Right-angled triangles

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1 13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Right-ngled tringles Syllus referene Mesurement 4 Right-ngled tringles In this hpter we will lern out right-ngled tringles nd how to pply your knowledge of them to prolem solving.

2 ARE You READY? Digitl do SkillSHEET 13.1 do-1621 Lelling sides of right-ngled tringle Try the questions elow. If you hve diffiulty with ny of them, etr help n e otined y ompleting the mthing SkillSHEET. Either lik on the SkillSHEET ion net to the question on the Mths Quest Preliminry Course ebookplus or sk your teher for opy. Lelling sides of right-ngled tringle 1 In eh of the following, nme the opposite side, djent side nd the otenuse. A P X Y Digitl do SkillSHEET 13.2 do-1622 Using Pythgors theorem B C Q R Z Using Pythgors theorem Digitl do SkillSHEET 13.3 do-1629 Rounding to given numer of deiml ples 2 Find the vlue of, orret to 1 deiml ple. 3.5 m 5.9 m 27 m 10 m Digitl do SkillSHEET 13.4 do-1630 Solving equtions of the type = to find Digitl do SkillSHEET 13.5 do-1631 Solving equtions of the = type to find Digitl do SkillSHEET 13.7 do-1634 Rounding ngles to the nerest degree Rounding to given numer of deiml ples 3 Round eh of the following numers to the numer of deiml ples indited in rkets [2] [1] [3] d [3] e [2] f [4] Solving equtions of the type = 4 Solve eh of the following equtions. 0.6 = = 47. Solving equtions of the type = 5 Solve eh of the following equtions = = ẋ Rounding ngles to the nerest degree to find to find 1.45 = = Round eh of the following to the nerest degree d Mths Quest Generl Mths Preliminry Course

3 13A Pythgors theorem PYTHAGORAS OF SAMOS (ir 580BC 500BC) During his life: Toism is founded Kung-Fu-tse (Confuius) is orn Buddhism is founded. Pythgors ws fmous Greek mthemtiin nd mysti who is now est known for his theorem out the sides of tringle. He ws orn on Smos Islnd, ner Greee. It is elieved tht he ws orn out 580 B C nd died out 500 B C, ut euse of the wy dtes were reorded then, vrious dtes re given for his life. Not muh is known out his personl life, ut we know tht he hd wife, son nd dughter. When Pythgors ws young mn, he trvelled to Egypt nd Byloni (Mesopotmi) where he lerned muh of his mthemtis nd developed n interest in investigting it further. He founded ult with the ide tht the essene of ll things is numer. This group elieved tht ll nture ould e epressed in terms of numers. They found, however, tht some numers ould not e epressed s rtionl numers, suh s 2. They kept this informtion to themselves nd there is story tht they killed one memer who told someody else out this prolem. Pythgors showed tht musil notes hd mthemtil pttern. He strethed string tightly nd found tht it produed ertin sound nd then found tht if he hlved the length of the string, it produed sound tht ws in hrmony with the first. He lso found tht if it ws not etly hlf, or multiple of hlf, then it ws lshing sound. This pproh is still used in musil instruments tody. He is redited with the disovery now known s Pythgors theorem. This sttes tht For right-ngled tringle, the squre of the otenuse (long side) is equl to the sum of the squres of the two short sides nd is normlly written s 2 = Other people knew of this ide long efore he nnouned it, nd there is Bylonin tlet known s Plimpton 322 (elieved to hve een mde out 1500 yers efore Pythgors ws orn), whih hs set of these sort of vlues (whih re now lled Pythgoren triples). He used string with knots in it to demonstrte nd mke right ngles. Some emples of Pythgoren triples re: 3, 4 nd 5 8, 15 nd 17 nd more diffiult emple: 20, 99 nd 101. Questions 1. Where ws Pythgors orn? 2. Where did he trvel to nd lern most of his mthemtis? 3. Wht is the formul for his fmous theorem? 4. Wht did he strt to investigte out ptterns in nture? 5. Wht is the nme of the nient tlet tht ontins Pythgoren triples? 6. Wht is Pythgoren triple? Pythgors theorem llows us to lulte the length A of side of right-ngled tringle, if we know the lengths of the other two sides. Consider ABC t right. AB is the otenuse (the longest side). It is opposite the right ngle. C otenuse B Chpter 13 Right-ngled tringles 415

4 Note tht the sides of tringle n e nmed in either of two wys. 1. A side n e nmed y the two pitl letters given to the verties t eh end. This is wht hs een done in the figure ove to nme the otenuse AB. 2. We n lso nme side y using the lower-se letter of the opposite verte. In the figure ove, we ould hve nmed the otenuse. WoRKED EAMPLE 1 Nme the otenuse in the tringle t right. P THINK 1 The otenuse is opposite the right ngle. 2 The verties t eh end or the lower se of the opposite verte n e used to nme the side. WRITE Q The otenuse is QR or p. R Intertivity int-2406 Pythgors Pythgors theorem In ny right-ngled tringle the squre of the otenuse is equl to the sum of the squres of the two shorter sides. Writing this result s formul we sy: 2 = This is the formul used to find the length of the otenuse in right-ngled tri ngle when we re given the lengths of the two shorter sides. WoRKED EAMPLE 2 Find the length of the otenuse in the tringle t right. THINK WRITE 1 Write the formul. 2 = Sustitute the lengths of the shorter sides. 2 = Evlute the epression for 2. = = Find the vlue of y tking the squre root. = 289 = 17 m 8 m 15 m In this emple, the nswer ws whole numer euse we re le to find 289 etly. In most emples this will not e possile. In suh ses, we re sked to write the nswer orret to given numer of deiml ples or signifint figures. By rerrnging Pythgors theorem, we n write the formul to find the length of shorter side of tringle. If 2 = then 2 = 2-2 nd 2 = 2-2. The method of solving this type of question is the sme s in the previous emple, eept we use sutrtion insted of ddition. For this reson, it is importnt to look t eh question refully to determine whether you re finding the length of the ot enuse or one of the shorter sides. 416 Mths Quest Generl Mths Preliminry Course

5 WoRKED EAMPLE 3 Find the length of side PQ in tringle PQR, orret to 3 signifint figures. THINK WRITE 16 m R 9 m 1 Write the formul. 2 = 2-2 P Q 2 Sustitute the lengths of the known sides. 2 = Evlute the epression. = = Find the nswer y finding the squre root. = 175 = 13.2 m Pythgors theorem sttes tht in right-ngled tringle, 2 = The onverse of this theorem sttes tht if 2 = then the tringle is right ngled. This is method used y uilders to ensure tht struture is squre. WoRKED EAMPLE 4 Determine whether the tringle t right is right ngled. THINK WRITE 1 Clulte 2 nd seprtely. 2 = 7 2 = = = = 41 4 m 5 m 7 m s Tutoril int-2336 Worked emple 4 2 Write down n equlity or inequlity sttement Write onlusion. Therefore the tringle is not right ngled. These formuls n e used to solve more prtil prolems. In these ses, it is neessry to drw digrm tht will help you to see whih method for finding solu tion is required. The digrm simply needs to represent the tringle; it does not need to show detils of the sitution desried. WoRKED EAMPLE 5 The fire rigde ttends lze in tll uilding. They need to resue person from the 6th floor of the uilding, whih is 30 metres ove ground level. Their ldder is 32 metres long nd must e t lest 10 metres from the foot of the uilding. Cn the ldder e used to reh the people needing resue? THINK WRITE 1 Drw digrm nd show ll given informtion. urning uilding 30 fire engine 10 Chpter 13 Right-ngled tringles 417

6 2 Write the formul fter deiding if you re finding the otenuse or shorter side. 2 = Sustitute the lengths of the known sides. 2 = Evlute the epression. = = Find the nswer y tking the squre root. = m 6 Give written nswer. The ldder will e long enough to mke the resue. REMEMBER 1. Mke sure tht you n identify the otenuse of right-ngled tringle. It is the side opposite the right ngle. 2. If you re finding the length of the otenuse use 2 = If you re finding the length of shorter side use either 2 = 2-2 or 2 = Red the question refully to mke sure tht you give the nswer in the orret form. 5. Begin written prolems with digrm nd finish with n nswer written in words. EERCISE 13A Pythgors theorem Digitl do SkillSHEET 13.1 do-1621 Lelling sides of right-ngled tringle 1 WE 1 Nme the otenuse in eh of the following tringles. P X Y Q R Z A B C Digitl do SkillSHEET 13.2 do-1622 Using Pythgors theorem Digitl do EXCEL Spredsheet do-1623 Pythgors 2 WE2 Find the length of the otenuse in eh of the following tringles. 5 m 80 mm m z 11 m 12 m 150 mm 60 m 3 In eh of the following, find the length of the otenuse, orret to 2 deiml ples. 4.9 m 8.6 km 11.3 km 9 m 4.9 m 6 m 418 Mths Quest Generl Mths Preliminry Course

7 4 WE3 Find the length of eh shorter side in the right-ngled tringles elow, orret to 1 deiml ple m 6 m p 2.2 m 2.9 m 4.37 m t 2 m q Digitl do GC progrm Csio do-1624 Pythgors 5 In eh of the following right-ngled tringles, find the length of the side mrked with pronumerl, orret to 1 deiml ple m 8 m m 24.5 m z 4 m Digitl do GC progrm TI do-1625 Pythgors 33 mm 34 mm d m p m 6 WE4 Use the onverse of Pythgors theorem to determine if the following tringles re right ngled. 31 m 8 m 10 m 41 m 40 m 38 m 16 m 6 m 9 m 7 MC The otenuse in XYZ t right is: Z A XY B XZ C YZ D impossile to tell 8 MC Whih of the following tringles is right ngled? A B 5 m 24.5 m Y X 24.5 m 32.5 m 25 m 20.5 m C 24.5 m 84 m D 25.4 m 87.5 m 24.5 m 35.3 m 9 WE5 A television ntenn is 12 m high. To support it, wires re tthed to the ground 5 m from the foot of the ntenn. Find the length of eh wire. Chpter 13 Right-ngled tringles 419

8 10 Susie needs to len the guttering on her roof. She ples her ldder 1.2 m k from the edge of the guttering tht is 3 m ove the ground. How long will Susie s ldder need to e (orret to 2 deiml ples)? 11 A retngulr gte is 3.5 m long nd 1.3 m wide. The gte is to e strengthened y digonl re s shown t right. How long should the re e (orret to 2 deiml ples)? 12 A 2.5 m ldder lerns ginst rik wll. The foot of the ldder is 1.2 m from the foot of the wll. How high up the wll will the ldder reh (orret to 1 deiml ple)? 13 Use the mesurements in the digrm t right to find the height of the flgpole, orret to 1 deiml ple. 14 An isoseles, right-ngled tringle hs otenuse of 10 m. Clulte the length of the shorter sides. (Hint: Cll oth shorter sides.) 15 A lok of lnd ws surveyed nd the field digrm is shown. Drw sle digrm of the lok of lnd. Use Pythgors theorem to lulte the perimeter of the lok of lnd, orret to the nerest metre. Further development 16 An eroplne is flying t n ltitude of 5000 m when it is 4 km from the end of the runwy. Wht is the distne in metres from the eroplne to the runwy? 17 A gte is 2 m wide y 900 mm high. Wht is the length of digonl re? 18 Eh of the following figures hs sloping side whose length is unknown. Find the missing side length y first onstruting right-ngled tringle mm 16 mm 60 mm m m C B A 1.3 m 7.9 m 50 D d e f 12 m 28 m 6 m 16 m 24 m 19 Clulte the vlue of the pronumerls in eh of the following shpes m 920 m m 420 Mths Quest Generl Mths Preliminry Course

9 12.6 mm 9.6 mm d d B 10.6 mm 20 Consider the figure elow. Find the length of: AC F AD AE E d AF. A D 1 m B C Leve eh nswer in squre root form. 21 The heights of the uildings shown in the digrm t right re 40 metres nd 80 metres respetively. The uildings re 40 metres prt. Find the length of le tht would e needed to onnet the roofs of the two uildings. Clulting trigonometri rtios In the previous setion we looked t Pythgors theorem. This enled us to find the length of one side of right-ngled tringle given the length of the other two. To use Pythgors theorem, we hd to reognise the otenuse in right-ngled tringle. In trigonometry, we need to e le to nme the two shorter sides s well. We do this with referene to given ngle, nd lel them opposite nd djent. They re the sides opposite nd djent to the given ngle. The digrm shows the sides lelled with respet to the ngle, q. opposite otenuse djent Digitl do EXCEL Spredsheet do-1626 Tngent INVESTIGATE: Looking t the tngent rtio The tngent rtio is rtio of sides in similr rightngled tringles, suh s those in the digrm. BAC G I is ommon to eh tringle nd is equl to 30. We re E going to look t the rtio of the opposite side to the C djent side in eh tringle. You n do this either on your lultor or y ompleting the spredsheet Tngent from the Mths Quest Generl Mthemtis Preliminry Course eook. Complete eh of the following mesurements nd lultions. H F D B 1 BC = mm AB = mm BC AB = 2 DE = mm AD = mm DE AD = 3 FG = mm AF = mm FG AF = 4 HI = mm AH = mm HI AH = A Chpter 13 Right-ngled tringles 421

10 Rememer tht BAC is ommon to eh tringle. In eh of the ove, prt is the rtio of the opposite side to the djent side of BAC. Wht do you notie out eh of these nswers? Trigonometry uses the rtio of side lengths to lulte the lengths of sides nd the size of ngles. The rtio of the opposite side to the djent side is lled the tngent rtio. This rtio is fied for ny prtiulr ngle. The tngent rtio for ny ngle, q, n e found using the result: opposite side tn q = djent side In the investigtion on pge 421, we found tht for 30 ngle the rtio ws We n find more urte vlue for the tngent rtio on lultor y pressing nd entering 30. For ll lultions in trigonometry you will need to mke sure tht your lultor is in DEGREES MODE. For most lultors you n hek this y looking for DEG in the disply. When mesuring ngles: 1 degree = 60 minutes 1 minute = 60 seonds You need to e le to enter ngles using oth degrees nd minutes into your lu ltor. Most lultors use DMS (Degrees, Minutes, Seonds) utton or $ utton. Chek with your teher to see how to do this. Worked Emple 6 Using your lultor, find the following, orret to 3 deiml ples. 8 tn tn 75 d tn tn 69 THINK WRITE/DISPLAY Press nd enter 60. tn 60 = Enter 15, press * nd, enter tn 75 = Enter 8, press / nd, enter 69. Method 1 8 tn 69 = d Press, enter 49, press DMS, enter 32, press DMS. d tn = Method 2 1 From the MENU selet RUN. 2 To set the lultor up to ept degrees, minutes nd even seonds, press OPTN, 6 for more options, then 5 for ANGL. The sreen should pper s shown. 3 Now to find the trigonometri rtio, enter tn 49 nd press 4 for degrees; enter 32 nd press 4 for minutes; then press w. 422 Mths Quest Generl Mths Preliminry Course

11 The tngent rtio is used to solve prolems involving the opposite side nd the djent side of right-ngled tringle. The tngent rtio does not llow us to solve prolems tht involve the otenuse. The sine rtio (revited to sin) is the nme given to the rtio of the opposite side nd the otenuse. Digitl do EXCEL Spredsheet do-1627 Sine INVESTIGATE: Looking t the sine rtio The tngent rtio is the rtio of the opposite side nd the djent side in right-ngled tringle. The sine rtio is the rtio of the opposite side nd the otenuse. Look k to the right-ngled tringles used in the tngent investigtion on pge 421. Complete eh of the following mesurements nd lultions y using your lultor or the spredsheet Sine from the Mths Quest Generl Mthemtis Preliminry Course ebookplus. As we sw erlier, BAC is ommon to ll of these similr tringles, nd so in this eerise, we look t the rtio of the side opposite BAC to the otenuse of eh tringle. 1 BC = mm AC = mm BC AC = 2 DE = mm AE = mm DE AE = 3 FG = mm AG = mm FG AG = 4 HI = mm AI = mm HI AI = In this eerise, prt is the rtio of the opposite side to BAC to the otenuse. You should gin notie tht the nswers re the sme (or very lose, llowing for mesurement error). In ny right-ngled tringle with equl ngles, the rtio of the opposite side to the otenuse will remin the sme, regrdless of the size of the tringle. The formul for the sine rtio is: opposite side sin q = otenuse The vlue of the sine rtio for ny ngle is found using the sin funtion on the lu ltor. sin 30 = 0.5 Chek this on your lultor. WoRKED EAMPLE 7 Find, orret to 3 deiml ples: sin 57 9 sin sin 44 THINK d 9.6 sin WRITE/DISPLAY Press s nd enter 57. sin 57 = Enter 9, press * nd s, enter sin 45 = Enter 18, press / nd s, enter sin 44 = d Enter 9.6, press * nd s, enter 26, press DMS, enter 12, press DMS. d 9.6 sin = A third trigonometri rtio is the osine rtio. This rtio ompres the length of the djent side nd the otenuse. Chpter 13 Right-ngled tringles 423

12 Digitl do EXCEL Spredsheet do-1628 Cosine INVESTIGATE: Looking t the osine rtio Look k to the right-ngled tringles used in the tngent investigtion on pge 421. Complete eh of the following mesurements nd lultions. You my do so y using the spredsheet Cosine from the Mths Quest Generl Mthemtis Preliminry Course ebookplus. 1 AB = mm AC = mm AB AC = 2 AD = mm AE = mm AD AE = 3 AF = mm AG = mm AF AG = 4 AH = mm AI = mm AH AI = WoRKED EAMPLE 8 Agin for prt, you should get the sme nswer for eh tringle. In eh se, this is the osine rtio of the ommon ngle BAC. The osine rtio is found using the formul: djent side os q = otenuse To lulte the osine rtio for given ngle on your lultor, use the os fun tion. On your lultor hek the lultion: os 30 = Find, orret to 3 deiml ples: os 27 6 os os 74 THINK d 4.5 os WRITE/DISPLAY Press nd enter 27. os 27 = Enter 6, press * nd, enter os 55 = Enter 21.3, press / nd, enter os 74 = d Enter 4.5, press * nd, enter 82, press DMS, enter 46, press DMS. d 4.5 os = Similrly, if we re given the sin, os or tn of n ngle, we re le to lulte the size of tht ngle using the lultor. We do this using the inverse funtions. On most lultors these re the 2nd funtion of the sin, os nd tn funtions nd re denoted sin -1, os -1 nd tn -1. WoRKED EAMPLE 9 Find q, orret to the nerest degree, given tht sin q = THINK WRITE/DISPLAY 1 Press 2nd F [sin 1 ] nd enter Round your nswer to the nerest degree. q = Mths Quest Generl Mths Preliminry Course

13 So fr, we hve delt only with ngles tht re whole degrees. You need to e le to mke lultions using minutes s well. On most lultors, you will use the DMS (Degrees, Minutes, Seonds) funtion or the $ funtion. Worked Emple 10 Given tht tn q = 1.647, lulte q to the nerest minute. THINK Method 1 1 Press 2nd F [tn 1 ] nd enter Convert your nswer to degrees nd minutes y pressing DMS. Method 2 1 From the MENU selet RUN. WRITE/DISPLAY q = As with sientifi lultor, press! [tn -1 ] nd enter 1.647, then press w. 3 Disply the ngle options y pressing K, 6 for more hoies, nd then 5 for ANGL. 4 The funtion for getting the nswer displyed in degrees, minutes nd seonds is essed y pressing 5. REMEMBER 1. The tngent rtio is the rtio of the opposite side nd the djent side. opposite side tn q = djent side 2. The sine rtio is the rtio of the opposite side nd the otenuse. opposite side sin q = otenuse 3. The osine rtio is the rtio of the djent side nd the otenuse. djent side os q = otenuse 4. The vlue of the trigonometri rtios n e found using the sin, os nd tn funtions on your lultor. 5. The ngle n e found when given the trigonometri rtio using the sin -1, os -1 nd tn -1 funtions on your lultor. Chpter 13 Right-ngled tringles 425

14 EERCISE 13B Digitl do SkillSHEET 13.3 do-1629 Rounding to given numer of deiml ples Clulting trigonometri rtios 1 WE6 Clulte the vlue of eh of the following, orret to 3 deiml ples. 8.6 tn 57 9 tn 63 d tn tn 12 2 WE7 Clulte the vlue of eh of the following, orret to 3 deiml ples. sin sin sin 72 3 WE8 Clulte the vlue of eh of the following, orret to 3 deiml ples. os os 9 6 os 24 4 Clulte the vlue of eh of the following, orret to 4 signifint figures. sin 30 os 15 tn 45 d 48 tn 85 e 128 os 60 f 9.35 sin g h i os 32 tn 20 sin 72 5 Clulte the vlue of eh of the following, orret to 2 deiml ples. sin tn os d 48 sin d 5.9 os 2 3 d 9 os e 4.9 sin f 2.39 tn g h i tn os sin WE9 Find q, orret to the nerest degree, given tht sin q = Find q, orret to the nerest degree, given tht: sin q = os q = tn q = WE10 Find q, orret to the nerest minute, given tht os q = Find q, orret to the nerest minute, given tht: tn q = os q = sin q = Further development 10 Find the vlue of eh of the following trigonometri rtio pirs. Give your nswers orret to four deiml ples. sin 40, os 50 sin 70, os 20 sin 13, os 77 d sin 84, os 6 11 Wht did you notie out the reltionship etween sin nd os in question 10? Use this to omplete eh of the following. sin 30 = os os 75 = sin sin 28 = d os 45 = sin 12 Find: sin 23 sin 23 os 23 os Find: d tn 23. sin 67 os 67 (sin 67 ) 2 + (os 67 ) Use your nswer to question 13 to find (sin 34 ) 2 + (os 34 ) 2. Chek your nswer with your lultor. 15 Fred tries to solve sin q = 1.2 on his lultor, however n error sttement is returned. Eplin why there is no solution to this question. Wht is the only trigonometri rtio tht n possily equl 1.2? 426 Mths Quest Generl Mths Preliminry Course

15 13C Finding n unknown side We n use the trigonometri rtios to find the length of one side of right-ngled tringle if we know the length of nother side nd n ngle. Consider the tringle t right. In this tringle we re sked to find the length of the opposite side nd hve een given the length of the djent side. We know from the formul tht: tn q = opposite. In this emple, tn 30 =. From our djent 14 lultor we know tht tn 30 = We n set up n eqution tht will llow us to find the vlue of. tn q = opp dj tn 30 = 14 = 14 tn 30 = m m dj opp Worked Emple 11 Use the tngent rtio to find the vlue of h in the tringle t right, orret to 2 deiml ples. Think 1 Lel the sides of the tringle opp, dj nd. Write m h m dj 2 Write the tngent formul. tn q = opp dj h opp 3 Sustitute for q (55 ) nd the djent side (17 m). tn 55 = h 17 4 Mke h the sujet of the eqution. h = 17 tn 55 5 Clulte. = m In the emple ove, we were told to use the tngent rtio. In prtie, we need to e le to look t prolem nd then deide if the solution is found using the sin, os or tn rtio. To do this we need to emine the three formuls. opposite side tn q = djent side We use the tn rtio when we re finding either the length of the opposite or djent side nd re given the length of the other. opposite side sin q = otenuse The sin rtio is used when we re finding the length of the opposite side or the otenuse nd re given the length of the other. os q = djent side otenuse Chpter 13 Right-ngled tringles 427

16 The os rtio is for prolems where we re finding the length of the djent side or the otenuse nd re given the length of the other. To mke the deision we need to lel the sides of the tringle nd mke deision sed on these lels. WoRKED EAMPLE 12 Find the length of the side mrked, orret to 2 deiml ples. THINK WRITE 1 Lel the sides of the tringle. 24 m 24 m 50 opp ebook plus s Tutoril int-2337 Worked emple 12 2 is the opposite side nd 24 m is the otenuse, therefore use the sin formul. 50 dj sin q = opp dj 3 Sustitute for q nd the otenuse. sin 50 = 24 Method 1 4 Mke the sujet of the eqution. = 24 sin 50 5 Clulte. = m Method 2 1 From the MENU selet EQUA. 2 Press 3 for Solver. 3 Delete ny eisting eqution nd enter sin 50 = y pressing sin 50! = X 24 w. 24 Do not worry out different vlue of X in the disply t this stge s it is previously stored vlue. 4 Press 6 for SOLV to solve this eqution. 428 Mths Quest Generl Mths Preliminry Course

17 Intertivity int-2405 SOHCAHTOA To rememer eh of the formuls more esily, we n use this ronym: SOHCAHTOA We pronoune this ronym s Sok toe her. The initils of the ronym represent the three trigonometri formuls. sin q = opp os q = dj tn q = opp dj Cre needs to e tken t the sustitution stge. In the ove emples, the unknown side ws the numertor in the frtion, hene we multiplied to find the nswer. If fter sustitution, the unknown side is in the denomintor, the finl step is done y division. WoRKED EAMPLE 13 Find the length of the side mrked z in the tringle t right ' z THINK WRITE 12.5 m 1 Lel the sides opp, dj nd ' z opp 12.5 m dj 2 Choose the osine rtio euse we re finding the otenuse nd hve een given the djent side. 3 Write the formul. os q = dj 4 Sustitute for q nd the djent side. os = 12.5 z 5 Mke z the sujet of the eqution. z os = z = os Clulte. = m If you re using the grphis lultor to solve n eqution tht involves the use of degrees nd minutes, the ngle needs to e entered s frtion. In worked emple 13 ove, os needs to e entered s os s the eqution solver does not llow ess to the degrees, 60 minutes nd seonds funtion. Trigonometry is used to solve mny prtil prolems. In these ses, it is neessry to drw digrm to represent the prolem nd then use trigonometry to solve the prolem. With written prolems tht require you to drw the digrm, it is neessry to give the nswer in words. WoRKED EAMPLE 14 A flying fo is used in n rmy trining mp. The flying fo is supported y le tht runs from the top of liff fe to point 100 m from the se of the liff. The le mkes 15 ngle with the horizontl. Find the length of the le used to support the flying fo. Tutoril int-2338 Worked emple 14 Chpter 13 Right-ngled tringles 429

18 THINK WRITE 1 Drw digrm nd show informtion. 2 Lel the sides of the tringle opp, dj nd. f m 3 Choose the osine rtio euse we re finding the otenuse nd hve een given the djent side. 4 Write the formul. os q = dj 5 Sustitute for q nd the djent side. os 15 = 100 f 6 Mke f the sujet of the eqution. f os 15 = f = os 15 7 Clulte. = m 8 Give written nswer. The le is pproimtely m long. REMEMBER 1. Trigonometry n e used to find the length of side in right-ngled tringle when we re given the length of one side nd the size of n ngle. 2. The trigonometri formuls re: sin q = opp os q = dj tn q = opp dj 3. Tke re to hoose the orret trigonometri rtio for eh question. 4. Sustitute refully nd note the hnge in the lultion, depending upon whether the unknown side is in the numertor or denomintor. 5. Before using your lultor, hek tht it is in degrees mode. 6. Be sure tht you know how to enter degrees nd minutes into your lultor. 7. Worded prolems will require you to drw digrm nd give written nswer. EERCISE 13C Digitl do SkillSHEET 13.4 do-1630 Solving equtions of Finding n unknown side 1 Lel the sides of eh of the following tringles, with respet to the ngle mrked with the pronumerl. α γ the type = to find 430 Mths Quest Generl Mths Preliminry Course

19 2 WE 11 Use the tngent rtio to find the length of the side mrked (orret to 1 deiml ple). Digitl do SkillSHEET 13.5 do-1631 Solving equtions of mm the type = to find 3 Use the sine rtio to find the length of the side mrked (orret to 2 deiml ples). 13 m 23 Digitl do SkillSHEET 13.6 do-1632 Rerrnging formuls 4 Use the osine rtio to find the length of the side mrked d (orret to 3 signifint figures). 5 WE 12 The following questions use the tn, sin or os rtios in their solution. Find the size of the side mrked with the pronumerl, orret to 3 signifint figures. 13 m 48 m km 35 m 31 d 68 y 41 z 6 WE 13 Find the length of the side mrked with the pronumerl in eh of the following (orret to 1 deiml ple). 21 t p 87 mm 77 q m 4.8 m 7 Find the length of the side mrked with the pronumerl in eh of the following (orret to 3 signifint figures) km 8.5 km m d m 116 mm 9 e 11 d m f f m 13 Chpter 13 Right-ngled tringles 431

20 g 44.3 m 83 h g km i 84 9' m 2.34 m j q 60 32' k t 75 19' 21.4 m l r 84.6 km 26.8 m 29 32' 8 MC Look t the digrm t right nd stte whih of the following is orret. 9.2 A = 9.2 sin 69 B = sin 69 C = 9.2 os 69 D = 9.2 os 69 9 MC Study the tringle t right nd stte whih of the following is orret. A tn φ = 8 15 C sin φ = B tn φ = 15 8 D os φ = MC Whih of the sttements elow is not orret? A The vlue of tn q n never e greter thn 1. B The vlue of sin q n never e greter thn 1. C The vlue of os q n never e greter thn 1. D tn 45 = φ MC Study the digrm t right nd stte whih of the sttements is orret. A w = 22 os B w = sin 36 C w = 22 os 54 D w = 22 sin 54 w 22 mm WE 14 A tree sts 3.6 m shdow when the sun s ngle of elevtion is 59. Clulte the height of the tree, orret to the nerest metre. 13 A 10 m ldder just rehes to the top of wll when it is lening t 65 to the ground. How fr from the foot of the wll is the ldder (orret to 1 deiml ple)? 14 The digrm t right shows the pths of two ships, A nd B, fter they hve left port. If ship B sends distress signl, how fr must ship A sil to give ssistne (to the nerest kilometre)? 15 A retngle 13.5 m wide hs digonl tht mkes 24 ngle with the horizontl. Drw digrm of this sitution. Clulte the width of the retngle, orret to 1 deiml ple. Port km A B 432 Mths Quest Generl Mths Preliminry Course

21 16 A wooden gte hs digonl re uilt in for support. The gte stnds 1.4 m high nd the digonl mkes 60 ngle with the horizontl. Drw digrm of the gte. Clulte the length tht the digonl re needs to e. 17 The wire support for flgpole mkes 70 ngle with the ground. If the support is 3.3 m from the se of the flgpole, lulte the length of the wire support (orret to 2 deiml ples). 18 A ship drops nhor vertilly with n nhor line 60 m long. After one hour the nhor line mkes 15 ngle with the vertil. Drw digrm of this sitution. Clulte the depth of wter, orret to the nerest metre. Clulte the distne tht the ship hs drifted, orret to 1 deiml ple. Further development 19 Find the length of the side mrked in the tringle t right. 5 m 3 m In the digrm t right find the size of ngle (to the nerest degree) nd the side lengths nd y orret to one deiml ple. 8 m y 21 Find the vlue of the unknown sides in eh of the following: m 20 m m 6.5 m 35 Chpter 13 Right-ngled tringles 433

22 Digitl do WorkSHEET 13.1 do D 22 A ot is moored in lm wter with depth of 29 m. If the nhor line mkes 28 ngle with the surfe of the wter, wht is the length of the nhor line? 23 A person hopes to swim ross river of width 43 metres. When she swims ross the urrent pushes her downstrem to point suh tht the swim line mkes 67 ngle with the river nk. Clulte: how fr she swm how fr she finished from the plnned finishing point. 24 Peter sys tht whenever finding the otenuse of right-ngled tringle, the solution will involve dividing y the trigonometri rtio. Is Peter orret? Eplin your nswer. Finding ngles In this hpter so fr, we hve onerned ourselves with finding side lengths. We re lso le to use trigonometry to find the sizes of ngles when we hve een given side lengths. We need to reverse our previous proesses. Consider the tringle t right. We wnt to find the size of the ngle mrked q. Using the formul sin q = opp we know tht in this tringle sin q = m 5 m = 1 2 = 0.5 We then lulte sin -1 (0.5) to find tht q = 30. As with ll trigonometry it is importnt tht you hve your lultor set to degrees mode. WoRKED EAMPLE 15 Find the size of ngle q, orret to the nerest degree, in the tringle t right. 6.5 m 4.3 m Tutoril int-2339 Worked emple 15 THINK WRITE Method 1 1 Lel the sides of the tringle nd hoose the tn rtio. 2 Sustitute for the opposite nd djent sides in the tringle nd simplify. 6.5 dj tn q = opp dj = = Mke q the sujet of the eqution. q = tn -1 (0.6615) 4 Clulte. = 33 (to the nerest degree) 4.3 opp 434 Mths Quest Generl Mths Preliminry Course

23 Method 2 1 From the MENU selet EQUA. 2 Press 3 for Solver. 3 Delete ny eisting eqution nd enter tn = y pressing tn X! [=] 4.3 / 6.5 w. 4 Press 6 for SOLV to solve this eqution. In mny ses we will need to lulte the size of n ngle, orret to the nerest minute. The sme method for finding the solution is used; however, you will need to use your lultor to onvert to degrees nd minutes. WoRKED EAMPLE 16 Find the size of the ngle q, orret to the nerest minute. 4.6 m THINK Method 1 WRITE 7.1 m 1 Lel the sides of the tringle nd hoose the sin rtio. opp 4.6 m dj 7.1 m 2 Sustitute for the opposite side nd djent in the tringle nd simplify. sin q = opp = = Mke q the sujet of the eqution. q = sin -1 (0.6479) 4 Clulte nd onvert your nswer to degrees nd = (to the nerest minute) minutes. Chpter 13 Right-ngled tringles 435

24 Method 2 1 Solve the eqution s shown previously, whih gives n nswer in degrees s deiml. 2 Press m nd selet RUN. 3 Press X w to rell the vlue of X from the eqution. 4 To ess the ngle funtions, press K 6, 5 for ANGL, nd 5 gin to onvert to degrees, minutes nd seonds. The sme methods n e used to solve prolems. As with finding sides, we set the question up y drwing digrm of the sitution. WoRKED EAMPLE 17 A ldder is lent ginst wll. The foot of the ldder is 4 m from the se of the wll nd the ldder rehes 10 m up the wll. Clulte the ngle tht the ldder mkes with the ground. THINK WRITE 1 Drw digrm nd lel the sides. opp 10 m 4 m dj 2 Choose the tngent rtio nd write the formul. tn q = opp dj 3 Sustitute for the opposite nd djent side, then simplify. = 10 4 = Mke q the sujet of the eqution. q = tn -1 (2.5) 5 Clulte. = Give written nswer. The ldder mkes n ngle of with the ground. 436 Mths Quest Generl Mths Preliminry Course

25 REMEMBER 1. Mke sure tht the lultor is in degrees mode. 2. To find n ngle given the trigonometri rtio, press! nd then the pproprite rtio utton. 3. Be sure to know how to get your lultor to disply n nswer in degrees nd minutes. When rounding off minutes, hek if the numer of seonds is greter thn When solving tringles rememer the SOHCAHTOA rule to hoose the orret formul. 5. In worded prolems drw digrm nd give n nswer written in words. EERCISE 13D Digitl do SkillSHEET 13.7 do-1634 Rounding ngles to the nerest degree Digitl do SkillSHEET 13.8 do-1635 Rounding ngles to the nerest minute Digitl do SkillSHEET 13.9 do-1636 Rounding ngles to the nerest seond Finding ngles 1 In eh of the following, use the tngent rtio to find the size of the ngle mrked with the pronumerl, orret to the nerest degree. 25 mm γ 162 mm 7 m 11 m 12 m φ 3 m 2 In eh of the following, use the sine rtio to find the size of the ngle mrked with the pronumerl, orret to the nerest minute. 4.6 m 24 m 13 m α 9.7 km 6.5 m 5.6 km 3 In eh of the following, use the osine rtio to find the size of the ngle mrked with the pronumerl, orret to the nerest minute. 15 m 9 m 4.6 m 2.6 m 27.8 m β 19.5 m 4 WE15 In the following tringles, you will need to use ll three trigonometri rtios. Find the size of the ngle mrked q, orret to the nerest degree. 14 m 11 m 15 m 9 m 7 m 8 m Chpter 13 Right-ngled tringles 437

26 d 9.2 m 3.6 m e 196 mm 32 mm f 26.8 m 14.9 m 5 WE 16 In eh of the following, find the size of the ngle mrked q, orret to the nerest minute. 2.5 m 0.6 m 30 m 63 m d 19.2 m 3.5 m 18.5 m e 10 m 8.3 m 16.3 m f 18.9 m 6.3 m 6 MC Look t the tringle drwn t right. Whih of the sttements elow is orret? A ABC = 30 B ABC = 60 C CAB = 30 D ABC = 45 7 MC Consider the tringle drwn t right. q is losest to: A B C 48 4 D MC The et vlue of sin q =. The ngle q = 2 A 30 B 45 C 60 D 90 9 WE 17 A 10 m ldder lens ginst 6 m high wll. Find the ngle tht the ldder mkes with the horizontl, orret to the nerest degree. 5 m 10 A kite is flying on 40 m string. The kite is flying 10 m wy from the vertil s shown in the figure t right. Find the ngle the string mkes with the horizontl, orret to the nerest minute. 11 A ship s ompss shows ourse due est of the port from whih it sils. After siling 10 nutil miles, it is found tht the ship is 1.5 nutil miles off ourse s shown in the figure elow. A C 10 m 9.3 m 40 m 10 m B 12.5 m kite 10 nm 1.5 nm Find the error in the ompss reding, or ret to the nerest minute. 7 m 12 The digrm t right shows footller s shot t gol. By dividing the isoseles tringle in hlf, lulte, to the nerest degree, the ngle within whih the footller must kik to get the ll to go etween the posts. 30 m 438 Mths Quest Generl Mths Preliminry Course

27 13 A golfer hits the ll 250 m, ut 20 m off entre. Clulte the ngle t whih the ll devited from stright line, orret to the nerest minute. Further development 14 MC The figure elow shows BMX iyle rmp. All mesurements re shown in metres The orret epression for the ngle of elevtion,, of the rmp is: sin os tn d os E 15 MC A flgpole tht is 2 metres tll sts shdow tht is 0.6 metres long. The ngle of the sun to the ground is: d A jvelin tht is 1.95 m long is thrown nd stiks 20 m into the ground. Given tht the sun is diretly overhed nd tht the jvelin sts 90 m shdow, find the ngle tht the jvelin mkes with the ground. 17 A hot ir lloon is hovering in strong winds 10 vertil metres ove the ground. The lloon is eing held in ple y rope tht is 15 m long. Wht ngle does the rope mke with the ground? 18 A le r follows diret line from mountin pek (ltitude 1250 m) to ridge (ltitude 840 m). If the horizontl distne etween the pek nd the ridge is 430 m, lulte the ngle through whih the le r desends. 19 A rmp joins two points 1.2 metres prt. One point is 25 m higher thn the other. Find the length of the rmp. Find the ngle of inlintion of the rmp. Angles of elevtion nd depression The ngle of elevtion is mesured upwrds from horizontl nd refers to the ngle t whih we need to look up to see n ojet. Similrly, the ngle of depression is the ngle t whih we need to look down from the horizontl to see n ojet. We re le to use the ngles of elevtion nd depression to lulte the heights nd distnes of ojets tht would otherwise e diffiult to mesure. Chpter 13 Right-ngled tringles 439

28 Worked Emple 18 From point 50 m from the foot of uilding, the ngle of elevtion to the top of the uilding is mesured s 40. Clulte the height, h, of the uilding, orret to the nerest metre. THINK 1 Lel the sides of the tringle opp, dj nd. 2 Choose the tngent rtio euse we re finding the length of the opposite side nd hve een given the length of the djent side. WRITE m dj 3 Write the formul. tn q = opp dj h opp m h 4 Sustitute for q nd the djent side. tn 40 = h 50 5 Mke h the sujet of the eqution. h = 50 tn 40 6 Clulte. = 42 m 7 Give written nswer. The height of the uilding is pproimtely 42 m. In prtil situtions, the ngle of elevtion is mesured using linometer. Therefore, the ngle of elevtion is mesured from person s height t eye level. For this reson, the height t eye level must e dded to the lulted nswer. Worked Emple 19 Bryn mesures the ngle of elevtion to the top of tree s 64, from point 10 m from the foot of the tree. If the height of Bryn s eyes is 1.6 m, lulte the height of the tree, orret to 1 deiml ple. THINK 1 Lel the sides opp, dj nd. 2 Choose the tngent rtio euse we re finding the length of the opposite side nd hve een given the length of the djent side. WRITE h opp 10 m dj 3 Write the formul. tn q = opp dj m 1.6 m 4 Sustitute for q nd the djent side. tn 64 = h 10 5 Mke h the sujet of the eqution. h = 10 tn 64 6 Clulte h. = 20.5 m 7 Add the eye height = Give written nswer. The height of the uilding is pproimtely 22.1 m. 440 Mths Quest Generl Mths Preliminry Course

29 A similr method for finding the solution is used for prolems tht involve n ngle of depression. WoRKED EAMPLE 20 When n eroplne is 2 km from runwy, the ngle of depression to the runwy is 10. Clulte the ltitude of the eroplne, orret to the nerest metre. h 2 km 10 ebook plus s Tutoril int-2340 Worked emple 20 THINK WRITE 1 Lel the sides of the tringle opp, dj nd. 2 Choose the tn rtio, euse we re finding the length of the opposite side given the length of the djent side. opp h dj 2 km 3 Write the formul. tn q = opp dj 4 Sustitute for q nd the djent side, h tn 10 = onverting 2 km to metres Mke h the sujet of the eqution. h = 2000 tn 10 6 Clulte. = 353 m 7 Give written nswer. The ltitude of the eroplne is pproimtely 353 m. 10 Angles of elevtion nd depression n lso e lulted y using known mesure ments. This is done y drwing right-ngled tringle to represent sitution. WoRKED EAMPLE 21 A 5.2 m uilding sts 3.6 m shdow. Clulte the ngle of elevtion of the sun, orret to the nerest degree. 5.2 m 3.6 m THINK WRITE 1 Lel the sides opp, dj nd. 2 Choose the tn rtio euse we re given the length of the opposite nd djent sides. opp 5.2 m 3.6 m dj Chpter 13 Right-ngled tringles 441

30 3 Write the formul. tn q = opp dj 4 Sustitute for opposite nd djent. tn q = Mke q the sujet of the eqution. q = tn Clulte. = 55 7 Give written nswer. The ngle of elevtion of the sun is pproimtely REMEMBER 1. The ngle of elevtion is the ngle t whih you look up to see n ojet. 2. The ngle of depression is the ngle t whih you look down to see n ojet. 3. Prolems n e solved y using ngles of elevtion nd depression with the id of digrm. 4. Worded prolems should e given n nswer written in words. To pture the top of the uilding in this photo the photogrpher hd to tilt the mer upwrds, hene, inrese the ngle of elevtion. 442 Mths Quest Generl Mths Preliminry Course

31 Eerise 13E Angles of elevtion nd depression 1 WE 18 From point 100 m from the foot of uilding, the ngle of elevtion to the top of the uilding is 15. Clulte the height of the uilding, orret to 1 deiml ple m 2 The ngle of elevtion from ship to n eroplne is 60. The eroplne is 2300 m due north of the ship. Clulte the ltitude of the eroplne, orret to the nerest metre m 3 From point out to se, ship sights the top of lighthouse t n ngle of elevtion of 12. It is known tht the top of the lighthouse is 40 m ove se level. Clulte the distne of the ship from the lighthouse, orret to the nerest 10 m. 4 WE 19 From point 50 m from the foot of uilding, Rod sights the top of uilding t n ngle of elevtion of 37. Given tht Rod s eyes re t height of 1.5 m, lulte the height of the uilding, orret to 1 deiml ple m 1.5 m m 1.8 m m 5 Rihrd is flying kite nd sights the kite t n ngle of elevtion of 65. The ltitude of the kite is 40 m nd Rihrd s eyes re t height of 1.8 m. Clulte the length of string the kite is flying on, orret to 1 deiml ple. 6 WE 20 Bettin is stnding on top of liff, 70 m ove se level. She looks diretly out to se nd sights ship t n ngle of depression of 35. Clulte the distne of the ship from shore, to the nerest metre. 7 From n eroplne flying t n ltitude of 4000 m, the runwy is sighted t n ngle of depression of 15. Clulte the distne of the eroplne from the runwy, orret to the nerest kilometre. 8 There is fire on the fifth floor of uilding. The losest fire truk n get to the uilding is 10 m. The ngle of elevtion from this point to where people need to e resued is 69. If the fire truk hs 30 m ldder, n the ldder e used to mke the resue? 9 From nvy vessel, eon whih is 80 m ove se level is sighted t n ngle of elevtion of 5. The vessel siled towrds the eon nd thirty minutes lter the eon is t n ngle of elevtion of 60. Use the digrm on the right to omplete the following. Clulte the distne tht the vessel ws from the eon when the ngle of elevtion to the eon ws 5 (the distne AC). Clulte the distne tht the vessel siled in the 30 minutes etween the two redings. A 70 m m 5 60 B 4000 m D 80 m C Chpter 13 Right-ngled tringles 443

32 10 WE 21 A 12 m high uilding sts shdow 15 m long. Clulte the ngle of elevtion of the sun, to the nerest degree. 12 m 15 m 11 An eroplne tht is t n ltitude of 1500 m is 4000 m from ship in horizontl diretion, s shown elow. Clulte the ngle of depression from the eroplne to the ship, to the nerest degree m 1500 m Digitl do WorkSHEET 13.2 do The ngle of elevtion to the top of tower is 12 from point 400 m from the foot of the tower. Drw digrm of this sitution. Clulte the height of the tower, orret to 1 deiml ple. Clulte the ngle of elevtion to the top of the tower from point 100 m from the foot of the tower. Further development 13 A lifesver sits in tower 2 m ove se level. He sees swimmer hving diffiulty t n ngle of depression of 12. How fr is the swimmer from the tower? 14 From the top of lookout 50 m ove the ground, the ngle of depression to mpsite is 37. How fr is the mp from the se of the lookout? 15 A heliopter hovers 1800 metres ove the ground. The ngle of depression to two lost ushwlkers is 40 nd 60 respetively. Clulte the distne etween the two ushwlkers. 16 A ship sights lighthouse t n ngle of elevtion of 50. The lighthouse is 50 m ove se level. Find the distne of the ship from the lighthouse. After drifting 200 m wy from shore find the ngle of elevtion tht the lighthouse is now t. 17 Two uildings 50 m nd 75 m tll re seprted y 70 m. Find the ngle of elevtion from the top of the shorter uilding to the top of the tller uilding. 18 Wyne sys tht the ngle of elevtion from A to B will e equl to the ngle of depression from B to A. Is Wyne orret? Eplin your nswer. INVESTIGATE: Clultion of heights To mesure the heights of trees nd uildings round your shool, try the following. 1 Mesure your height t eye level. 2 Tke linometer nd from point mesure the ngle of elevtion to the top of the tree or uilding. 3 Mesure your distne from the foot of the tree or uilding. 4 Use trigonometry to lulte the height, rememering to dd your height t eye level to the result of the lultion. 444 Mths Quest Generl Mths Preliminry Course

33 Proportionl digrms In mny ses, we need only n pproimte 4 km mesurement for prtil prolem. This type of nswer n e otined y drwing sle digrm. Consider the sitution where hiker wlks 6 km due north, turns nd wlks 4 km due est. By drwing digrm using sle of 1 m = 1 km, we n otin n pproimte mesurement for the 6 km distne the hiker is from the strting point. By mesurement, we n see tht the distne the hiker is from the strting point is pproimtely 7.2 km. With protrtor, we n drw sle digrm to solve prolems involving ngles. Suppose tht the ngle of elevtion to the top of tree is 40 from point 12 m from the foot of the tree. By mesurement, the height of the tree is pproimtely 10 m. h In mny situtions quik hek of the ury of n nswer is useful nd n e mde y using sle 40 drwing. In suh ses the drwing would need to e 12 m only pproimtely to sle. Suppose tht you were told tht the ngle of depression from the top of 50 m liff to ship out to se ws 15. You were then told tht this ship is 1 km from shore m Using this digrm, we would estimte tht the ship is only 190 m from shore. Suh digrm is useful hek to lultion. INVESTIGATE: Cheking with proportionl digrm Drw digrms roughly to sle to hek the results to the previous investigtion. Suh digrms re used to develop r rlly ourses, ross-ountry running ourses nd orienteering events. INVESTIGATE: using proportionl digrms Pln trk for ross-ountry run or orienteering event round your shool. 1 Mesure the length of eh leg nd the ngle involved in eh turn. 2 On sle digrm, drw the ourse. 3 By mesuring your digrm, lulte the pproimte length of the ourse. Chpter 13 Right-ngled tringles 445

34 Summry Pythgors theorem When finding the otenuse of right-ngled tringle, use the formul: 2 = To find shorter side of right-ngled tringle use: Clulting trigonometri rtios tn q = opp dj sin q = opp 2 = 2-2 or 2 = 2-2. os q = dj SOHCAHTOA this ronym will help you rememer trigonometri formuls. Finding n unknown side Lel the sides of the tringle opposite, djent nd otenuse. Choose the orret rtio. Sustitute given informtion. Mke the unknown side the sujet of the eqution. Clulte. Finding ngles Lel the sides of the tringle opposite, djent nd otenuse. Choose the orret rtio. Sustitute given informtion. Mke the unknown ngle the sujet of the eqution. Clulte y using the inverse trigonometri funtions. Angles of elevtion nd depression The ngle of elevtion is the ngle we look up from the horizontl to see n ojet. The ngle of depression is the ngle we look down from the horizontl to see n ojet. Prolems re solved using ngles of elevtion nd depression y the sme methods s for ll right-ngled tringles. Proportionl digrms A sle digrm n e drwn to otin resonle estimte of distne or ngle. A digrm tht is drwn roughly to sle n e used to hek tht n nswer is resonly urte. 446 Mths Quest Generl Mths Preliminry Course

35 CHAPTER REVIEW MuLTIPLE CHoICE 1 MC Whih of the tringles drwn elow is not right ngled? A 13 m B 5 m 5 m 3 m C 7 m 12 m 19 m 9 m 2 MC Look t the tringle elow. 9 m 40 m 41 m D 8 m 4 m 15 m 17 m Sttement 1. os q = 9 41 Sttement 2. tn q = 9 40 Whih of the ove sttements is true? A 1 only B 2 only C oth 1 nd 2 D neither sttement 3 MC Whih of the following sttements is orret? A os 30 = tn 60 B os 30 = sin 60 C os 30 = sin 30 D os 60 = sin MC The et vlue of os q = 2. The ngle q = A 30 B 45 C 60 D 90 SHoRT ANSWER 1 Find the length of the side mrked with pronumerl, in eh se writing your nswer orret to 2 deiml ples. 9.2 m n m 9.2 m 26 m 32 m d e f 4.8 m 3.2 m 7.25 m 0.6 m 1.9 km r 2.4 m p t q m 1.3 km 2 To trvel etween the towns of Bolong nd Molong, you need to trvel west long rod for 45 km, then north long nother rod for nother 87 km. Clulte the stright-line distne etween the two towns. 3 A rope is 80 m long nd runs from liff top to the ground, 45 m from the se of the liff. Clulte the height of the liff, to the nerest metre. 4 Clulte eh of the following, orret to 4 deiml ples. sin 46 tn os 56 d 8.9 sin e f os 75 tn Clulte q, orret to the nerest degree, given tht: os q = tn q = 1.23 sin q = Clulte q, orret to the nerest minute, given tht: os q = tn q = 0.5 sin q = Chpter 13 Right-ngled tringles 447

36 7 Find the length of eh side mrked with pronumerl, orret to 1 deiml ple. 6 m i z 83 30' 138 mm 9 q j 4.32 m 3.9 m 29 51' j d e f g h t m m 12.6 m 22 q 7.8 m 2.9 m n 6.8 m 65 g 26 42' k l 38.5 m 16 8' 63 km 85 12' m 8 A rope tht is used to support flgpole mkes n ngle of 70 with the ground. If the rope is tied down 3.1 m from the foot of the flgpole, find the height of the flgpole, orret to 1 deiml ple. 9 A dirt trk runs off rod t n ngle of 34 to the rod. If I trvel for 4.5 km long the dirt trk, wht is the shortest distne k to the rod (orret to 1 deiml ple)? 10 A fire is urning in uilding nd people need to e resued. The fire rigde s ldder must reh height of 60 m nd must e ngled t 70 to the horizontl. How long must the ldder e to omplete the resue? 11 Find the size of the ngle mrked q in eh of the following, giving your nswer orret to the nerest degree. 2.3 m 4.6 m k 16 m 19 m 4.8 m h 77 18' 43 m 116 m 448 Mths Quest Generl Mths Preliminry Course

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