SAMPLE. Trigonometry. Naming the sides of a right-angled triangle

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1 H P T E R 7 Trigonometry How re sin, os nd tn defined using right-ngled tringle? How n the trigonometri rtios e used to find the side lengths or ngles in right-ngled tringles? Wht is ment y n ngle of elevtion or n ngle of depression? How re ompss erings nd true erings mesured? How n the sine nd osine rules e used to solve non-right-ngled tringles? Wht re the three rules tht n e used to find the re of tringle? Trigonometry n e used to solve mny prtil prolems. How high is tht tree? Wht is the height of the mountin we n see in the distne? Wht is the et lotion of the fire tht hs just een seen y fire spotters? How wide is the lke? Wht is the re of this irregulr-shped pddok? 7.1 Trigonometry sis lthough you re likely to hve studied some trigonometry, it my e helpful to review few si ides. Nming the sides of right-ngled tringle The hypotenuse is the longest side of the right-ngled tringle nd is lwys opposite the right ngle (90 ). Hypotenuse The opposite side is diretly opposite the ngle. The djent side is eside the ngle, utit is not djent the hypotenuse. It runs from to the right ngle. SMPLE Opposite The opposite nd djent sides re loted in reltion to the position of ngle. If ws in the other orner, the sides would hve to swp their lels. The letter is the Greek letter thet. Itisommonly used to lel n ngle. 268 mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

2 hpter 7 Trigonometry 269 Emple 1 Identifying the sides of right-ngled tringle Give the vlues of the hypotenuse, the opposite side nd the djent side in the tringle shown. Solution The hypotenuse is opposite the right ngle. The hypotenuse, h = 5 The opposite side is opposite the ngle. The opposite side, o = 3 The djent side is eside,ut is not the hypotenuse. The djent side, = 4 The trigonometri rtios The trigonometri rtios sin, os nd tn n e defined in terms of the sides of right-ngled tringle. Hypotenuse h opposite sin = hypotenuse sin = o h Opposite o Hypotenuse h djent djent os = hypotenuse os = h djent opposite tn = djent o tn = s oh h to This mnemoni is often used y students to help them rememer the rule for eh trigonometri rtio. Or you my prefer: Sir Oliver s Horse me mling Home To Oliver s rms The mening of the trigonometri rtios Opposite o Using lultor, we find, for emple, tht sin 30 = 0.5. This mens tht in ll right-ngled tringles with n ngle of 30, the rtio of the side opposite the 30 to the hypotenuse is lwys 0.5. SMPLE opposite 1 hypotenuse = 2 = 0.5 opposite 2 hypotenuse = 4 = 0.5 opposite 3 hypotenuse = 6 = mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

3 270 Essentil Stndrd Generl Mthemtis Try drwing ny right-ngled tringle with n ngle of 30 nd hek tht the rtio opposite hypotenuse = 0.5 Similrly, for ny right-ngled tringle with n ngle of 30 the rtios os 30 nd tn 30 lwys hve the sme vlues: os 30 = djent 3 is lwys = (to 4 deiml ples) hypotenuse 2 tn 30 = opposite djent is lwys 1 3 = (to 4 deiml ples). lultor gives the vlue of eh trigonometri rtio for ny ngle entered. TI-Nspire S tip When solving prolems in trigonometry, your lultor should e kept in Degree mode. Press /8:System Info/2:System Settings. Use the keytohighlight the ngle entry o. Press to ess the hoies nd use or rrows to highlight Degree. Press enter. twie to ept this hnge. Press enter SMPLE In ddition, it is reommended tht you lwys press / + ' to insert the degree symol fter ny ngle. This overrides ny mode hnges nd reminds you tht you should e entering n ngle, not length. mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

4 hpter 7 Trigonometry 271 lsspd tip When solving prolems in trigonometry, your lultor should e kept in Degree mode. Open the Min pplition. The sttus line t the ottom of the pplition sreen is used to set your lultor to work with ngles in degrees nd to disply nswers s deimls. The settings you require re, reding from the left: lg, Deiml, Rel nd Degree. If Stndrd not Deiml shows, tp to hnge. If Gr or Rd, not Deg show, tp to hnge. In ddition, it is reommended tht you lwys insert the degree symol fter ny ngle. This overrides ny lultor settings nd reffirms n ngle mesurement, not length. To ess the degree symol, press k on the front of the lultor. Tp the 9 t nd then the < menu item t the ottom of the keyord window. fter entering the ngle size, tp the degree symol (*)toinsert its symol. SMPLE Emple 2 Finding the vlues of trigonometri rtios Use your grphis lultor to find, orret to 4 deiml ples, the vlue of: sin 49 os 16 tn 27.3 mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

5 272 Essentil Stndrd Generl Mthemtis Solution On the lultion sreen 1 For TI-nspire S, press m 4 / ' enter. Notes: 1 Pressing / ' enters the degree sign ( ). 2 If your nswer is not deiml, press / enter. lterntively, set your lultor to pproimte (deiml) mode (see the ppendi). 2 For lsspd, disply the keyord (k), tp the 9 t, then the < menu. To enter nd evlute the epression, tp s ej* ) E. 3 Write your nswer, orret to 4 deiml ples. Si n 49 = On the lultion sreen 1 For TI-nspire S, press n16/ ' enter. 2 For lsspd, tp g* ) E. 3 Write your nswer, orret to 4 deiml ples. os 16 = On the lultion sreen 1 For TI-nspire S, press o2 7 ^ 3/ ' enter. 2 For lsspd, tp h.d* ) E. 3 Write your nswer, orret to 4 deiml ples. tn 27.3 = In the following two setions we will see tht if n ngle nd side re known we n find one of the other sides y using the required trigonometri rtio. If two sides of the right-ngled tringle re known we n find one of the ngles. SMPLE mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

6 hpter 7 Trigonometry 273 Eerise 7 1 Stte the vlues of the hypotenuse, the opposite side nd the djent side in eh tringle. d e Write the rtios for sin, os nd tn for eh tringle in Question 1. 3 Find the vlues of the following trigonometri rtios, orret to 4 deiml ples. sin 27 os 43 tn 62 d os 79 e tn 14 f sin 81 g os 17 h tn 48 i sin 80 j sin 49.8 k tn 80.2 l os Finding n unknown side in right-ngled tringle 8 10 The trigonometri rtios n e used to find unknown sides in right-ngled tringle, given n ngle nd one side. When the unknown side is in the numertor (top) of the trigonometri rtio, proeed s follows. Emple 3 Finding n unknown side Find the length of the unknown side in the tringle shown, orret to 2 deiml ples. SMPLE Solution 1 The sides involved re the opposite nd the hypotenuse, so use sin. 6 f sin = opposite hypotenuse 2 Sustitute in the known vlues. sin 38 = 65 5 mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

7 274 Essentil Stndrd Generl Mthemtis 3 Multiply oth sides of the eqution y 65 to otin n epression for. Use lultor to evlute. 65 sin 38 = = 65 sin 38 = Write your nswer orret to 2 deiml ples. = Finding n unknown side in right-ngled tringle 1 Drw the tringle with the given ngle nd side shown. Lel the unknown side s. 2 Use the trigonometri rtio tht inludes the given side nd the unknown side. If given the opposite nd the hypotenuse, use sin = opposite hypotenuse If given the djent nd the hypotenuse, use os = djent hypotenuse If given the opposite nd the djent, use tn = opposite djent 3 Rerrnge the eqution to mke the sujet. 4 Use the pproprite funtion key to find. n etr step is needed when the unknown side is in the denomintor (t the ottom) of the trigonometri rtio. Emple 4 Finding n unknown side whih is in the denomintor of the trig rtio Find the vlue of in the tringle shown, orret to 2 deiml ples Solution 1 The sides involved re the djent nd the os = djent hypotenuse hypotenuse, so use os. 2 Sustitute in the known vlues. os 34 = 72 3 Multiply oth sides y. os34 = 72 4 Divide oth sides y os 34 to otin n = 72 os 34 epression for. Use lultor to evlute. = Write your nswer orret to 2 deiml ples. = SMPLE mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

8 hpter 7 Trigonometry 275 Eerise 7 1 In eh right-ngled tringle elow, find the unknown side, orret to 2 deiml ples. d g j e h k f i Find the unknown side in eh right-ngled tringle elow, orret to 2 deiml ples SMPLE l mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

9 276 Essentil Stndrd Generl Mthemtis d e g j h k 65 3 Find the length of the unknown side shown in eh tringle, orret to 1 deiml ple. d e SMPLE g h 20 i f i l f mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

10 hpter 7 Trigonometry Finding n ngle in right-ngled tringle Wrning!! Mke sure tht your lultor is set in DEGREE mode efore ttempting this setion. Finding n ngle from trigonometri rtio vlue efore we look t how to find n unknown ngle in right-ngled tringle, it will e useful to see how to find the ngle when we know the vlue of the trigonometri rtio. If we re sked to find when sin = it is s if we hve to find reverse ger to undo the effet of the W key (or utton), so tht we n go k to see the ngle tht ws used when W ws pressed (or tpped) to get The reverse ger for the W key (or utton) is lled the inverse of sine, written sin 1. The supersript 1 is not power. It s just sying let s undo, or tke one step kwrds from, pplying the sine funtion. The request to find when sin = n e written s sin = In the following emple we will see how to find when sin = Similrly, the inverse of osine is written os 1, nd the inverse of tngent is written tn 1. Emple 5 Finding n ngle from trigonometri rtio Find the ngle, orret to 1 deiml ple, given: sin = os = 0.5 tn = 1.67 Solution We need to find sin 1 (0.8480). 1 For TI-nspire S, press / m 0^84 80 enter. 2 For lsspd, tp S.iei ) E. SMPLE 3 Write your nswer, orret to 1 deiml ple = 58.0 We need to find os 1 (0.5). 1 For TI-nspire S, press / n0^5 enter. 2 For lsspd, tp.f)e. 3 Write your nswer, orret to 1 deiml ple. = 60 mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

11 278 Essentil Stndrd Generl Mthemtis We need to find tn 1 (1.67). 1 For TI-nspire S, press / o1^6 7 enter. 2 For lsspd, tp.gh ) E. 3 Write your nswer, orret to 1 deiml ple. = 59.1 We n think of the results in Emple 5 s follows: For sin = 58, think the ngle whose sine is equls 58. For os = 60, think the ngle whose osine is 0.5 equls 60. For tn = 59.1, think the ngle whose tngent is 1.67 equls Emple 6 Find the ngle,inthe right-ngled tringle shown, orret to 1 deiml ple. Finding n ngle given two sides in right-ngled tringle Solution 1 The sides involved re the opposite nd sin = opposite hypotenuse the hypotenuse, so use sin. 2 Sustitute in the known vlues. sin = ( ) 19 3 Write the eqution to find n epression for. = sin 1 42 Use lultor to evlute. = Write your nswer orret to 1 deiml ple. = 26.9 The three ngles in tringle dd to 180.sthe right ngle is 90, the other two ngles must dd to mke up the remining 90. When one ngle hs een found, just sutrt it from 90 to find the other ngle. In Emple 6, the other ngle must e = SMPLE 42 mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

12 hpter 7 Trigonometry 279 Finding n ngle in right-ngled tringle 1 Drw the tringle with the given sides shown. Lel the unknown ngle s. 2 Use the trigonometri rtio tht inludes the two known sides. If given the opposite nd hypotenuse, use sin = opposite hypotenuse If given the djent nd hypotenuse, use os = djent hypotenuse If given the opposite nd djent, use tn = opposite djent 3 Divide the side lengths to find the vlue of the trigonometri rtio. 4 Use the pproprite inverse funtion key to find the ngle. Eerise 7 1 Find the ngle, orret to 1 deiml ple. sin = os = tn = d sin = e tn = f os = g sin = h tn = i os = j sin = k os = l tn = m sin = n tn = o os = p os = Find the unknown ngle in eh tringle, orret to 1 deiml ple. d g e SMPLE 3 4 h f i mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

13 280 Essentil Stndrd Generl Mthemtis j m Find the vlue of in eh tringle, orret to 1 deiml ple. d pplitions of right-ngled tringles Emple 7 k n e pplition requiring length SMPLE flgpole sts shdow 7.42 m long. The sun s rys mke n ngle of 38 with the level ground. Find the height of the flgpole, orret to 2 deiml ples l f 10 6 o m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

14 hpter 7 Trigonometry 281 Solution 1 Drw digrm showing the rightngled tringle. Inlude ll the known detils nd lel the unknown side. 2 The opposite nd djent sides tn = opposite 38 djent re involved, so use tn m 3 Sustitute in the known vlues. tn 38 = Multiply oth sides y tn 38 = 5 Use your lultor to find the = vlue of. 6 Write your nswer orret to Theheight of the flgpole is 5.80 m. 2 deiml ples. Emple 8 pplition requiring n ngle sloping roof uses sheets of orrugted iron 4.2 m long on shed 4 m wide. There is no overlp of the roof pst the sides of the wlls. Find the ngle the roof mkes with the horizontl, orret to 1 deiml ple. Solution 1 Drw digrm showing the right-ngled tringle. Inlude ll known detils nd lel the required ngle. 2 The djent nd hypotenuse re involved, so use os. 3 Sustitute in the known vlues. 4 Write the eqution to find. 4.2 m os = djent hypotenuse SMPLE os = ( ) 4 = os m 4.2 m 4 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

15 282 Essentil Stndrd Generl Mthemtis 5 Use your lultor to find the vlue of. = Write your nswer orret to 1 deiml ple. The roof mkes n ngle of 17.8 with the horizontl. Wrning!! lwys evlute mthemtil epression s whole, rther thn reking it into severl smller lultions. Rounding-off errors umulte s more pproimte nswers re fed into the lultions. Surprisingly, vlue of the trigonometri rtio orret to 4 deiml ples n still give n ngle tht is not orret to 3 deiml ples! 4 In Emple 8, if we used the vlue of 4.2 orret to 4 deiml ples (0.9524), the ngle otined (17.749) would not even e orret to 3 deiml ple ( ). Eerise 7D 1 pole is supported y wire tht runs from the top of the pole to point on the level ground 6 m from the se of the pole. The wire mkes n ngle of 47 with the ground. Find the height of the pole, orret to 2 deiml ples. SMPLE 2 3mlogrests with one end on the top of post nd the other end on the level ground 2.8 m from the se of the post. Find the ngle the log mkes with the ground, orret to 1 deiml ple m 6 m 2.8 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

16 3 lloon is tied to string 20 m long. The other end of the string is seured y peg to the surfe of level sports field. The wind lows so tht the string forms stright line mking n ngle of 30 with the ground. Find the height of the lloon ove the ground. 4 Peter notied tht tree ws diretly opposite him on the fr nk of the river. fter he wlked 30 m long his side of the river, he found tht his line of sight to the tree mde n ngle of 28 with the river nk. Find the width of the river, to the nerest metre. 5 ldder rests on wll2mhigh. The foot of the ldder is3mfrom the se of the wll on level ground. opy the digrm nd inlude the given informtion. Lel s the ngle the ldder mkes with the ground. Find the ngle the ldder mkes with the ground, orret to 1 deiml ple. hpter 7 Trigonometry 283 Tree m m 6 The distne mesured up the sloping fe of mountin ws 3.8 km. The sloping fe ws t n ngle of 52 with the horizontl. Mke opy of the digrm nd show the known detils. Show the height of the mountin s. Find the height of the mountin, orret to 1 deiml ple. 7 n eroplne mintins flight pth of 17 with the horizontl fter it tkes off. It trvels for 2 km long tht flight pth. Show the given nd required informtion on opy of the digrm. Find, orret to 2 deiml ples, the horizontl distne of the eroplne from its tke-off point nd the height of the eroplne ove ground level. SMPLE 8 3mldder rests ginst n internl wll. The foot of the ldder is 1mfrom the se of the wll. Find the ngle the ldder mkes with the floor, orret to 1 deiml ple. Peter mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

17 284 Essentil Stndrd Generl Mthemtis 9 The entrne to horizontl mining tunnel hs ollpsed, trpping the miners inside. The resue tem deide to drill vertil espe shft from position 200 m further up the hill. If the hill slopes t 23 from the horizontl, how deep does the resue shft need to e to meet the horizontl tunnel? nswer orret to 1 deiml ple. 10 strong rope needs to e fied with one end tthed to the top of 5 m pole nd the other end pegged t n ngle of 60 with the level ground. Find the required length of the rope, orret to 2 deiml ples. 7.5 ngles of elevtion nd depression The ngle of elevtion is the ngle through whih you rise your line of sight from the horizontl when you re looking up t something. The ngle of depression is the ngle through whih you lower your line of sight from the horizontl when you re looking down t something. ngle of elevtion = ngle of depression The digrm shows tht the ngle of elevtion nd the ngle of depression re lternte ngles ( Z ngles), so they re equl. Horizontl Horizontl ngle of depression ngle of depression ngle of elevtion pplitions of ngles of elevtion nd depression Emple 9 ngle of elevtion prk rnger mesured the top of plume of volni sh to e t n ngle of elevtion of 29.From her mp she noted tht the volno ws 8kmwy. Show how she lulted the height ove level ground of the plume of volni sh, orret to 1 deiml ple. ngle of elevtion SMPLE 29 8 km mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

18 Solution 1 Drw right-ngled tringle showing the given informtion. Lel the required height. 2 The opposite nd djent sides re involved, so use tn. 3 Sustitute in the known vlues. 4 Multiply oth sides y 8. 5 Use your lultor to find the vlue of. 6 Write your nswer orret to 1 deiml ple. Emple 10 ngle of depression From the top of liff 61 m ove se-level, hen sw psized yht. He estimted the ngle of depression to e out 10.Howfr ws the yht from the se of the liff, to the nerest metre? hpter 7 Trigonometry km tn = opposite djent tn 29 = 8 8 tn 29 = = The height of the sh plume ws 4.4 km. Solution 1 Drw digrm showing the given m informtion. Lel the required 10 distne. 2 Mrk in the ngle t the yht orner of the tringle. This is lso 10, euse it nd the ngle of depression re lternte (or Z ) ngles. Wrning: The ngle etween the liff 3 fe nd the line of sight is not 10. The opposite nd djent sides re tn = opposite involved, so use tn. djent 4 Sustitute in the known vlues. tn 10 = 61 5 Multiply oth sides y. tn 10 = 61 6 Divide oth sides y tn 10. = 61 tn 10 7 Do the division using your lultor. = Write your nswer to the nerest metre. The yht ws 346 m from the se of the liff. SMPLE 61 m 10 mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

19 286 Essentil Stndrd Generl Mthemtis Emple 11 pplition with two right-ngled tringles le 100 m long mkes n ngle of elevtion of 41 with the top of tower. Find the height h of the tower, to the nerest metre. Find the ngle of elevtion,tothe nerest degree, tht le 200 m long would mke with the top of the tower. Solution h m 200 m Strtegy: Find h in tringle, then use this vlue to find in tringle D. 1 Drw tringle showing the given nd required informtion. h α 100 m 41 2 The opposite nd hypotenuse re involved, so use sin. sin = opposite hypotenuse 3 Sustitute in the known vlues. sin 41 = h Multiply oth sides y 100. h = 100 sin 41 5 Evlute 100 sin(41 ) using your lultor nd store the nswer s the vlue of the vrile h for lter use. TI-nspire Type in 100 m 4 1 / ' nd press enter to evlute 100 sin(41 )(= ). Press / H enter to store sthevlue of the vrile h. h = lsspd From the < menu of the 9 t, tp the following sequene of uttons: s e* ) nd selet h. Press E to lulte 100 sin(41 )(= ) nd store the nswer s the vlue of the vrile h. SMPLE D 6 Write your nswer to the nerest metre. The height of the tower is 66 m. mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

20 1 Drw tringle D showing the given nd required informtion t hpter 7 Trigonometry m 2 The opposite nd hypotenuse re sin = opposite hypotenuse involved, so use sin. 3 Sustitute in the known vlues. In sin α = t 200 prt we stored the height of the tower s T. ( ) 4 Write the eqution to find. t α = sin Use your lulte to evlute. α = TI-nspire Type in / m H 200 enter nd press enter to find the vlue of (= ). 6 Write your nswer to the nerest degree. lsspd Tp the following sequene of uttons: S h /)nd press E to find the vlue of (= ). The 200 m le would hve n ngle of elevtion of 19. SMPLE Eerise 7E 1 fter wlking 300 m wy from the se of tll uilding, on level ground, Elise mesured the ngle of elevtion to the top of the uilding to e 54.Find the height of the uilding, to the nerest metre m α D mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

21 288 Essentil Stndrd Generl Mthemtis 2 The pilot of n eroplne sw n irport t se-level t n ngle of depression of 15. His ltimeter showed tht the eroplne ws t height of 3000 m. Find the horizontl distne of the eroplne from the irport, to the nerest metre m 15 irport 3 The ngle of elevtion mesured from ground level to the top of tll tree ws 41. The distne of the mesurer from the se of the tree ws 38 m. How tll ws the tree? Give your nswer to the nerest metre. 4 When Dry looked from the top of liff, 60 m high, he notied his girlfriend t n ngle of depression of 20 on the ground elow. How fr ws she from the liff? nswer orret to 1 deiml ple. 5 From the top of mountin I ould see town t n ngle of depression of 1.4 ross the level plin. Looking t my mp I found tht the town ws 10 km wy. Find the height of the mountin ove the plin, to the nerest metre. 6 Wht would e the ngle of elevtion to the top of rdio trnsmitting tower 100 m tll nd 400 m from the oserver? nswer to the nerest degree. 7 Find the length, orret to 1 deiml ple. Find the ngle,tothe nerest degree. 8 Find the length, orret to 1 deiml ple. Find the ngle,tothe nerest degree. 9 From the top of liff 45 m high, n oserver looking long n ngle of depression of 52 ould see mn swimming in the se. The oserver ould lso see ot t n ngle of depression of 35. lulte to the nerest metre: the distne of the mn from the se of the liff the distne y of the ot from the se of the liff the distne from the mn to the ot. 50 m 75 m m SMPLE 45 m y 35 m α mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

22 10 polie heliopter hovering in fied position t n ltitude of 500 m moved its spotlight through n ngle of depression of 57 onto lost hild. The pilot sighted the resue tem t n ngle of depression of 31.Ifthe terrin ws level, how fr, to the nerest metre, ws the resue tem from the hild? 7.6 erings nd nvigtion ompss erings hpter 7 Trigonometry 289 ompss ering gives the diretion y stting the ngle either side of north or south. For emple, ompss ering of N40 Eisfound y fing north nd then swinging 40 towrds the est side. Emple 12 Determining ompss erings Give the ompss erings of the points,, nd D. N Solution To find the diretion of,fe north nd swing 30 est. isinthediretion N30 E. To find the diretion of,fe south nd swing 65 est. isinthe diretion S65 E. To find the diretion of,fe south nd swing 20 west. isinthediretion S20 W. To find the diretion of D,fe north nd swing west. ngle from north = = 75 Disinthe diretion N75 W. SMPLE Diretions midwy etween the four diretions of the ompss omine the letters of the diretions they re etween. For emple, the diretion midwy etween north nd est is often lled north-est (NE). It ould lso e lled N45 E. D W W NW SW S N S NE SE E E mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

23 290 Essentil Stndrd Generl Mthemtis True erings true ering is the ngle mesured lokwise from north round to the required diretion. True erings re sometimes lled three-figure erings euse they re written using three numers or figures. For emple, 090 Tisthe diretion mesured 90 lokwise from north, etter known s est! Emple 13 Determining true erings from ompss erings Desrie the ompss erings elow s true erings: S20 E N80 W Solution 1 Show the diretion on the digrm of the ompss points. 2 dd the ngles lokwise from north to the required diretion. Note tht the four points of ompss re 90 prt. W 20 N S T 3 Write your nswer. ering = = 160 T Thetrueering is 160 T. 1 Show the diretion on the digrm of the ompss points. 2 dd the ngles lokwise from north to the required diretion. or The diretion is 80 less thn one full sweep (360 )ofthe ompss. 3 Write your nswer. 280 T W N S ering = = 280 T or = = 280 T Thetrueering is 280 T. SMPLE Emple 14 Determining ompss nd true erings Give the ompss ering nd true ering for the diretion shown. W E E 25 N E S mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

24 hpter 7 Trigonometry 291 Solution N ompss ering 1 lulte the ngle from the diretion of south. Notie tht the swing is towrds west. W E Write your nswer. ngle from south = = 65 True ering 1 lulte the totl ngles swept out lokwise from north. There is n ngle of 90 etween eh of the four points of the ompss. 2 Write your nswer. Nvigtion prolems S The ompss ering is S65 W. W 25 N S True ering = = 245 T or = 245 T The true ering is 245 T. Nvigtion prolems usully involve onsidertion of not only the diretion of trvel, given s ering, ut lso the distne trvelled. In mny prtil pplitions we need to know the distne tht hs een trvelled fter moving t prtiulr speed for given time. Ifr moved t 60 km/h for 2 hours, the distne trvelled would e 2 60 = 120 km. Distne trvelled nd speed When trvelling t onstnt speed: SMPLE Distne trvelled = time tken speed Mke sure tht the sme units of length nd time re used for the speed, distne nd time. If r moved t 60 km/h for 90 minutes, onvert 90 minutes to 1.5 hours efore multiplying y the speed. The distne trvelled would e = 90 km. E mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

25 292 Essentil Stndrd Generl Mthemtis Emple 15 Nvigting using ompss ering group of ushwlkers leve point P,whih is on rod tht runs north south, nd wlk for 6 hours in the diretion N20 E to reh point Q. They wlk t 5 km/h. Wht is the shortest distne from Q k to the rod orret to 1 deiml ple? Looking from point Q,wht would e the ompss ering nd true ering of their strting point? Solution 1 Show the given nd required informtion in right-ngled tringle. 2 lulte the distne trvelled, PQ. Distne = time tken speed. 3 The opposite nd hypotenuse re involved, so use sin. P 20 P 20? Distne PQ = 6hours 5km/h = 30 km sin = opposite hypotenuse 4 Sustitute in the known vlues. sin 20 = 30 5 Multiply oth sides y sin 20 = 6 Find the vlue of using your lultor. = Write your nswer orret to 1 deiml ple. Theshortest distne to the rod is 10.3 km. 1 Drw the ompss points t Q. N 2 Enter the lternte ngle 20. SMPLE W Q E S P Q Q N mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

26 3 The diretion of P, looking from Q,isgiven y swing of 20 from south towrds west. 4 Stnding t Q, dd ll the ngles when fing north nd then turning lokwise to look t P. This gives the true ering of P when looking from Q. Eerise 7F hpter 7 Trigonometry 293 The ompss ering is S20 W. The true ering is = 200 T. 1 Give the ompss ering (from north or south) nd the true ering of eh of the diretions: SE SW NW 2 Stte the ompss ering nd true ering of eh of the points,, nd D. N N N d W S 25 E W 3 Eddie mped overnight t point eside river tht rn est west. He wlked in the diretion N65 E for 3 hours to point. Eddie wlks t 6 km/h. Wht ngle did his diretion mke with the river? How fr did he wlk from to? Wht is the shortest distne from to the river, orret to 2 deiml ples? S 70 E W 60 S N E 65 D W 10 River 4 ship siled 3 km west, then 2 km south. Give its ompss ering from n oserver who styed t its strting point, orret to 1 deiml ple. Forperson on the ship, wht would e the ompss ering looking k to the strting point? SMPLE 5 n eroplne flew 500 km south, then 600 km est. Give its true ering from its strting point, to the nerest degree. N S E mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

27 294 Essentil Stndrd Generl Mthemtis 6 ship left port nd siled est for 5 km, then siled north. fter some time n oserver t the port ould see the ship in the diretion N50 E. How fr north hd the ship trvelled? nswer orret to 1 deiml ple. Looking from the ship, wht would e the true ering of the port? 7 womn wlked from point for 2 hours in the diretion N60 Etoreh point. Then she wlked for 3 hours heding south until she ws t point D. The womn wlked t onstnt speed of 5 km/h. Give the following distnes orret to 1 deiml ple nd diretions to the nerest degree. Find the distnes wlked from to nd from to D. How fr south did she wlk from to? N Find the distne from to. 60 d Wht is the distne from to D? e Find the ompss ering nd distne she would need to wlk to return to her strting point. 8 ship left port P nd siled 20 km in the diretion 230 T. It then siled north for 30 km to reh point.give the following distnes orret to 1 deiml ple nd diretions to the nerest degree. Find the distne. Find the distne P. N Find the distne. d Find the ngle t point. 30 km e Stte the true ering nd distne of P 230 the port P from the ship t. 7.7 The sine rule SMPLE Stndrd tringle nottion The onvention for lelling non-right-ngled tringle is to use the upper se letters,, nd for the ngles t eh orner. The sides re nmed using lower se letters so tht side is opposite ngle, nd so on. This nottion is used for the sine rule nd osine rule (see Setion 7.8). oth rules n e used to find ngles nd sides in tringles tht do not hve right ngle. D 20 km mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

28 How to derive the sine rule In tringle, show the height h of the tringle y drwing perpendiulr line from D on the se of the tringle to. In tringle D, So In tringle D, So We n mke the two rules for h equl to eh other. Divide oth sides y sin. Divide oth sides y sin. If the tringle ws redrwn with side s the se, then using similr steps we would get: We n omine the two rules s shown in the following o. hpter 7 Trigonometry 295 D h sin = h h = sin sin = h h = sin sin = sin = sin sin sin = sin sin = sin The sine rule In ny tringle ; sin = sin = sin The sine rule n e used to find the sides nd ngles in non-right-ngled tringle when given: two sides nd n ngle opposite one of the given sides or two ngles nd one side*. If neither of the given ngles is opposite the given side, find the third ngle using + + = 180. SMPLE The sine rule is relly three possile equtions: sin = sin sin = sin sin = sin Eh eqution hs two sides nd two ngles opposite those sides. If we know three of the prts, we n find the fourth. So if we know two ngles nd side opposite one of the ngles, we n find the side opposite the other ngle. Similrly, if we know two sides nd n ngle opposite one of the sides, we n find the ngle opposite the other side. mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

29 296 Essentil Stndrd Generl Mthemtis Using the sine rule Emple 16 Find ngle in the tringle shown, orret to 1 deiml ple. Solution 1 We hve the pirs = 7 nd = 120 = 6 nd =? with only unknown. So use sin = sin. 2 Sustitute in the known vlues. Using the sine rule given two sides nd n opposite ngle 6 7 sin = sin 7 sin 120 = 6 sin 3 ross-multiply. 7 sin = 6 sin Divide oth sides y 7. 6 sin 120 sin = 7 ( 5 Write the eqution to find ngle. 6 sin 120 = sin 1 ) 6 Use your lultor to evlute the epression for. = Write your nswer orret to 1 deiml ple. ngleis47.9. In Emple 16, now tht we know tht = 120 nd = 47.9,wen use the ft tht the ngles in tringle dd to 180 to find. SMPLE + + = = = 180 = = 12.1 s we now know tht = 120, = 7 nd = 12.1,wen find side using sin = sin. The steps re similr to those in the emple. Finding ll the ngles nd sides of tringle is lled solving the tringle. 7 mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

30 hpter 7 Trigonometry 297 Emple 17 Find side in the tringle shown, orret to 1 deiml ple. Using the sine rule given two ngles nd one side = 8 =? + + = 180 Solution 1 Find the ngle opposite the given side y using + + = = = 180 = 30 2 We hve the pirs = 8 nd = 30 =? nd = 50 with only unknown. So use sin = sin. sin = sin 3 Sustitute in the known vlues. 8 sin 30 = sin 50 4 Multiply oth sides y sin sin 50 = sin 30 5 Use your lultor to find. = Write your nswer orret to 1 deiml ple. Side is 12.3 units long In some speil ses it is possile to drw two different tringles tht oth fit the given informtion. This is lled the miguous se of the sine rule. It is overed in the Essentil Further Mthemtis tetook. Emple 18 pplition of the sine rule SMPLE Leo wnts to tie rope from tree t point to tree t point on the other side of the river. He needs to know the length of rope required. When he stood t, the ompss ering of ws N40 E. Leo wlked 200 m est long the river nk to,where the ompss ering of ws N60 W. Find the length of rope required to reh from to, orret to 2 deiml ples. Tree Tree mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

31 298 Essentil Stndrd Generl Mthemtis Solution 1 Inlude the given informtion in sketh. N 2 Use the ompss erings to find the ngle nd the ngle of the tringle. 3 To use the sine rule, we need to know n ngle nd its opposite side. We know side = 200. Use + + = 180 to find ngle. 4 We hve the pirs: = 200 nd = 100 =? nd = 30 with only unknown. So use sin = sin. 5 Sustitute in the known vlues = 200 m =? ngle = = 50 ngle = = = = 180 sin = sin sin 30 = 200 sin 100 = Multiply oth sides y sin sin 30 = sin Use your lultor to find. = Write your nswer orret to 2 deiml ples. The rope must e m long. Tips for solving trigonometry prolems lwys mke rough sketh in penil s you red the detils of prolem. Youmy need to mke hnges s you red more, ut it is very helpful to hve sketh to guide your understnding. SMPLE In ny tringle, the longest side is opposite the lrgest ngle. The shortest side is opposite the smllest ngle. When you hve found solution, re-red the question nd hek tht your nswer fits well with the given informtion nd your digrm. N mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

32 hpter 7 Trigonometry 299 Eerise 7G In this eerise, lulte lengths orret to 2 deiml ples nd ngles orret to 1 deiml ple where neessry. 1 In eh tringle, stte the lengths of sides, nd Find the vlue of the unknown ngle in eh tringle. Use + + = In eh of the following student ws using the sine rule to find n unknown prt of tringle, ut ws unle to omplete the finl steps of the solution. Find the unknown vlue y ompleting eh prolem. sin 40 = 8 sin 60 sin 50 = 15 sin 72 sin 110 = 24 sin 30 d 17 sin = 16 e sin 70 4 Find ngle Find ngle sin = 37 f sin 95 Find ngle sin = 47 sin 115 SMPLE d Find ngle mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

33 300 Essentil Stndrd Generl Mthemtis 5 Find side. 103 Find side Find side Find side Find side d Find side Find side d Find side. 7 Solve (find ll the unknown sides nd ngles of) the following tringles d SMPLE 8 In the tringle, = 105, = 39 nd = 60. Find side. 9 In the tringle, = 112, = 65 nd = 48. Find ngle mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

34 hpter 7 Trigonometry In the tringle, = 50, = 45 nd = 70. Find side. 11 In the tringle, = 59, = 74 nd = 41. Find sides nd nd ngle. 12 In the tringle, = 60, = 100 nd = 130.Find ngles nd nd side. 13 In the tringle, = 130, = 30 nd = 69. Find sides nd nd ngle. 14 firespotter loted in tower t sw fire in the diretion N10 E. Five kilometres to the est of nother firespotter t sw the fire in the diretion N60 W. opy the digrm nd inlude the given informtion. Find the distne of the fire from eh tower. 15 surveyor stnding t point mesured the ngle of elevtion to the top of the mountin s 30. She moved 150 m loser to the mountin nd t point mesured the ngle of elevtion to the top of the mountin s 45. There is proposl to hve strong le from point to the top of the mountin to rry tourists in le r. Wht is the length of the required le? N m 16 nvl offier sighted the smoke of volni islnd in the diretion N44 E. nvigtor on nother ship 25 km due est of the first ship sw the smoke in the diretion N38 W. Find the distne of eh ship from the volno. If the ship losest to the volno n trvel t 15 km/h, how long will it tke it to reh the volno? 17 n ir-trffi ontroller t irport reeived distress ll from n eroplne low on fuel. The ering of the eroplne from ws 070 T. From irport, 80 km north of irport, the ering of the eroplne ws 120 T. SMPLE Whih irport ws losest for the eroplne? Find the distne to the losest irport. N mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

35 302 Essentil Stndrd Generl Mthemtis 18 Holly ws reording the heights of tll trees in Stte forest to hve them registered for protetion. river prevented her from mesuring the distne from the se of prtiulr tree. She reorded the ngle of elevtion of the top of the tree from point s 25. Holly wlked 80 m towrds the tree nd reorded the ngle of elevtion from point s 50. D opy the digrm shown nd dd the given informtion. Find the ngle t in tringle. Find the ngle t in tringle. d Find the length (from to ). e Use the length s the hypotenuse in right-ngled tringle D, nd the ngle t, to find distne D, the height of the tree. 7.8 The osine rule The osine rule n e used to find the length of side in ny non-right-ngled tringle when two sides nd the ngle etween them re known. When you know the three sides of tringle, the osine rule n e used to find ny ngle. How to derive the osine rule In the tringle, show the height h of the tringle y drwing line perpendiulr from D on the se of the tringle to. Let D = s =, then D =. h D In tringle D, os = Multiply oth sides y. = os 1 Using Pythgors Theorem in tringle D. 2 + h 2 = 2 2 Using Pythgors Theorem in tringle D. ( ) 2 + h 2 = 2 Epnd (multiply out) the squred rket h 2 = 2 Use 1 to reple with os. 2 2 os h 2 = 2 Use 2 to reple 2 + h 2 with os + 2 = 2 Reverse nd rerrnge the eqution. 2 = os Repeting these steps with side s the se, we get: 2 = os Repeting these steps with side s the se, we get: 2 = os SMPLE The three versions of the osine rule n e rerrnged to give rules for os, os, nd os. mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

36 hpter 7 Trigonometry 303 The osine rule In ny tringle : when given two sides nd the ngle etween them, the third side n e found using one of the rules: 2 = os 2 = os 2 = os when given three sides, ny ngle n e found using one of the following rerrngements of the osine rule: Emple 19 os = Find side, orret to 2 deiml ples, in the tringle shown. os = os = Using the osine rule given two sides nd the ngle etween them =? Solution 1 Write down the given vlues nd = 34, = 27, =?, = 50 the required unknown vlue. 2 We re given two sides nd the ngle 2 = os etween them. To find side use 2 = os 3 Sustitute the given vlues into the rule. 2 = os 50 SMPLE 4 Tke the squre root of oth sides. = ( os 50 ) 5 Use your lultor to find. = Write your nswer orret to 2 deiml ples. Thelength of side is units. mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

37 304 Essentil Stndrd Generl Mthemtis Emple 20 Using the osine rule to find n ngle given three sides Find the lrgest ngle, orret to 1 deiml ple, in the tringle shown. Solution 1 Write down the given vlues. = 6, = 4, = 5 2 The lrgest ngle is lwys opposite the lrgest side, so find ngle. 3 We re given three sides. To find ngle use os = =? 4 6 os = Sustitute the given vlues into the rule. os = ( 4 5 Write the eqution to find ngle. = os ) Use your lultor to evlute the epression = for. Mke sure tht your lultor is in DEGREE mode. Tip: Wrp ll the terms in the numertor (top) within rkets. lso put rkets round ll of the terms in the denomintor (ottom). 7 Write your nswer. The lrgest ngle is Emple 21 pplition of the osine rule ushwlker left his se mp nd wlked 10 km in the diretion N70 E. His friend lso left the se mp ut wlked 8kminthe diretion S60 E. Find the ngle etween their pths. How fr prt were they when they stopped wlking? Give your nswer orret to 2 deiml ples. SMPLE N = 10 km = 8 km =? mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

38 hpter 7 Trigonometry 305 Solution 1 ngles lying on stright line dd to = = 180 = 50 2 Write your nswer. The ngle etween their pths ws Write down the known vlues nd the =?, = 8, = 10, = 50 required unknown vlue. 2 We hve two sides nd the ngle etween them. To find side use 2 = os 2 = os 3 Sustitute in the known vlues. 2 = os 50 4 Tke the squre root of oth sides. 2 = ( os 50 ) 5 Use your lultor to find the vlue of. = Write your nswer orret to 2 deiml ples. The distne etween them ws 7.82 km. Eerise 7H In this eerise, lulte lengths orret to 2 deiml ples nd ngles orret to 1 deiml ple. 1 Find the unknown side in eh tringle SMPLE d e f mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

39 306 Essentil Stndrd Generl Mthemtis 2 Find ngle in eh tringle d e In the tringle, = 27, = 22 nd = 40.Find side. 4 In the tringle, = 18, = 15 nd = 110.Find side. 5 In the tringle, = 42, = 38 nd = 80.Find side. 6 In the tringle, = 9, = 10 nd = 11. Find ngle. 7 In the tringle, = 31, = 47 nd = 52. Find ngle. 8 In the tringle, = 66, = 29 nd = 48. Find ngle Find the smllest ngle in the tringle, with = 120, = 90 nd = In the tringle, = 16, = 21 nd = 19. Find the lrgest ngle. 11 ship left port nd trvelled 27 km in the diretion N40 Etoreh point. nother ship left the sme port nd trvelled 49 km in the diretion S80 Etorrivetpoint. Find the ngle etween the diretions of the two ships. How fr prt were the two ships when they stopped? SMPLE 12 ttleship deteted sumrine on ering of 050 T nd t distne of 8 km. rgo ship ws 5kmdue est of the ttleship. How fr ws the sumrine from the rgo ship? N f km km 8 =? 6 13 frm hs tringulr shpe with fenes of 5 km, 7 km nd 9 km in length. Find the size of the smllest ngle etween the fenes. N 50 8 km 5 km =? mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

40 hpter 7 Trigonometry From lookout tower,fire-spotter sw ushfire t distne of 15 km in the diretion N45 W. township ws loted 12 km in the diretion S85 W. How fr ws the ushfire from the township? 15 Pssengers in r trvelling west, long rod tht runs est west, see mountin 9 km wy in the diretion N70 W. When they hve trvelled further 5 km west long the rod, wht will e the distne to the mountin? 16 t point on the ground, the ngle of elevtion to the top of rdio trnsmission tower is 60.From tht point 40 m le ws tthed to the top of the tower. t point, further 10 m wy from the se of the tower, nother le is to e pegged to the ground nd tthed to the top of the tower. Wht length is required for the seond le? 7.9 The re of tringle re of tringle = 2 1 se height From the digrm, we see tht the re of tringle with se nd height h is equl to hlf the re of the retngle h tht it fits within. re of tringle = 1 se height 2 SMPLE se, Emple 22 Height, h = 1 2 h se, Finding the re of tringle using 1 se height 2 Find the re of the tringle shown, orret to 1 deiml ple. h 3 m h Height, h 7 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

41 308 Essentil Stndrd Generl Mthemtis Solution 1 s we re given vlues for the se se, = 7 Height, h = 3 nd height of the tringle, use re of tringle = 1 2 h re = 1 se height 2 2 Sustitute the given vlues. = Evlute. = 10.5m 2 4 Write your nswer. The re of the tringle is 10.5m 2. re of tringle = 2 1 sin In tringle D, sin = h h = sin So we n reple h with sin in the rule: re of tringle = 1 2 h re of tringle = 1 2 sin Similrly, using side or for the se, we n mke omplete set of three rules: re of tringle = 1 sin 2 re of tringle = 1 sin 2 re of tringle = 1 sin 2 Notie tht eh version of the rule follows the pttern: re of tringle = 1 (produt of two sides) sin(ngle etween those two sides) 2 Emple 23 Finding the re of tringle using 1 sin 2 SMPLE Find the re of the tringle shown, orret to 1 deiml ple. Solution 1 We re given two sides, nd the ngle etween them, so use: re of tringle = 1 sin re 2 6 m = 5, = 6, = 135 of tringle = 1 sin 2 D h m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

42 hpter 7 Trigonometry Sustitute vlues for, nd into the rule. = sin Use your lultor to find the re. = Write your nswer orret to 1 deiml ple. The reofthetringle is 10.6 m 2. Heron s rule for the re of tringle Heron s rule n e used to find the re of ny tringle when we know the lengths of the three sides. Heron s rule for the re of tringle re of tringle = s(s )(s )(s ) where s = 1 ( + + ) 2 The vrile s is lled the semi-perimeter euse it is equl to hlf the sum of the sides. Emple 24 Finding the re of tringle using Heron s formul The oundry fenes of frm re shown in the digrm. Find the re of the frm, to the nerest squre kilometre. 6 km 11 km Solution 1 s we re given the three sides of the tringle, use Heron s rule. Strt Let = 6, = 9, = 11 s = 1 ( + + ) 2 y finding s, the semi-perimeter. 2 Write Heron s rule. = 1 ( ) = 13 2 re of tringle = s(s )(s )(s ) SMPLE 3 Sustitute the vlues of s,, nd into Heron s rule. = 13(13 6)(13 9)(13 11) 4 Use your lultor to find the = re. = km 2 5 Write your nswer. The reofthefrm, to the nerest squre kilometre, is 27 km 2. 9 km mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

43 310 Essentil Stndrd Generl Mthemtis Eerise 7I In this eerise, lulte res orret to 1 deiml ple where neessry. 1 Find the re of eh tringle. d 12 m 17 m 13 m 8 m e 5 m 2 Find the res of the tringles shown. d 6 m m m 8 m 3 Find the re of eh tringle. d 7 km 20 km 15 km 11 km 18 km 21 km e e 8 m 11 m 18 m 5 m 7 m 10 m m 4 m 9 m 6 m 12 m 16 m f 3 m 4 m f 8 m 5 m 5 m m 12 m 60 5 m 5 m 6 m m SMPLE 9 m f 8 m 9 m 8 m 8 m 8 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

44 hpter 7 Trigonometry Find the re of eh tringle shown. 4 m 5 m d 11 m 9 m 12 m g 16 km km 9 m e h 5 m 7 km m 7 km 12 m 7 km 5 Find the re of tringle with se of 28 m nd height of 16 m. f i 16 m 17 m 8 m In tringle, side is 42 m, side is 57 m nd ngle is 70.Find the re of the tringle. 7 Find the re of tringle with sides of 16 km, 19 km nd 23 km. 8 The kite shown is mde using two stiks, nd D. The length of is 100 m nd the length of D is 70 m. Find the re of the kite. 9 Three students, nd strethed rope loop 12 m long into different shpes. Find the re of eh shpe. 5 m 3 m 4 m SMPLE 5 m 5 m 2 m 5 km D 8 m 4 m 4 m 4 m 8 m 3 km mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

45 312 Essentil Stndrd Generl Mthemtis 10 frmer needs to know the re of his property with the oundry fenes s shown. The mesurements re orret to 2 deiml ples. Hint: Drw line from to D to divide the property into two tringles. Find the re of tringle D. Find the re of tringle D. Stte the totl re of the property. 11 regulr hegon with sides 10 m long n e divided into si smller equilterl tringles. (Rememer, n equilterl tringle hs ll sides of equl length.) Find the re of eh tringle. Wht is the re of the hegon? SMPLE 8 km D 70 9 km 6 km km 10 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

46 hpter 7 Trigonometry 313 Key ides nd hpter summry Right-ngled tringles Nming the sides of right-ngled tringle Trigonometri rtios Finding n unknown side in the denomintor of the trigonometri rtio Finding n unknown side in the denomintor of the trigonometri rtio Hypotenuse djent Opposite The hypotenuse is the longest side nd is lwys opposite the right ngle (90 ). The opposite side is diretly opposite the ngle (the ngle eing onsidered). The djent side is eside ngle nd runs from to the right ngle. The trigonometri rtios re sin, os nd tn : sin = opposite os = djent tn = opposite hypotenuse hypotenuse djent Use the trigonometri rtio tht hs the given side nd the unknown side. Finding : os = djent hypotenuse 40 os 35 = 40 = 40 os 35 = Use the trigonometri rtio tht hs the given side nd the unknown side. Finding : sin = opposite hypotenuse sin 36 = sin = 20 = 20 sin 36 = SMPLE Finding n unknown Use the trigonometri rtio tht hs ngle in right-ngled oth known sides. fter working out tringle the vlue of the rtio, use sin 1 15, os 1 or tn 1 on your lultor 18 to find the ngle. tn = opposite djent tn = Review mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

47 314 Essentil Stndrd Generl Mthemtis Review SOH H TO Degree mode pplitions of right-ngled tringles ngle of elevtion ngle of depression ngle of elevtion = ngle of depression ompss erings True erings tn = = tn 1 (0.8333) = 39.8 This helps you to rememer the trigonometri rtio rules. Mke sure your lultor is in DEGREE mode when doing lultions with trigonometri rtios. lwys drw well-lelled digrms showing ll known sides nd ngles. lso lel ny sides or ngles tht need to e found. The ngle of elevtion is the ngle through whih you rise your line of sight from the horizontl, looking up t something. The ngle of depression is the ngle through whih you lower your line of sight from the horizontl, looking down t something. ngle of elevtion horizontl horizontl ngle of depression The ngles of elevtion nd depression re lternte ( Z ) ngles so re equl. ompss erings re N N 60 E mesured y the swing towrds west or est from north or 60 south, e.g. N60 E, S40 W. W True erings re mesured lokwise from north nd lwys given with three digits, e.g. 060 T, 220 T. S 40 W 40 S N 060 Τ SMPLE Distne, speed nd time S =220 T Nvigtion prolems usully involve distne, speed nd time, s well s diretion. Distne trvelled = time tken speed W E E mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

48 hpter 7 Trigonometry 315 Non-right-ngled tringles Lelling non-right-ngled Side is lwys opposite ngle, tringle nd so on. Sine rule osine rule sin = sin = sin Use the sine rule when given: two sides nd n ngle opposite one of those sides two ngles nd one side. If neither ngle is opposite the given side, find the third ngle using + + = 180. Finding side orret to one deiml ple: sin = sin 80 sin 80 = sin 60 sin 80 = = sin 80 = = 11.4 The osine rule hs three versions. When given two sides nd the ngle etween them, use the rule tht strts with the required side: 2 = os 2 = os 2 = os To find n ngle when given the three sides, use one of: SMPLE os = os = Finding ngle. os = os = os = = os 1 (0.1666) = 80.4 os = Review mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

49 316 Essentil Stndrd Generl Mthemtis Review re of tringle re of = 1 se height 2 re of = 1 sin 2 Heron s rule Use this formul if the se nd height of the tringle re known: re of tringle = 1 2 h Finding the re. re of = 1 2 h = = 52 m 2 h 13 m 8 m Use this formul if two sides nd the ngle etween them re known. There re three versions of the formul: re of tringle = 1 sin 2 re of tringle = 1 sin 2 re of tringle = 1 sin 2 Finding the re: re of = 1 sin 2 = sin = m = 10 m = 7 m Use this formul if the lengths of the three sides of the tringle re known: SMPLE re of tringle = s(s )(s )(s ) where s = 1 ( + + ) nd is lled the 2 semi-perimeter. Finding the re: s = 1 ( ) = 12 2 re = 12(12 7)(12 8)(12 9) = = m 2 9 m h 7 m 8 m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

50 hpter 7 Trigonometry 317 Skills hek Hving ompleted this hpter you should e le to: use trigonometri rtios to find n unknown side or ngle in right-ngled tringle show the ngle of elevtion or ngle of depression on well-lelled digrm show diretions on digrm y using ompss erings or true erings use the sine rule nd osine rule in non-right-ngled tringles to find n unknown side or ngle use the pproprite rule from the three rules for finding the re of tringle solve prtil prolems involving right-ngled nd non-right-ngled tringles. Multiple-hoie questions 1 In the tringle shown, sin equls: D 12 E The length is given y: 24 sin tn os 36 sin 36 os 36 D E To find length we should use: 17 sin tn os 62 D tn E sin The side is given y: 95 tn os 46 SMPLE D 95 sin 46 E 95 sin 46 sin To find the side we need to lulte: tn D 20 os 43 E 20 sin tn tn Review 43 mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

51 318 Essentil Stndrd Generl Mthemtis Review 6 To find the ngle we need to use: ( ) ( ) ( ) os 1 os sin D 15 sin (19) E 19 os (15) 7 The ngle, orret to 1 deiml ple, is: D 38.7 E The diretion shown hs the ompss ering: N30 S S30 E S60 W D S60 E E N30 E 9 The diretion shown ould e desried s the true ering: 030 T 060 T 210 T D 150 T E 030 T S 10 r tht trvelled for 3 hours t speed of 60 km/h would over distne of: 20 km 180 km 63 km D 90 km E 60 km 11 To find ngle we should use the rule: sin = sin sin = sin sin = sin D os = E os = To find side we should use the rule: 2 = = os SMPLE E sin = sin = sin D sin = sin sin W W N 30 S N E E mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

52 hpter 7 Trigonometry The rule needed to find side is: sin = sin sin = sin 2 = os D 2 = E 2 = os 14 To find ngle we should use the rule: 12 os = djent sin = opposite hypotenuse hypotenuse 14 os = D os = E sin = sin 15 The re of the tringle shown is: 108 m 2 54 m 2 36 m 2 D 90 m 2 E 67.5m 2 16 The re of the tringle shown, orret to 2 deiml ples, is: m m m 2 D m 2 E m 2 17 The re of the tringle shown, orret to 1 deiml ple, is: 29.5m m m 2 D 161.5m 2 E 158.6m 2 Short-nswer questions 1 Find the length of, orret to 2 deiml ples. 57 m 39 3 rod rises 15 m for every 2mtrvelled horizontlly. Find the ngle of slope,tothe nerest degree. 15 m 10 m m 19 m m m m 17 m 2 Find the length of the hypotenuse, orret to 2 deiml ples. SMPLE 104 m 28 4 Find the length of side, orret to 2 deiml ples. Review 17 m 15 m m mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

53 320 Essentil Stndrd Generl Mthemtis Review 5 Find the ngle, orret to 1 deiml ple. 28 m m 6 Find the smllest ngle in the tringle shown, orret to 1 deiml ple. 15 m 23 m 17 m 7 r trvelled 30 km est, then trvelled 25 km in the diretion N70 E. How fr ws the r from its strting point? nswer orret to 2 deiml ples. 8 pennnt flg is to hve the dimensions shown. Wht re of loth will e needed for the flg? 60 m nswer orret to 1 deiml ple m 9 Find the re of n equilterl tringle with sides of 8 m, orret to 1 deiml ple. Etended-response questions 1 Timws stnding t point when he sw tree T diretly opposite him on the fr T nk of the river. He wlked 100 m long the river nk to point nd notied tht his line of sight to the tree mde n ngle of 27 with the river nk. nswer the following orret to 2 deiml ples. How wide ws the river? Wht is the distne from point to the tree? Stnding t,tim mesured the ngle of elevtion to the top of the tree to e 18. Mke lerly lelled digrm showing distne T, the height of the tree nd the ngle of elevtion, then find the height of the tree. 2 One group of ushwlkers left rod running north south to wlk long ering of 060 T. seond group of wlkers left the rod from point 3 km further north. They wlked on ering of 110 T. The two groups met t the point, where their pths interseted. Find the ngle t whih their pths met. Find the distne wlked y eh group, orret to 2 deiml ples. If the ushwlkers deided to return to the rod y wlking k long the pth tht the seond group of wlkers hd tken, wht ompss ering should they follow? SMPLE mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

54 hpter 7 Trigonometry yht P left port nd siled in the diretion N70 Wt15km/h for 3 hours. nother yht Q left the sme port ut siled in the diretion N40 Et18km/h for 3 hours. How fr did yht P sil? How fr did yht Q sil? Wht ws the ngle etween their diretions? d How fr prt were they fter 3 hours (orret to 2 deiml ples)? 4 tringulr shdeloth must hve sides of 5 m, 6mnd7mtoovertherequired re of hildren s plyground. Wht ngle is required in eh of the orners (orret to 1 deiml ple)? The mnufturer hrges ording to the re of the shdeloth. Wht is the re of this shdeloth (orret to 2 deiml ples)? The ost of shdeloth is $29 per squre metre. Wht will e the ost of this shdeloth? 5 The pyrmid shown hs squre se with sides of 100 m. The line down the middle of eh side is 120 m long. Find the totl surfe re of the pyrmid. (s the 120 m 100 m pyrmid rests on the ground, the re of its se is not prt of its surfe re.) 100 m If 1 kg of gold n e rolled flt to over 0.5 m 2 of surfe re, how muh gold would e needed to over the surfe of the pyrmid? t tody s pries, 1 kg of gold osts $ How muh would it ost to over the pyrmid with gold? 6 surveyor mesured the oundries of property s shown in the digrm. The side D ould not e mesured euse it rossed through swmp. The owner of the property wnted to know the totl re nd the length of the side D.Toonsider the prolem s two tringles, line D ws drwn on the digrm. Find the re of tringle D. 6 km 65 Find the distne D. Find the ngle D. 5 km d Find the ngle D 8 km e Find the length D. 110 f Find the re of tringle D.? g Wht is the totl re of the property? D Give lengths nd res orret to 2 deiml ples, nd ngles orret to 1 deiml ple. SMPLE Review mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird

55 322 Essentil Stndrd Generl Mthemtis Review Tehnology tip On the Internet you n find some eellent TI-83 Plus nd TI-84 Plus progrms for solving non-right-ngled tringles. Mke sure, however, tht you test ny progrm using wide vriety of prolems, s some progrms ville on the Internet re fulty. The progrm TRISOLVE y Ross Levine t ompletely solves ny tringle when you enter the known sides or ngles. Enter zero for the unknown vlues. The progrm solves the miguous se of the sine rule. It lso finds the perimeter nd the re of eh tringle solved. Foremple, when = 27, = 19 nd = 110 were entered, the unknown vlues were found nd displyed. SMPLE mridge University Press Unorreted Smple Pges Evns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in ollortion with Jn Honnens & Dvid Hird