elow. Summry 1 ¼ 60 0 (1 degree ¼ 60 minutes) 1 0 ¼ (1 minute ¼ 60 seonds) For exmple, n ngle size of is 48 degrees, 35 minutes nd 56 seonds, out hlfwy etween 48 nd 49. When rounding n ngle to the nerest degree or minute, use 30 s the hlfwy mrk. Exmple 7 Round eh ngle orret to the nerest degree Solution < 30 0, so round down , so round up

16 NEW CENTURY MATHS for the Austrlin Curriulum9 Exmple 8 Stge 5.2 Round eh ngle orret to the nerest minute Solution , so round up , so round up. Degrees nd minutes on lultor To enter degrees nd minutes (nd seonds) into sientifi lultor, use the (Degrees-Minutes-Seonds) key. or DMS Exmple 9 Evlute eh expression orret to two deiml ples. sin 46 tn os 20 d 68.3 sin e 23 os Solution Mke sure tht your lultor is in the degrees mode (D or DEG) or your nswer will e inorret. sin 46 ¼ 0: :72 On lultor: sin 46 = tn 57:4 ¼ 1: :56 On lultor: tn 57.4 = This ngle is 57.4 ( deiml), not os 20 ¼ 3: :76 d 68:3 sin ¼ 42: :44 e 23 os ¼ 24: :30 On lultor: 4 os 20 = This mens 4 3 os 20. On lultor: 68.3 sin = On lultor: 23 4 os = Stge

17 Chpter Stge 5.2 Exmple 10 Convert eh ngle size to degrees nd minutes, orret to the nerest minute Solution 82.5 ¼ On lultor: 82.5 = 60:81 ¼ On lultor: = Exerise 4-04 on lultor See Exmple 7 See Exmple 8 See Exmple 9 Stge 5.2 See Exmple 10 1 Round eh ngle size orret to the nerest degree d e f 56.4 g h Round eh ngle size orret to the nerest minute d e f g h Evlute eh expression orret to two deiml ples. tn 84 os 15 tn 47 d sin 33 e sin 77 f os 60.1 g tn h os 18 i 8 tn 75 j 14 sin 56 k 12 4 tn 20 l m 50 3 sin n 66.2 os o sin 11.8 p q 44:5 tn r 200 sin 54:2 s tn 63 t 7 sin os 35 15:7 os Convert eh ngle size to degrees nd minutes, orret to the nerest minute d e f g h i j k l By guess-nd-heking with your lultor, find the ngle size, y (to the nerest degree), tht gives eh vlue. sin y ¼ tn y ¼ tn y ¼ d os y ¼ 0.5 e sin y ¼ f os y ¼ g tn y ¼ h sin y ¼

18 NEW CENTURY MATHS for the Austrlin Curriulum9 Mentl skills 4 Mths without lultors Estimting nswers A quik wy of estimting n nswer is to round eh numer in the lultion. 1 Study eh exmple. 631 þ 280 þ 51 þ 43 þ þ 300 þ 50 þ 40 þ 100 ¼ð600 þ 300 þ 100Þþð50 þ 40Þ ¼ 1000 þ 90 ¼ 1090 ðatul nswer ¼ 1101Þ 55 þ þ þ þ ¼ð60 þ 20 80Þþð130 30Þ ¼ 0 þ 100 ¼ 100 ðatul nswer ¼ 92Þ ¼ 560 ðatul nswer ¼ 546Þ d ¼ ¼ 25 ðatul nswer ¼ 21:25Þ 2 Now estimte eh nswer. 27 þ 11 þ 87 þ 142 þ þ þ þ 903 d 35 þ 81 þ 110 þ 22 þ 7 e f þ g h i j k l Study eh exmple involving deimls. 20:91 11:3 þ 2: þ 3 ¼ 13 ðext nswer ¼ 12:11Þ 4: : ¼ 100 ðext nswer ¼ 91:776Þ 75:13 4 8: < < 10 9 ðext nswer ¼ 8: Þ d 37:6 þ 9:3 41:2 12:7 38 þ ¼ :6 (Ext nswer = )

19 Chpter Now estimte eh nswer þ þ 4.6 þ þ d e 11:07 þ 18:4 12:2 f 38:18 17:2 9:6 g h 18:46 3 4:9 39:72 15:2 i j (4.89) 2 Just for the reord Degrees, minutes nd seonds We re fmilir with using se 10 systems in numer, mesurement nd urreny: there re 100 entimetres in metre, 1000 grms in kilogrm nd 100 ents in dollr. So why re there 360 in revolution, 60 minutes in degree nd 60 seonds in minute? In 2000 BCE, the Bylonins used se 60 or sexgesiml system of numertion, euse 60 is rounder, more onvenient numer thn 10. This is euse 60 hs more ftors nd is divisile y 3, 4 nd 6. Furthermore, ¼ 360, whih ws the Bylonin pproximtion for the numer of dys in yer, so tht eh dy the Erth would trvel 1 round the Sun. As mesuring devies nd lultions required greter preision, eh degree ws sudivided into 60 equl prts lled minutes, nd these were further divided into 60 prts lled seonds. This level of ury is essentil in nvigtion nd mpping. 1 A minute is smll prt of degree. Investigte how n lterntive mening (nd pronunition) of minute is tiny. 2 A seond is the seond sudivision of degree. Explin how there re two different menings of seond Finding n unknown side Sine the trigonometri rtio of ny ngle is onstnt numer, we n use it to lulte the length of n unknown side in right-ngled tringle if one other side is known. We need to selet the orret rtio tht links the given ngle to the unknown side nd known side

20 NEW CENTURY MATHS for the Austrlin Curriulum9 Exmple 11 Find the vlue of eh pronumerl, orret to two deiml ples m p 20 m 33 Video tutoril Finding n unknown side MAT09MGVT10008 Video tutoril MAT09MGVT d Solution SOH, CAH or TOA? The mrked sides re the djent (A) side nd the hypotenuse (H), so use os. os 58 ¼ djent hypotenuse ¼ d 15:2 os :2 ¼ d 15:2 3 15:2 15:2 os 58 ¼ d d ¼ 15:2 os 58 ¼ 8: :05 From the digrm, length of 8.05 m looks resonle. SOH, CAH or TOA? The mrked sides re the opposite (O) side nd the hypotenuse (H), so use sin. sin 33 ¼ opposite hypotenuse ¼ p 20 sin ¼ p sin 33 ¼ p p ¼ 20 sin 33 opposite ¼ 10: :89 From the digrm, length of m looks resonle. p hypotenuse 15.2 m 58 d djent Multiply oth sides y m hypotenuse

21 Chpter Summry Finding n unknown side in right-ngled tringle 1 identify the two lelled sides nd deide whether to use sin, os or tn 2 write n eqution using the rtio, the given ngle nd the vrile 3 solve the eqution to find the vlue of the vrile Exmple 12 Find the vlue of q orret to the nerest entimetre. Solution q is opposite, 47 m is djent, so use tn. tn 23 ¼ opposite djent ¼ q 47 q ¼ 47 tn 23 ¼ 19: m From the digrm, length of 20 m looks resonle m q Exmple 13 njkl is right-ngled t K, JK ¼ 35 m nd \J ¼ 63. Find the length of LK orret to the nerest metre. Solution Drw digrm. Let the length of LK e x. x is opposite, 35 m is djent, so use tn. tn 63 ¼ x 35 x ¼ 35 tn 63 ¼ 68: LK 69 m L x K m J From the digrm, length of 69 m looks resonle

22 NEW CENTURY MATHS for the Austrlin Curriulum9 Exerise 4-05 Finding n unknown side 1 For eh tringle, whih trigonometri rtio (sin y, os y or tn y) is equl to? d e f 2 Find the vlue of the pronumerl in eh tringle, orret to two deiml ples. 28 x m m 25.3 m mm mm d k m 36 y m e f 150 mm 35.2 m m 57 33' 20.7 See Exmple 11 See Exmple 12 Stge m mm 3 Find the vlue of the pronumerl in eh tringle, orret to one deiml ple. d e m 74 m 26 y m m 77 f m e 8.5 m v m 50 3' m f 34 g mm 263 mm w mm mm Stge

23 Chpter See Exmple 11 4 Find the vlue of the pronumerl in eh tringle, orret to two deiml ples. z m 52 m m r m 85.3 mm t mm 29 Stge 5.2 d 7.3 m 45 18' n m e p m f m 73 37' 315 mm 40.1 q mm 5 Find the vlue of the pronumerl in eh tringle, orret to one deiml ple. 33 n m 200 m p m m mm q mm Stge 5.2 d r m 67 36' 0.9 m e m 50 11' f t m 8.7 m s m Wht is the height of this tree? Selet the orret nswer A, B, C or D. A 2.68 m B 3.27 m C 3.83 m D 6.68 m 7 Find eh length or distne orret to one deiml ple. How high the ldder rehes up the wll mm 355 mldder

24 NEW CENTURY MATHS for the Austrlin Curriulum9 The distne etween the ot nd the strt. river 200 m ot 45 strt The distne from the oserver to the se of the uilding m oserver d The height of the ot s mst m 47 e The distne etween: i hekpoints 1 nd 2 ii hekpoint 1 nd the strt km 30 strt/finish 8 nabc is right-ngled t B, AC ¼ 14.8 m nd \C ¼ 56. Find the length of side AB, orret to one deiml ple. 9 nmnr is right-ngled t M, MN ¼ 19 m nd \N ¼ 27. Find the length of MR, orret to the nerest entimetre. 10 In nxyw, \X ¼ 90, \Y ¼ 43.7 nd WY ¼ 8.34 m. Find the length of XW, orret to two deiml ples. 11 nahk is right-ngled t K, \H ¼ 76 nd AH ¼ 13.9 m. Find the length of HK, orret to one deiml ple. 12 A tree sts shdow 20 m long. If the Sun s rys meet the ground t 25, find the height of the tree, orret to the nerest m. 13 A 6 m ldder is pled ginst pole. If the ldder mkes n ngle of 17 with the pole, how fr up the pole does the ldder reh? Answer to the nerest mm. 14 A golfer is 180 m (in stright line) from the eighth hole. The ll is hit 15 to the right of the hole ut still ends up level with the hole. How fr is the ll from the hole? Answer to the nerest metre See Exmple

25 Chpter Worked solutions Finding n unknown side MAT09MGWS A prk is in the shpe of retngle. A pth 450 m in length rosses the prk digonlly. If the pth mkes n ngle of 36 with the longer side, find the dimensions of the prk. Answer to the nerest metre. 16 A wheelhir rmp is 6 m long nd mkes n ngle of 4.5 with the ground. How high is the top of the rmp ove the ground (orret to two deiml ples)? 17 A ot is nhored y rope 5.5 m long. If the nhor rope mkes n ngle of 23 with the vertil, lulte the depth of the wter (orret to one deiml ple). 18 A retngulr gte hs digonl re tht mkes n ngle of 60 with the ottom of the gte. If the length of the digonl re is 1860 mm, lulte the height of the gte. Selet the orret nswer A, B, C or D. A 2148 mm B 930 mm C 1610 mm D 3221 mm 19 Jo is flying kite tht is tthed to string 155 m long. The string mkes n ngle of 35 to the horizontl. Clulte, orret to the nerest metre, the height of the kite ove Jo. 35 Shutterstok.om/pirit Investigtion: Clulting the height of n ojet You will need: tpe mesure or trundle wheel, linometer (or protrtor) to mesure the ngle. n e used to find the heights of uildings, flgpoles nd trees without tully mesuring them. This n e done y mesuring the distne long the ground from the se of the ojet to person. The person then mesures the ngle to the top of the ojet. For exmple, the height of flgpole n e lulted using the set-up shown in the digrm elow. x A H where H = x + h h L

26 NEW CENTURY MATHS for the Austrlin Curriulum9 h is the eye height of the person who mesures the ngle, A, to the top of the flgpole. L is the distne the person is from the se of the flgpole, x is the height of the flgpole ove the person s eye height, H ¼ x þ h is the height of the flgpole ove the ground. 1 Selet tll ojet outside to mesure. 2 Work with prtner to mesure (in m) the distne, L, long the ground, the height, h, of the person, nd the ngle (in degrees) to the top of the ojet. Copy the tle elow nd reord your informtion in the first row. Distne, L (m) Angle, A Height of person, h (m) Clulted height, x m Height of flgpole, H m 3 Use the tn rtio to lulte the vlue of x to the nerest whole numer. 4 Hene find H, the height of the kite to the nerest entimetre. Write your nswers in the tle. 5 Repet the mesurements nd lultions three more times from different positions, with different persons mesuring the ngle. This will help to improve the ury of your results nd minimise errors. Write your results in the tle. 6 Did you find similr vlues for H? Do they seem resonle for the height of the ojet? 7 Clulte the verge vlue for H Finding more unknown sides In the following exmples, the unknown ppers in the denomintor of the eqution. Using sin or os to find the hypotenuse Exmple 14 Worksheet Finding n unknown side MAT09MGWK10043 Puzzle sheet equtions 1 MAT09MGPS00041 Find the vlue of w, orret to two deiml ples. Solution 80 m is the opposite side, w m is the hypotenuse, so use sin. sin 55 ¼ 80 w Note tht the vrile w ppers in the denomintor of the eqution. w m m

27 Chpter sin 55 3 w ¼ 80 w 3 w Multiply oth sides y w. w sin 55 ¼ 80 w sin 55 sin 55 ¼ 80 sin 55 w ¼ 80 sin 55 ¼ 97: :66 Divide oth sides y sin 55. Note tht when the unknown ppers in the denomintor of n eqution, it n swp positions with the trigonometri rtio, so tht sin 55 ¼ 80 eomes w ¼ 80 w sin 55. Exmple 15 npqr is right-ngled t Q, QR ¼ 41 m nd \R ¼ 25. Find RP, orret to the nerest metre. Q R Solution Let x ¼ RP. 41 m is the djent side, x is the hypotenuse, so use os. os 25 ¼ 41 x x ¼ 41 os 25 ¼ 45: m Note tht the vrile x ppers in the denomintor of the eqution. P x Swp the position of x with os 25. Using tn to find the djent side Exmple 16 Find the length of x, orret to two deiml ples. Solution 18 m is the opposite side, x is the djent side, so use tn. x 34 tn 34 ¼ 18 x x ¼ 18 tn 34 ¼ 26: :69 m x ppers in the denomintor 18 m Swp the position of x with tn

28 NEW CENTURY MATHS for the Austrlin Curriulum9 Alterntive method To void hving x in the denomintor, we ould use tn with the third ngle of the tringle. Third ngle ¼ ¼ 56 tn 56 ¼ x 18 x ¼ 18 tn 56 ¼ 26: :69 m Exerise 4-06 Finding more unknown sides 1 Find the vlue of eh pronumerl, orret to one deiml ple. See Exmple 14 x m 35 m 18.4 m 73 y m 20 z mm 78.3 mm 35.7 d m r m 43 43' e 15 25' s m m f t mm 58 5' 200 mm Stge nxyz is right-ngled t Z, ZY ¼ 230 mm nd \Y ¼ 45. Find the length of XY, orret to the nerest millimetre. 3 In nklw, \L ¼ 90, KL ¼ 12 m nd \W ¼ Find KW, orret to the nerest metre. 4 Find the vlue of eh pronumerl, orret to two deiml ples. n e h d 8 k See Exmple 15 See Exmple 16 5 ncde is right-ngled t D, \E ¼ 36 nd CD ¼ 5 m. Find the length of side DE, orret to two deiml ples

29 Chpter Stge In nhmt, \T ¼ 90, \M ¼ nd side HT ¼ 18.4 m. Find the length of side HM, orret to one deiml ple. 7 Find the vlue of eh pronumerl, orret to one deiml ple n m p m 40 40' 83 m 93.1 mm q mm 17.6 d 5.27 m 25 m e f m mm 23 6' r m s m 38 11' t mm 8 Find the length of this ldder. Selet the orret nswer A, B, C or D. A 159 m C 243 m B 171 m D 638 m ldder wll 75 9 Find eh length or distne orret to one deiml ple. How fr the person is from eing diretly under the irds. 165 m 2450 m 62 d 187 m The length of the rmp. 255 mm rmp

30 NEW CENTURY MATHS for the Austrlin Curriulum9 The length of one of the support wires. 2 m 7 m d The length of: i the shortest rod ii the longest rod e The slnt height of the roof. 7.5 km w m 10 nfgw is right-ngled t F, \W ¼ 84 nd WF ¼ 42.1 m. Find the length of WG, orret to one deiml ple. 11 A ldder rests ginst wll. The foot of the ldder is 355 m from the wll nd mkes n ngle of 63 with the ground. How long (to the nerest m) is the ldder? 12 A supporting wire is tthed to the top of flgpole. The wire meets the ground t n ngle of 51 nd the flgpole is 15 m high. How fr from the se of the flgpole is the wire nhored to the ground? (Give your nswer to the nerest 0.1 m.) 13 Agliderisflyingtnltitude (height) of 1.5 km. To lnd, it desends t n ngle of 18 to the ground. How fr must the glider trvel efore lnding? (Give your nswer to the nerest 0.1 km.) See Exmple 15 Worked solutions Finding more unknown sides MAT09MGWS10019 Shutterstok.om/Pixhi 14 The entrne to the shool lirry is 60 m ove ground level. A wheelhir rmp is uilt to the entrne t n ngle of 5 with the ground. How long (to the nerest 0.01 m) is the rmp? 15 A shooter ims diretly t trget, ut just efore firing, the rifle is lifted 1 off trget. The shot misses the trget y 67 mm. How fr is the shooter stnding from the trget? Selet the orret nswer A, B, C or D. A 1169 mm B 3838 mm C 3839 mm D 6701 mm 16 A hot ir lloon is nhored to the ground y rope. When it drifts 20 m sidewys, it mkes n ngle of 75 with the ground. How long is the rope (orret to one deiml ple)?

31 Chpter Investigtion: Finding n ngle, given trigonometri rtio You will need: ruler, ompsses nd lultor. 1 Copy nd omplete this tle, lulting eh rtio s deiml orret to three deiml ples. u sin u os u tn u undefined 2 Cn you work out why there is no nswer for tn 90? 3 Wht re the minimum nd mximum vlues of sin y? 4 Wht re the minimum nd mximum vlues of os y? 5 Is there pttern etween the vlues of sin y nd os y? 6 Chek the vlue of sin 30 y onstruting right-ngled tringle with one ngle tht is 30, mesuring the opposite side nd hypotenuse nd dividing them. 7 Chek the vlue of tn 45 y onstruting right-ngled tringle with one ngle tht is 45, mesuring the opposite nd djent sides nd dividing them. 8 Use the tle to estimte eh trigonometri rtio nd hek your estimte using lultor. sin 80 os 34 tn 55 9 If sin u ¼ 3, find the vlue of the unknown ngle y to the nerest degree: 8 y using the tle nd estimting (hnge 3 to deiml first) 8 using lultor to guess-nd-hek onstruting right-ngled tringle with one ngle y, opposite side 3 m nd hypotenuse 8 m, then mesuring the size of y. C 8 m 3 m A B

32 N E W C E N T U R Y M AT H S for the A ustrlin Curriulum 9 10 If os u ¼ 2, find the vlue of the unknown ngle y to the nerest degree: 5 y using the tle nd estimting (hnge 2 to deiml first) 5 using lultor to guess-nd-hek onstruting right-ngled tringle with one ngle y, djent side 2 m nd hypotenuse 5 m, then mesuring the size of y. 11 If tn u ¼ 7, find the vlue of the unknown ngle y to the nerest degree: 10 y using the tle nd estimting (hnge 7 to deiml first) 10 using lultor to guess-nd-hek onstruting right-ngled tringle with one ngle y, opposite side 7 m nd djent side 10 m, then mesuring the size of y. Worksheet 4-07 Finding n unknown ngle Finding n unknown ngle A sientifi lultor n e used to evlute trigonometri rtio suh s sin 38, ut it n lso e used to find n unknown ngle, y, if the trigonometri rtio of the ngle is known, for exmple, if sin y ¼ An unknown ngle n e found using the sin 1, os 1 nd tn 1 keys on the lultor. These re lled the inverse sin, inverse os nd inverse tn funtions, found y pressing the SHIFT or 2ndF key efore the sin, os or tn keys. MAT09MGWK10044 Puzzle sheet squresw MAT09MGPS10042 Homework sheet 2 Exmple 17 MAT09MGHS10033 Homework sheet If sin y ¼ , find ngle y, orret to the nerest degree. If tn X ¼ 3.754, find ngle X, orret to the nerest minute. 4 If os ¼, find ngle, orret to the nerest degree. 7 revision MAT09MGHS10034 Animted exmple Solution MAT09MGAE00009 sin u ¼ 0:9063 u ¼ 64: tn X ¼ 3:754 X ¼ 75: ¼ :6200 On lultor: SHIFT sin = Worksheet prolems MAT09MGWK10045 On lultor: SHIFT On lultor: tn or Stge 5.2 = DMS ¼ 55: os ¼ On lultor: SHIFT os 4 / 7 =

33 Chpter Exmple 18 Video tutoril Finding n unknown ngle Find the size of ngle y, orret to the nerest degree. W MAT09MGVT10009 Video tutoril 13 m 9 m MAT09MGVT00009 M T Puzzle sheet : Finding ngles MAT09MGPS00044 Puzzle sheet equtions 2 Solution SOH, CAH or TOA? The known sides re the opposite (O) side nd the hypotenuse (H), so use sin. sin u ¼ m hypotenuse W opposite 9 m MAT09MGPS00042 M T u ¼ 43: On lultor: SHIFT sin 9 / 13 = From the digrm, n ngle size of 44 looks resonle. Summry Finding n unknown ngle in right-ngled tringle 1 Identify the two known sides nd deide whether to use the sin, os or tn rtio. 2 Write n eqution using the rtio, the ngle vrile nd the two sides s frtion. 3 Use the lultor s inverse trigonometri funtion to find the size of the ngle

34 NEW CENTURY MATHS for the Austrlin Curriulum9 Exmple 19 nxyz is right-ngled t Y, with XY ¼ 35 m nd YZ ¼ 47 m. Find \Z, orret to the nerest degree. Solution Sketh digrm. SOH, CAH or TOA? The known sides re the opposite (O) nd the djent (A), so use tn. X 35 m opposite tn u ¼ Z 47 m djent Y u ¼ 36: On lultor: SHIFT tn 35 / 47 = From the digrm, n ngle size of 37 looks resonle. Exerise 4-07 Finding n unknown ngle 1 Find the size of ngle y orret to the nerest degree. pffiffiffi 3 os y ¼ 0.76 tn y ¼ sin u ¼ d tn y ¼ 6 2 e sin u ¼ 7 f os u ¼ 13 g sin u ¼ 1 h os u ¼ p 1 ffiffiffi p i tn u ¼ ffiffiffi 3 j os y ¼ k tn y ¼ l sin u ¼ 1 4 See Exmple 17 2 Find the size of ngle A orret to the nerest minute. tn A ¼ 15 7 sin A ¼ os A ¼ 4 5 d os A ¼ Stge 5.2 e tn A ¼ f os A ¼ 3 10 i tn A ¼ j os A ¼ 1 7 g sin A ¼ 5 11 pffiffiffi 2 k tn A ¼ 2 h sin A ¼ 0.88 l sin A ¼

35 Chpter See Exmple 18 3 Find the size of ngle y, orret to the nerest degree d 28 e f Stge Find the size of ngle orret to the nerest minute. Selet the orret nswer A, B, C or D. A B C D α 8.7 m 5.2 m 5 Find the size of ngle orret to the nerest minute. 8 m α 8.7 m 5.2 m 17 m 20 m α α 12 m d 81 mm e α f α α 7.1 m 1.2 m 0.8 m 95 mm 3.2 m

36 NEW CENTURY MATHS for the Austrlin Curriulum9 6 Find the size of ngle y, orret to the nerest degree. gte 15 m rmp 3.2 m two-storey r prk 80 m string 2500 mm 1250 mm 30 m d e f shdow 340 m 175 m 5 m rope 2.5 m rok limer 90 m lening tower 3 m 7 In nxyw, \X ¼ 90, XY ¼ 8 m nd XW ¼ 10 m. Find \W orret to the nerest degree. 8 In nfgh, \G ¼ 90, GH ¼ 3.7 m nd FH ¼ 19.5 m. Find the size of ngle F, orret to the nerest minute. 9 nhtm is right-ngled t T, HM ¼ 45 m nd MT ¼ 35 m. Find \M, orret to one deiml ple. 10 ntsv is right-ngled t S, TV ¼ 9.5 m, nd ST ¼ 8.4 m. Find \V, orret to the nerest degree. For questions 11 to 18, write your nswers orret to the nerest degree. 11 A streth of freewy rises 55 m for every 300 m trvelled long the rod. Find the ngle t whih the rod is inlined to the horizontl. 12 A ldder 20 m long is pled ginst uilding. If the ldder rehes 16 m up the uilding, find the ldder s ngle of inlintion to the uilding. 13 An irrft is desending in stright line to n irport. At height of 1270 m, it is 1500 m horizontlly from the irport. Find its ngle of desent to the horizontl. Selet the orret nswer A, B, C or D. A 32 B 40 C 50 D A tree 8.5 m high sts shdow 3 m long. Wht is the ngle of the Sun from the ground? 15 At resort, n rtifiil eh slopes down t stedy ngle. After wlking 8.5 m down the slope from the wter s edge, the wter hs depth of 1.6 m. At wht ngle is the eh inlined to the horizontl? 16 A pile of whet is in the shpe of one tht hs dimeter of 35 m nd mesures 27 m up the slope to the pex. Clulte the ngle of repose of the whet (the ngle the sloping side mkes with the horizontl se) See Exmple 19 Stge 5.2 Worked solutions Finding n unknown ngle MAT09MGWS

37 Chpter A 2.8 m vertil tent pole is supported y 3.1 m rope. Wht ngle does the rope mke with the pole? 18 A ship is nhored in wter 40 m deep y 65 m nhor hin. Find the ngle t whih the hin is inlined to the se floor. Power plus 1 Copy nd omplete eh pir of trigonometri rtios orret to three deiml ples. i sin 20 ¼, os 70 ¼ ii sin 47 ¼, os 43 ¼ iii sin 55 ¼, os 35 ¼ iv sin 85 ¼, os 5 ¼ Wht do you notie out eh pir of nswers in prt? Wht do you notie out eh pir of ngles in prt? d If os nd sin y , wht is the vlue of y? e Copy nd omplete eh eqution. i sin 75 ¼ os ii 80 ¼ os 10 iii os ¼ sin 72 iv sin 30 ¼ 60 v os 65 ¼ sin vi sin ¼ os 58 f g Copy nd omplete this generl rule: sin x ¼ os ( ). Use right-ngled tringle with one ngle x nd sides, nd to prove tht the ove rule is true. 2 A plne is flying t n ngle of 15 inlined to the horizontl. How fr to the nerest metre will the plne hve to trvel long its line of flight to inrese its ltitude (height) y 500 m? At wht ngle to the nerest degree must the plne lim to hieve n inrese in ltitude of 500 m in hlf the distne needed t n ngle of 15? 3 If sin 30 ¼ 1, find, s surd, the vlue of: 2 os 30 tn 30 4 Find the vlue of ngle y, orret to the nerest seond Chpter 4 review 5 By drwing n pproprite tringle, prove tht: tn 45 ¼ 1 sin 45 ¼ p 1 ffiffiffi os 45 ¼ p 1 ffiffiffi

38 Chpter 4 review n Lnguge of mths djent lph () linometer osine (os) degree ( ) denomintor horizontl hypotenuse inverse ( 1 ) minute ( 0 ) mnemoni opposite phi (f) Pythgors theorem right-ngled seond ( 00 ) similr tringles sine (sin) tngent (tn) thet (y) trigonometry trigonometri rtio unknown vertil Puzzle sheet rossword MAT09MGPS10049 Quiz MAT09MGQZ When mesuring ngle size, wht is seond nd wht is its symol? 2 Wht word mens next to? 3 Whih side of right-ngled tringle is fixed nd does not depend on the position of n ngle? 4 Wht re the first two letters of the Greek lphet? 5 The word minute hs n lterntive pronunition nd mening. Wht is its lterntive mening? 6 Wht does inverse men nd how is it used in trigonometry? n Topi overview Copy nd omplete this mind mp of the topi, dding detil to its rnhes nd using pitures, symols nd olour where needed. Ask your teher to hek your work. Right-ngled tringles The trigonometri rtios Worksheet Mind mp: MAT09MGWK10050 H O TRIGONOMETRY on lultor A Finding n unknown ngle Finding n unknown side 50 9 m

39 Chpter 4 revision See Exerise For ngle U, nme the opposite nd djent sides nd the hypotenuse. 25 V U See Exerise For ngle V, nme the opposite nd djent sides nd the hypotenuse. U W v w u See Exerise For this tringle, write s frtion: sin Y tn Y sin X d os X V Y See Exerise 4-02 See Exerise 4-03 See Exerise 4-04 Stge 5.2 See Exerise 4-04 See Exerise 4-04 See Exerise 4-04 X 4 If sin ¼ 36, write the vlues of os nd tn s frtions. (Drw digrm.) 85 5 Construt lrge right-ngled tringle with n ngle of 42, then y mesurement nd lultion, find the vlue of eh trigonometri rtio, orret to three deiml ples. tn 42 os 42 sin 42 Compre your nswers to the lultor s nswers. 6 Round eh ngle to the nerest degree Round eh ngle to the nerest minute Evlute eh expression, orret to four deiml ples. os 32 sin tn d 200 tn 18 e 14 sin f 13 os Convert eh ngle size to degrees nd minutes, orret to the nerest minute

40 Chpter 4 revision 10 Find the vlue of eh pronumerl, orret to two deiml ples r m 3.6 mm t m 85.3 m n mm See Exerise m For eh tringle, find the length of side AC, orret to one deiml ple. A A m C 23 m 47 29' 12 Find the size of ngle y, orret to the nerest degree. tn y ¼ 2.57 os u ¼ 4 sin u ¼ 1:5 7 1:6 13 Find the size of ngle, orret to the nerest degree. α 11.7 m α 6.3 m 1.5 m C A 0.8 m mm 2500 mm 14 In naec, \C ¼ 90, CE ¼ 3.9 m nd AE ¼ 4.2 m. Find \A, orret to the nerest minute. C α 1975 mm See Exerise 4-06 See Exerise 4-07 See Exerise 4-07 See Exerise 4-07 Stge