x = (c) = -----

Transcription

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2 R ainbows hav bn a sourc of aw and inspiration throughout history. Many ancint popls saw thm as bridgs to th gods. ncint Pruvians hld thm in such wondr that thy rmaind silnt whil rainbows wr in th sky. In Nw Zaland, dad Maori chifs wr said to travl up rainbows to thir nw hom, and borigins s th Rainbow Srpnt as a powrful traditional mythological figur. For cnturis, mathmaticians trid to plain why thy occurrd. Th scrt lay in trigonomtry and a vrsion of a formula calld th sin rul. This vrsion, calld Snll s law, hlps plain why w s th colours, and why w nd to stand with our back to th sun and look at an angl of 42 to s a rainbow. So what is this magical formula? sin = n sin B whr = th angl of light as it ntrs th raindrop B = th angl of light as it is transmittd back out of th raindrop n = th ind of rfraction for watr. Startr 6 281

3 Worksht R6.1 Worksht R6.2 Worksht R6.3 Prpar for this chaptr by attmpting th following qustions. If you hav difficulty with a qustion, click on th Rplay Worksht icon on your Studnt DVD or ask your tachr for th Rplay Worksht. 1 (a) Writ th following corrct to two dcimal placs. (i) (ii) (b) Writ th following corrct to four dcimal placs. (i) (ii) Solv ths quations to find. Whr ncssary, prss your answr as a dcimal corrct to four dcimal placs. (a) 3 = 24.5 (b) = (c) = (d) = 12 3 Copy ach right-angld triangl shown blow and labl th hypotnus, th opposit sid and th adjacnt sid for th angl givn. (a) (b) (c) (d) 24.8 Worksht R Find th valu of th unknown angl in ach of th triangls abov. angl of dprssion angl of lvation convntional baring cosin minut quadrant scond sin tangnt tru baring unit circl 282 Hinmann MTHS ZONE 10

4 Trigonomtry was dvlopd from th simpl masurmnts of gomtry and survying and uss th fact that th ratios of pairs of sids in triangls ar linkd to th angls within th triangls. For a givn angl θ (pronouncd thta), w can labl a right-angld triangl as shown on th right. For right-angld triangls, th basic trigonomtric ratios ar sin, cosin and tangnt (abbrviatd to sin, cos and tan). Ths ar th ratios btwn two of th thr sid lngths. Hypotnus (th longst sid) θ djacnt (th sid adjacnt or nt to th angl) Opposit sin of angl θ = sin θ = or O Hypotnus ---- H djacnt cosin of angl θ = cos θ = or Hypotnus H --- Opposit tangnt of angl θ = tan θ = or O djacnt Ā --- Rmmbr, th hypotnus is th longst sid in a right-angld triangl and is always opposit th right angl. In th triangl shown at right, th sin of th givn 3 angl, θ, is -- or 0.6. That is, sin θ = cos θ = -- 3 = 0.8 and tan θ = -- = Each of ths ratios involvs an angl (othr than th right angl) and two sid lngths. Thy ar usful in finding an unknown sid θ lngth or angl if th othr two quantitis ar known. To idntify which trigonomtric ratio to us: 1. Labl ach sid of th triangl with O, and H, rlativ to th angl considrd. 2. Dcid which two sids ar involvd in th problm. 3. Us SOH-CH-TO: O sin θ = ---- cos θ = --- O tan θ = ---- H H H 5 4 Opposit (th sid opposit th angl) 3 O workd ampl 1 Labl this right-angld triangl and hnc dcid which trigonomtric ratio to us to find θ θ 6 trigonomtry 283

5 Stps 1. Labl th triangl. Th two sids w ar intrstd in ar th sid adjacnt to th angl and th hypotnus. Solution O H From SOH-CH-TO, and H indicat th ratio cos θ. θ 7.1 Th cos ratio should b usd. 7.1 (cos θ = ) 9.4 workd ampl 2 Masur th sid lngths of this triangl to th narst millimtr and hnc calculat th valu (to two dcimal placs) of: (a) sin 55 (b) cos 55 (c) tan Stps (a) 1. Masur all sids 2. Labl th triangl. For sin 55 w ar intrstd in th sid opposit th angl and th hypotnus. Solutions (a) cm 6.1 cm H O 5 cm 5 3. Calculat sin 55 using SOH. sin 55 = = (b) 1. For cos 55 w ar intrstd in th sid (b) adjacnt to th angl and th hypotnus Calculat cos 55 using CH. cos 55 = = (c) 1. For tan 55 w ar intrstd in th sids (c) opposit and adjacnt to th angl Calculat tan 55 using TO. tan 55 = = Hinmann MTHS ZONE 10

6 rcis 6.1 Trigonomtric ratios Skills 1 Labl ths right-angld triangls and hnc dcid which trigonomtric ratio should b usd to solv for th unknown in ach cas. (a) (b) (c) 20 a θ 9 θ 15 Workd Eampl 1 (d) () (f) d (a) Masur th sid lngths of this triangl to th narst millimtr. Calculat th valu (to two dcimal placs) of: (i) sin 47 (ii) cos 47 (iii) tan 47 (iv) sin 47 cos 47 (b) What do you notic about th valus of sin 47 and tan 47? cos Workd Eampl 2 3 (a) Masur th sid lngths of this triangl to th narst millimtr. Hnc calculat th valu (to two dcimal placs) of: (i) cos 60 (ii) tan 60 (iii) sin 60 (iv) sin 60 (v) sin 30 cos 60 (: What is th valu of th third angl in this triangl?) sin 30 (vi) tan 30 (vii) cos 30 (viii) cos 30 (b) What do you notic about th valus for cos 60 and sin 30? (c) What do you notic about th valus for sin 60 and cos 30? sin 30 (d) What do you notic about th valus for tan 30 and ? cos trigonomtry 285

7 4 (a) Masur th sid lngths of th triangl shown at right to th narst millimtr and calculat th valu (to two dcimal placs) of: (i) sin 52 (ii) tan 52 (iii) cos 52 (b) From your answrs to part (a), writ down th valu of: (i) cos 38 (ii) sin 38 (c) Using your answrs to part (b), calculat tan (a) Masur th sid lngths of th triangl blow to th narst millimtr and calculat th valu (to two dcimal placs) of: (i) tan 15 (ii) cos 15 (iii) sin (b) From your answrs to part (a), writ down th valu of: (i) cos 75 (ii) sin 75 (c) Us your answrs to part (b) to calculat tan 75. pplications 6 For ach of th following triangls, say which trigonomtric ratio (sin, cosin or tangnt) w could us to solv for th unknown quantity. (a) (b) (c) (d) () (f) (a) Th cosin of an angl is always lss than 1 bcaus: th djacnt sid is always lss than th Opposit sid B th djacnt sid is always shortr than th Hypotnus C th djacnt sid is always longr than th Hypotnus D th djacnt sid is always longr than th Opposit sid E all trigonomtric ratios ar lss than 1. θ θ Tstr 286 Hinmann MTHS ZONE 10

8 (b) Which of th following statmnts is not tru? sin 53 = cos 37 B tan 48 = sin 48 cos 48 C cos 15 = sin 75 D sin 32.5 = cos 57.5 E cos 44.3 = sin Masur th sid lngths of triangls, B and C to th narst millimtr. a b c Triangl Triangl B Triangl C (a) Calculat th valu for sin a. (b) Calculat th valu for sin b. (c) Calculat th valu for sin c. (d) r ths sin ratios about th sam? Why? () Masur th siz of angls a, b and c with a protractor. (f) Would th cosin ratios b th sam for ach of th triangls? (g) Would th tangnt ratios b th sam for ach of th triangls? 9 Draw a right-angld triangl with sids of any lngth. (a) Carfully masur all th angls of th triangl. (b) Carfully masur all th sids of th triangl. (c) Find th sin, cosin and tangnt ratios for th two acut angls. (d) Writ down as many rlationships btwn th angls and th ratios as you can find. 10 ngl θ in a triangl has a sin ratio of 0.4. Us Pythagoras to hlp you draw thr diffrnt triangls that could contain angl θ and find th valu of tan θ to on dcimal plac. nalysis 3 11 right-angld triangl has a sin ratio of -- for on angl. (a) Calculat th sin ratio for th othr acut angl. (b) Calculat th othr ratios, laving your answrs in surd form. 7 6 trigonomtry 287

9 In th last sction w workd out th trigonomtric ratios sin, cosin and tangnt for th givn angl, θ, in a right-angld triangl by masuring th sid lngths of th triangl. sin θ = Opposit cos θ = djacnt tan θ = Hypotnus Hypotnus Opposit djacnt or SOH CH TO scintific calculator can also b usd to find sin, cosin and tangnt for a givn angl. Not that som calculators may nd diffrnt ky squncs from thos givn blow. Th ampls in this sction ar basd on DL calculators, which oprat in a similar way to graphics calculators. workd ampl 3 Us a calculator to obtain valus, corrct to four dcimal placs, for th following. (a) sin 29 (b) cos 58 (c) tan 60 (d) cos Stps Mak sur that your calculator is in dgr mod, i.. that DEG or D is showing on th scrn. (a) 1. Entr th information into th calculator. Prss th following kys: sin 2 9 = 2. Round th answr to four dcimal placs. (b) 1. Entr th information into th calculator. Prss th following kys: cos 5 8 = 2. Round th answr to four dcimal placs. (c) 1. Entr th information into th calculator. 2. Round th answr off corrct to four dcimal placs. (d) 1. Entr th information into th calculator. 2. Round th answr to four dcimal placs. Solutions (a) Scrn shows sin (b) Scrn shows cos (c) Scrn shows tan (d) Scrn shows cos Hinmann MTHS ZONE 10

10 ngls in dgrs, minuts and sconds Dgrs ( ) can b brokn up into smallr units calld minuts ( ) and sconds ( ). Thr ar 60 minuts in a dgr and 60 sconds in a minut. That is, 60 = 1 and 60 = 1 So 3600 = 1 30 n angl of is th sam as 24.5, bcaus = Most scintific calculators hav a ky markd or D M S, which convrts angls in dgrs, minuts and sconds to dcimal dgrs. To chang into dcimal dgrs using a scintific calculator, prss: 2 4 D M S 3 0 D M S 2nd F D M S Th scrn will show workd ampl 4 Convrt th following angls into dcimal dgrs, corrct to two dcimal placs. (a) (b) Stps Mak sur that your calculator is in dgr mod, i.. that DEG or D is showing on th scrn. (a) 1. Entr th information into th calculator. Prss th following kys: 5 6 D M S 2 4 D M S Solutions (a) Scrn shows nd F D M S 2. Round th answr to two dcimal placs. θ 56.4 (b) 1. Entr th information into th calculator. (b) Scrn shows Prss th following kys: 2 3 D M S 1 6 D M S 4 5 D M S 2nd F D M S 2. Round th answr to two dcimal placs. θ For scintific calculators that do not follow Dirct lgbraic Logic (DL), th sin, cos or tan of an angl can b obtaind only whn th angl is prssd in dcimal dgrs. 6 trigonomtry 289

11 workd ampl 5 Us a calculator to find th valus, corrct to four dcimal placs, of th following. (a) sin (b) cos (c) tan Stps Mak sur that your calculator is in dgr mod, i.. that DEG or D is showing on th scrn. (a) 1. Entr th information into th calculator. Prss th following kys: Solutions (a) Scrn shows sin 4 6 D M S 3 6 D M S = 2. Round th answr to four dcimal placs. sin (b) 1. Entr th information into th calculator. (b) Scrn shows Prss th following kys: cos 6 D M S 1 2 D M S 3 4 D M S = 2. Round th answr to four dcimal placs. cos (c) 1. Entr th information into th calculator. (c) Scrn shows Round th answr to four dcimal placs. tan rcis 6.2 Using a scintific calculator Skills 1 Us your calculator to find, corrct to four dcimal placs, th valu of: (a) sin 23 (b) cos 17 (c) tan 39 (d) cos 5 () sin 41 (f) cos 49 (g) tan 19 (h) cos 23.5 (i) tan (j) sin (k) tan (l) tan Convrt ths angls to dcimal dgrs (to no mor than two dcimal placs). (a) (b) (c) (d) () (f) Calculat th following, corrct to four dcimal placs. (a) sin (b) tan (c) cos (d) tan () sin (f) cos Workd Eampl 3 Workd Eampl 4 Workd Eampl Hinmann MTHS ZONE 10

12 4 (a) Using a calculator, and valuating th rsult to four dcimal placs, th valu of sin is: B C D E (b) Using a calculator, and valuating th rsult to four dcimal placs, th valu of cos is: B C D E pplications 5 right-angld triangl has on acut angl of (a) Using pronumrals to rprsnt th sid lngths, draw a diagram of th triangl including th masurmnt of th othr angl. (b) Writ down sin and cos using pronumrals. (c) Find th valus of sin and cos using a calculator. (d) Show that whthr you us pronumrals or valus th valu of (sin ) 2 + (cos ) 2 quals 1. (Not: This is normally writtn as sin cos ) nalysis 6 6 Grtha has answrd a problm in trigonomtry and com up with an answr of 10. Writ thr trigonomtric prssions that could hav bn Grtha s problm. On of thm should involv th sin ratio, on th cosin ratio and on th tangnt ratio. For any right-angld triangl, w can find any sid lngth if w know th siz of an angl (apart from th right angl) and th lngth of on sid. Rmmbr: SOH-CH-TO H O θ workd ampl 6 Find th valu of t, corrct to two dcimal placs. 25 t 34 6 trigonomtry 291

13 Stps 1. Copy and labl th triangl. Solution H 25 t O Dcid which trigonomtric ratio to us. Opposit and hypotnus, so us sin. 3. Form an quation. t sin 34 = Mak t th subjct. t = 25 sin Entr th information into th calculator. Scrn shows Prss th following kys: 2 5 sin 3 4 = 6. Round th answr to two dcimal placs. t workd ampl m laddr rsts against a wall so that its angl with th wall is 65. (Vry unsaf!) How high up th wall dos th laddr rach, corrct to two dcimal placs? Stps 1. Draw a diagram and labl O, and H. Lt d b th vrtical distanc w nd to find. (Th diagram dos not nd to b to scal.) Solution 65 d H 3.2 m O 2. Dcid which trigonomtric ratio to us. djacnt and hypotnus, so us cosin. 3. Form an quation and substitut th known valus. cos 65 = 4. Mak d th subjct. d = 3.2 cos Entr th information into th calculator. Scrn shows Prss th following kys: 3. 2 cos 6 5 = 6. Round th answr to two dcimal placs. d Stat th answr. Th laddr rsts about 1.35 m up th wall. d Hinmann MTHS ZONE 10

14 In gnral, to solv for an unknown quantity in a right-angld triangl, follow ths stps: 1 Draw th right-angld triangl, showing all known information and th unknown quantity. 2 Labl th sids O, and H. 3 Us SOH-CH-TO to dcid which trigonomtric function to us. 4 Form an quation and solv for th unknown. rcis 6.3 Finding lngths Skills 1 Find th valu of ach pronumral (to two dcimal placs). (a) (b) b (c) c Workd Eampl 6 6 a (d) () (f) 25 2 To th narst cntimtr, find how far a tnt pg is away from a tnt pol if th 3 m guy is pggd so that it maks an angl of 50 with th ground. 3 (a) plan riss stadily at an angl of to th horizontal. Th altitud of th plan aftr it has travlld along a flight path of 2.6 km (to th narst mtr) is: 487 m B 496 m C 2554 m D m E 864 m 77 d m 2.6 km f Worksht C6.1 Workd Eampl trigonomtry 293

15 (b) To masur a tall building, th masurmnts shown in th diagram wr obtaind. Th hight of th building (corrct to two dcimal placs) is: m B m C m D m E m 68 pplications 4 Jssica is flying a kit attachd to a string 12 m long. How high is th kit if it maks an angl of with th horizontal? Stat your answr to two dcimal placs. ssum that Jssica is holding th string at a hight of 0.9 m from th ground. 0.9 m m 5 min shaft 350 m long is to b drilld at an angl of to th vrtical. How far blow th ground will th nd of th shaft b, corrct to two dcimal placs? nimation nalysis 6 Th angl that th roof of a hous is pitchd at (i.. th angl that th roof maks with th horizontal) is Find at last thr ralistic dimnsions for th roof, including hight at th cntr, lngth of iron ndd and distanc from th dg of th hous to th cntr. 7 Jams is trying to dcid which room to us for a party. H has two rooms in mind but nds to us th biggr on. On room is a rctangular shap with lngth 4 m and width 3 m. Th othr room is in th shap of a trapzium as shown in th diagram. (a) Calculat th ara of ach of th rooms. (b) Which room is biggr? 4 m 3 m Hinmann MTHS ZONE 10

16 circl of radius 1 unit is calld th unit circl. It provids a basis for finding trigonomtric valus. W call ach of th four quartrs of th circl a quadrant. Th triangl drawn on th diagram at right is in th first quadrant. Looking at th circl, you can s that th hypotnus of th triangl is always 1, as it is qual to th radius of th circl. If w tak a point P(, y) on th circumfrnc of th circl, thn sinθ = -- y = y (sinc y is th opposit sid and 1 is th hypotnus) 1 and cosθ = -- = (sinc is th adjacnt sid and 1 is th hypotnus). 1 vry usful rlationship in trigonomtry is sin 2 θ + cos 2 θ = 1 Th unit circl can b usd to prov this rlationship. By using Pythagoras, it is possibl to s that 2 + y 2 = 1. If w rplac y with sin θ and with cos θ, th quation bcoms sin 2 θ + cos 2 θ = 1. : sin 2 θ is th way that (sin θ) 2 is writtn and cos 2 θ is th way that (cos θ) 2 is writtn. Using th unit circl, it is possibl to s that if θ = 0 thn th y valu is zro and so sin 0 = 0, but th valu is 1 and so cos 0 = 1. Th tangnt ratio can also b found using th unit circl. Y 1 P(, y) 1 y θ 1 1 X 1 Y 1 θ 1 tan θ 1 θ = cos θ y = sin θ θ 1 1 X From th diagrams at right and abov, it is possibl to s that two similar triangls ar formd and, using your knowldg of similar triangls, that tan θ = sin θ y or tan θ = -- cos θ Th most significant fatur of th unit circl is that w can us it to find quivalnt angls in th first quadrant for angls in th othr thr quadrants. Th quadrants ar lablld in numrical ordr moving around th circl in an anticlockwis dirction, as shown at right. 1 Y quadrant 2 1 quadrant 1 θ θ 1 θ θ 1 X quadrant 3 1 quadrant 4 6 trigonomtry 295

17 By drawing th sam triangl in ach quadrant w can dtrmin valus for angls that ar not acut. To plain th procss w will us an angl θ of 30. For th triangl in th scond quadrant, w can s that, sinc th triangls ar idntical, just rflctions of ach othr, th point Q has coordinats of (, y). So if w want th valu of sin 150 it will b th sam as sin 30, and cos 150 will b qual to cos 30, sinc sin is th y valu and cosin is th valu. Chcking this with a calculator, w find it is corrct. For angls gratr than 90, a rul can b formd for ach of th quadrants to find th quivalnt angl in th first quadrant: In th scond quadrant, θ = (180 th obtus angl). In th third quadrant, θ = (th rfl angl 180 ). In th fourth quadrant, θ = (360 th rfl angl). By looking at th triangl in ach quadrant, as shown in th diagram at right, w can s that bcaus sin and cosin ar rlatd to th and y valus, and tan θ = sin θ, diffrnt ratios will cos θ b positiv in ach quadrant. In th first quadrant, all th ratios ar positiv bcaus both and y ar positiv. In th scond quadrant, only sin (y) is positiv. In th third quadrant, only tangnt is positiv (bcaus both and y ar ngativ). In th fourth quadrant, only cosin () is positiv. This can b asily rprsntd and rmmbrd by using th diagram shown at right. Intractiv Y 1 Q(, y) P(, y) X 1 Y 1 Q(, y) Sin ll P(, y) θ θ 1 θ θ 1 X R(, y) Tangnt Cosin S(, y) 1 S T C workd ampl 8 Find th quadrant that ach of th following angls would b found in. (a) 35 (b) 265 (c) 135 (d) 342 Stps (a) 35 is btwn 0 and 90. (b) 265 is btwn 180 and 270. (c) 135 is btwn 90 and 180. (d) 342 is btwn 270 and 360. Solutions (a) First quadrant (b) Third quadrant (c) Scond quadrant (d) Fourth quadrant workd ampl 9 For ach of th following rlationships, find th quivalnt trigonomtric ratio in th first quadrant, with th appropriat sign. (a) tan 353 (b) sin 124 (c) cos Hinmann MTHS ZONE 10

18 Stps (a) 353 is in th fourth quadrant, so th angl bcoms Tangnt is ngativ in th fourth quadrant. (b) 124 is in th scond quadrant, so th angl bcoms Sin is positiv in th scond quadrant. (c) 234 is in th third quadrant, so th angl bcoms Cosin is ngativ in th third quadrant. Solutions (a) tan 353 = tan ( ) = tan 7 (b) sin 124 = sin ( ) = sin 56 (c) cos 234 = cos ( ) = cos 54 rcis 6.4 Th unit circl Skills 1 For ach of th following, work out which quadrant th angl is in. (a) 355 (b) 222 (c) 111 (d) 102 () 191 (f) 23 (g) 263 (h) 97 (i) 286 (j) 342 (k) 156 (l) For ach of th following, find th quivalnt trigonomtric ratio in th first quadrant, taking car to includ th appropriat sign. (a) sin 135 (b) cos 224 (c) tan 111 (d) sin 264 () cos 321 (f) tan 123 (g) sin 263 (h) cos 97 (i) tan 186 (j) sin 342 (k) cos 166 (l) tan (a) Which of th following groups of angls ar all found in th third quadrant? 213, 182 and 156 B 187, 245 and 194 C 98, 124 and 156 D 342, 276 and 199 E 256, 280 and 234 (b) Find th quivalnt trigonomtric ratio to sin 25. sin 205 B sin 345 C sin 155 D sin 155 E sin For ach of sin, cosin and tangnt, us any angl to writ at last on quation using an angl in a diffrnt quadrant. pplications 5 Jy knows th valus of sin, cosin and tangnt whn th angls ar 30, 45 and 60. What othr angls lss than 360 can h also work out using what h alrady knows? Workd Eampl 8 Workd Eampl 9 nalysis 6 t a harbour, a sirn sounds vry tim th tid is at 5 m. Sinc tids ar basd on sin curvs, and th sirn first sounds whn th angl for th sin is 24, calculat th angl whn th sirn will sound nt. 6 trigonomtry 297

19 workd ampl 10 Find th valu of, corrct to two dcimal placs Stps 1. Copy and labl th triangl. Solution H 15 O O Form an quation using sin θ = ---. sin 36 = H 3. Transpos to mak th subjct. sin 36 = 15 Multiply both sids by. 15 = Divid both sids by sin 36. sin Entr th information into th calculator. Scrn shows Prss th following kys: 1 5 sin 3 6 = 5. Round th answr to two dcimal placs SOH-CH-TO must b applid in ordr, i.. from lft to right. This can man that th unknown sid will nd up as part of th dnominator. workd ampl 11 laddr lans against a fnc so that th bas of th laddr is 0.75 m from th fnc. How long is th laddr if it maks an angl of to th ground? nswr corrct to two dcimal placs. 298 Hinmann MTHS ZONE 10

20 Stps 1. Draw and labl a diagram. Solution H O m Form an quation using cos θ = ---. cos = H 3. Mak th subjct. cos = = cos Entr th information into th calculator. Scrn shows Prss th following kys:. 7 5 cos 6 4 D M S 3 6 D M S = 5. Round th answr to two dcimal placs Stat th answr. Th laddr is 1.75 m long. rcis 6.5 Skills Finding mor lngths 1 For ach of th following, form an quation whr th pronumral is th subjct and solv. (Stat your answrs corrct to two dcimal placs.) (a) (b) (c) (d) (g) d () (h) a c (f) (i) b 6.7 Intractiv Workd Eampl trigonomtry 299

21 (j) (k) 58 g (l) f rctangular gat has diagonal mtal supports which mak an angl of 25 to th horizontal sctions of th fram. If th gat is 1.6 m high, how long is on of th diagonal supports? 3 Th radius of th circl at right is closst to: 6.57 m B m C m D m E 5.48 m 4 right-angld triangl BC has th right angl at B. If th sid a has a lngth of 10 cm, writ down at last thr possibl lngths of th hypotnus if angl is btwn 35 and m 15 h nimation Worksht C6.2 Workd Eampl 11 pplications 5 Jssica wants to calculat th width of a rivr. Sh finds that lins joining hr point of obsrvation to th two posts on th opposit bank ar 42 apart. Hr sistr Naomi, who is on th opposit bank, masurs th distanc btwn th two posts to b m. How wid is th rivr? 6 road sign is in th shap of an quilatral triangl. Th lin from on vrt to th midpoint of th opposit sid is 34.6 cm long. How long is ach sid of th road sign? posts 42 Jssica nalysis 7 Mmbrs of a stunt tam wish to mak a daring climb along a rop, on nd of which has bn attachd to th top of a 23 m building and th othr joind to th top of a narby building 38 m high. Th rop is inclind at an angl of to th horizontal. (a) How far along th rop will ach mmbr of th stunt tam nd to climb to rach th othr building? (b) On of th stunt tam wants to drop a ball so that a man standing midway btwn th two buildings can catch it. From how far along th rop dos sh nd to drop it, and how far will it fall if th man catchs it 1.8 m abov th ground? Worksht C6.3 Homwork 6.1 Rstartr Hinmann MTHS ZONE 10

22 So far w hav usd a scintific calculator to find th sin, cosin or tangnt of a givn angl. W can also find a particular angl if w know a trigonomtric ratio for it. For ampl, w may want to find th angl θ whos sin is That is, if sin θ = 0.72, θ =? To find θ w nd to undo th sin ratio. On most calculators you will hav an INV, F, 2nd F or SHIFT ky. This ky usd bfor th sin ky applis th invrs sin ratio or sin 1. That is, θ= sin To find θ on a calculator, prss 2nd F sin. 7 2 =. (Mak sur your calculator is in dgr mod.) Th scrn will show So θ, to th narst dgr, is 46. workd ampl 12 If cos θ = , find: (a) θ in dcimal dgrs to two dcimal placs (b) θ in dgrs, minuts and sconds. Stps Mak sur that your calculator is in dgr mod, i.. that DEG or D is showing on th scrn. (a) 1. Entr th information into th calculator. Prss th following kys: 2nd F cos = Solutions (a) Scrn shows Lav th answr on th scrn. 2. Round th answr to two dcimal placs. θ (b) 1. Convrt th answr from part (a) to (b) Scrn shows dgrs, minuts and sconds. Prss 2nd F D M S. Not: Som calculators do not show symbols for dgrs minuts and sconds. 2. Round th answr to th narst scond. θ Rmmbr that 1 = 60 and 1 = 60. Thrfor, and or 31. Mak sur that rounding-off procdurs ar followd carfully. 6 trigonomtry 301

23 rcis 6.6 Finding angls Skills 1 Find th valu of ach pronumral to th narst dgr. (a) cos a = (b) sin b = (c) tan c = (d) sin d = () cos = (f) tan f = Find th valu of ach pronumral corrct to two dcimal placs. (a) cos g = 0.62 (b) tan h = 1.65 (c) sin j = (d) cos k = () tan m = (f) sin p = Find th valu of θ in dgrs, minuts and sconds. (a) tan θ = (b) sin θ = 0.4 (c) cos θ = (d) tan θ = () sin θ = 0.87 (f) cos θ = (a) Th valu of th pronumral a in th quation tan a = to th narst dgr is: 40 B 50 C 33 D 41 E 32 (b) Th valu of θ in dgrs, minuts and sconds in th quation sin θ = is: B C D E pplications 5 10 m laddr is to b usd to rach diffrnt windows in a building. (a) If th hights of th windows rang from 2 m to 6 m from th ground, calculat th rang of angls that th laddr would b on. (b) How fasibl is it to us th laddr for all th windows? nalysis 6 tr that is 4.2 m tall maks a shadow of 6 m. (a) Draw a diagram to show this situation. (b) To th narst dgr, calculat th angl that th sun maks with th ground to form th shadow. (c) Find th lngths of shadows that at last thr othr objcts would mak at th sam tim. Workd Eampl Hinmann MTHS ZONE 10

24 If w know th lngths of any two sids in a right-angld triangl, w can calculat th siz of any angl in it. gain w can us th trigonomtric ratios (SOH-CH-TO). workd ampl 13 Find th valu of θ in dgrs, minuts and sconds. θ Tutorial Tutorial Tutorial Stps Solution 1. Dcid which trigonomtric ratio to us. Opposit and hypotnus, so us sin. O Form an quation using sin θ = ---. sin θ = H Find θ. θ= sin Entr th information into th calculator. Scrn shows Prss th following kys: 2nd F sin ( ) = 5. Convrt th answr to dgrs, minuts Scrn shows and sconds. Prss 2nd F D M S. 6. Round th answr to th narst scond. θ workd ampl 14 Th top of a 1.8 m laddr, laning against a wall, rachs a hight of 1.2 m. What angl, to th narst dgr, dos th top of th laddr mak with th wall? Stps 1. Draw a diagram. Solution 1.8 m θ 1.2 m 2. Dcid which trigonomtric ratio to us. djacnt and hypotnus, so us cosin. 6 trigonomtry 303

25 Form an quation. cos θ = Find θ. θ= cos Entr th information into th calculator. Scrn shows Prss th following kys: 2nd F cos ( ) = 6. Round th answr to th narst dgr. θ Stat th answr. Th laddr maks an angl of about 48 with th wall. rcis 6.7 Skills 1 Find th valu of θ in ach triangl in dgrs, minuts and sconds. (a) (b) (c) θ On a slop, a railway riss 35 cm for vry 50 m of track. Find th angl at which th track is inclind to th horizontal, to two dcimal placs. 3 Scott is dragging a havy bo along th floor using a 1.5 m pic of rop. H is actly 1.0 m in front of th bo. To th narst dgr, what angl dos th rop mak with th floor? 4 To th narst dgr, find th inclination, θ, of th sun s rays whn a 5 m high tlphon pol casts a shadow 3.5 m long, as shown in th photograph at right. 3 Solving triangls to find angls (d) () (f) θ θ Worksht C Workd Eampl 14 θ 125 θ θ m Workd Eampl 13 θ 26.3 sun s rays 5 m 304 Hinmann MTHS ZONE 10

26 5 childrn s slid has a laddr that rachs 1.43 m vrtically abov th ground. In dgrs, minuts and sconds, th angl th 2.75 m long slid maks with th vrtical is: B C D E m 2.75 m pplications 6 furnitur rmoval company uss a ramp to whl furnitur into its vans. Th back of a van is 82 cm abov th ground and th ramp is 1.5 m long. (a) To th narst dgr, what is th angl of th ramp with th ground? (b) How much longr dos th ramp nd to b so that th angl is 30? 7 roof of hight 1.85 m has a span of 12.6 m as shown in th diagram at right. Find th angl at th ap of th roof, to th narst dgr. 8 sun loung can b adjustd from a flat, lying position through to an upright, sitting position. Th adjustabl back is 1.2 m long. Calculat at last fiv diffrnt hights that th top of th back could b placd at. Find th angl that th back would b at for ach m 12.6 m Qustions Qustions Qustions Insuranc claim Invstigating and dsigning During a rcnt storm a tr has falln on a hous, and so an insuranc inspctor has com to assss th damag. Th 10-m tr is laning at an angl of 78º to th horizontal and 2 m away from th bottom of th hous. 1 Draw an accurat diagram of th situation with all possibl valus calculatd. Producing 2 Th bottom of th tr slids a furthr 0.8 m away from th hous. Draw a diagram of this situation. nalysing and valuating 3 How much furthr out from th wall would th tr hav hit th wall of th hous instad of th roof? 6 trigonomtry 305

27 Th angl of lvation from to B is th angl that th lin B maks with th horizontal. If you wr positiond at, you would nd to rais your ys from a horizontal lin of sight to viw an objct at B. angl of lvation horizontal B Th angl of dprssion from C to D is th angl th lin CD maks with th horizontal. If you wr positiond at C, you would nd to lowr your ys from a horizontal lin of sight to viw an objct at D. C horizontal angl of dprssion D Th angl of dprssion is oftn not th angl insid th triangl that w draw. Howvr, w can us altrnat angls so as to includ its valu within th triangl. and y ar altrnat angls = y (angl of dprssion) y workd ampl 15 yacht is 1.8 km away from th bas of a 120 m cliff. To th narst dgr, what is th angl of lvation from th yacht to th top of a 4.2 m flagpol on th dg of th cliff? 306 Hinmann MTHS ZONE 10

28 Stps 1. Draw a diagram. Lt θ b th unknown angl. (Th diagram nd not b to scal.) Solution flagpol 4.2 m θ yacht 1.8 km 2. Dcid which trigonomtric ratio to us. Opposit and adjacnt, so us tangnt. 3. Form an quation. Rmmbr to hav tan θ = distancs in th sam units = Find θ. θ= tan Entr th information into th calculator. Scrn shows Prss th following kys: 2nd F tan ( ) = 120 m 6. Round th answr to th narst dgr. θ 4 7. Stat th answr. Th angl of lvation from th yacht to th top of th flagpol is about 4. workd ampl 16 Th angl of dprssion from th top of a 120 m cliff to a yacht out at sa is How far away is th yacht from th bas of th cliff? nswr corrct to two dcimal placs. Stps 1. Draw a diagram. Lt b th unknown distanc that w want to find. (Th diagram nd not b to scal.) 2. Rdraw th triangl using th angl of lvation. Rmmbr to us altrnat angls. Solution cliff m 120 m yacht trigonomtry 307

29 3. Dcid which trigonomtric ratio to us. Opposit and adjacnt, so us tangnt. 4. Form an quation. 120 tan = Mak th subjct. tan = = tan Entr th information into th calculator. Scrn shows Prss th following kys: tan 2 4 D M S 3 5 D M S = 7. Round th answr to two dcimal placs m 8. Stat th answr. Th yacht is about m from th bas of th cliff. ngls of lvation and dprssion ar always masurd from th horizontal. B angl of dprssion angl of lvation Th angl of lvation from objct to objct B is th sam as th angl of dprssion from objct B to objct. (Sinc th lin B crosss btwn two paralll lins, th angl of lvation and th angl of dprssion ar altrnat angls. Hnc thy ar qual.) rcis 6.8 Skills ngls of lvation and dprssion 1 During th day, Kristy notics that a post that is 3 m tall casts a shadow of 2.3 m. What is th angl of lvation of th sun at that tim? 2 Th angl of dprssion from th top of a pin tr to a point on th ground 50 m away is 38. Find th hight of th tr to th narst mtr. 3 traffic signal 3.95 m high casts a shadow across th road. If th angl of lvation of th sun is 24.8, find th lngth of th shadow corrct to two dcimal placs. Workd Eampl 15 Workd Eampl Hinmann MTHS ZONE 10

30 4 hlicoptr is flying at a hight of 160 m abov th ground. Th pilot can s anothr hlicoptr on a pontoon along th rivr locatd at an angl of dprssion of 50. Find, to th narst mtr: (a) th distanc from th first hlicoptr to th scond along th lin of sight (b) how far th first hlicoptr must fly to b dirctly ovr th scond. 5 (a) railway riss 400 cm for vry 2 km of track. From th lowr nd, th angl of lvation (to two dcimal placs) is: 0.11 B C D 25 E (b) From a hot-air balloon, th angl of dprssion to a markd landing sit is If th obsrvation dck in th balloon is 190 m abov th ground, th horizontal distanc th balloon nds to travl to b abov th landing sit is closst to: 1.67 km B 1.68 km C m D m E 1.58 km pplications 6 Rachl uss a protractor and pn to masur th angl of lvation from hr y lvl to th top of a billboard 25 m away. Sh obtains a valu of 42. Sh also masurs th distanc from th ground to hr y lvl and finds it to b 153 cm. Find th hight of th billboard to th narst mtr. 7 tlvision antnna is mountd on th top of a narby hill. From a position 4.5 km away, th angl of lvation to th bottom of th antnna is 17 30, whil th angl of lvation to th top of th antnna is Find th hight of th antnna in mtrs corrct to two dcimal placs m nalysis 8 Ptr viws a stationary btl on th ground, 2.6 m away, at an angl of dprssion of 32. Ptr s ys ar 15 cm blow th top of his had. H is standing upright. (a) Find Ptr s hight to th narst cntimtr. (b) Th btl movs dirctly ahad to a nw position. If th distanc from Ptr to th btl along his lin of sight is 4.7 m, find th nw angl of dprssion corrct to on dcimal plac. (c) How far, to th narst cntimtr, did th btl travl btwn th two sts of masurmnts? (d) Find th avrag spd of th btl if it took 30 sconds to covr th distanc. Giv your answr to th narst cm/s. 9 Janin looks at th top of a 15 m flagpol. Th angl of lvation to th top is btwn 30 and 60. Sh is an act numbr of mtrs from th pol. Calculat thr possibl angls of lvation from hr to th top of th pol. Worksht C6.5 6 trigonomtry 309

31 170 Masuring hight whn th bas lngth is unknown You will nd: Protractor, pn, tap masur, calculator. On mthod of masuring th hight of an objct is to hold a protractor at y lvl, with its bas horizontal, and lin up a pncil with th top of th objct. This will provid you with an stimatd angl from your y to th top of th objct. Thn, if you know how far from th objct you ar standing, you can us trigonomtry to calculat th hight of th objct. This is shown in th diagram blow. Lin pn up with th top of th objct masurmnt of y lvl 35 objct Kp th bas of th protractor horizontal Plac this point at y lvl distanc away from objct Howvr, if you ar trying to dtrmin th hight of a tr, and thr is a fnc btwn you and th tr, you cannot find th distanc from you to th tr using th abov mthod. diffrnt mthod will b ndd. 1 Using th protractor and pn, calculat th angl of lvation of a tr top on th othr sid of a fnc. 2 Masur th horizontal distanc to th fnc by following your lin of sight to th tr. Rcord this information on a diagram lik th on shown at right. 3 On th sam lin of sight as bfor, masur th angl of lvation to th top of th tr from th fnc. Includ th information on th sam diagram. h 310 Hinmann MTHS ZONE 10

32 4 Brak this diagram down into two right-angld triangls, whr h is a common sid. Triangl 1 rprsnts your first angl masurmnt (from your first position) and triangl 2 your scond angl masurmnt (from th fnc). rprsnts th distanc from you to th fnc and y th distanc from th fnc to th tr. tr top tr top θ h β h your 1st angl fnc + y your 2nd angl fnc y 5 Calculat θ and β, whr θ is th complmnt of your first angl and β is th complmnt of your scond angl. 6 Eprss θ and β in trms of tangnt. (Us your own masurmnt of and your own calculations of θ and β in th following quations.) tan θ = y tan β = -- y h h h tan θ = + y h tan β = y 7 Combin your two quations. h tan θ = + (h tan β) 8 Rarrang your formula to mak h th subjct. h tan θ h tan β = h(tan θ tan β) = h = tanθ tanβ 9 Calculat th hight of th tr. Rmmbr to add to h th hight of your ys abov ground lvl. 10 For ach of th following, calculat th valu of th pronumral corrct to two dcimal placs. (a) (b) h h m km 6 trigonomtry 311

33 Mrcator s projction Imagin a sphrical balloon placd insid a glass cylindr. Th balloon is blown up until it just touchs th sids of th cylindr. Now imagin that this is th Earth. Think about th shap of th countris drawn on th balloon. Now blow th balloon up som mor. What happns? Th cntr lin of th balloon dos not rally chang but th rst of th balloon strtchs out until it is touching all parts of th glass cylindr. Th furthr up or down from th cntr (quator) of th balloon you travl, th mor strtchd th balloon is to rach th cylindr. This is what happns whn w try to rprsnt th sphrical Earth on a flat map. Changs ar forcd onto th shaps and sizs of th countris to nabl th flat rprsntation. Thr ar actually svral diffrnt ways in which th sphrical Earth can b projctd onto a flat surfac. Th mthod dscribd abov is fairly clos to what happns in Mrcator s projction. Th map blow shows th nd rsult of this procss. This projction is namd aftr th Flmish cartographr Grhard Krämr who first drw it. Th latinisd form of his nam was Grhardus Mrcator, and this is th nam w associat with th projction. 312 Hinmann MTHS ZONE 10

34 On thing that you should notic immdiatly about th map is that th parallls of latitud ar not qually spacd. This is ncssary in this projction to tak account of th tra strtching rquird th furthr north or south w find ourslvs. On of th gratst impacts of this projction is that countris in th high northrn latituds look much biggr than thy actually ar. Th aras of th glob abov 85 N and blow 85 S cannot b shown using this projction. On application of mathmatics with Mrcator s projction is in working out th distanc btwn th parallls of latitud at any givn point. Th radius of th Earth is approimatly 6375 km, but to simplify th mathmatics w will call it 1. So th circumfrnc of th Earth at th Equator is 2π. W ar intrstd in th circumfrnc of th Earth around any othr paralll of latitud. Th radius of th circl at 60 N is found using th diagram shown. From it w can s that r = cos 60, which mans that th circumfrnc of this circl, or paralll of latitud, is 2π cos 60. rquirmnt of th projction is that th scals of th north south and ast wst lins ar qual at any particular point on th map, and that th north south lins and ast wst lins mt at right angls. This condition is calld conformality, and so Mrcator's projction producs a conformal map. Now, th mridians of longitud all nominally pass through both pols, so rprsnt th sam lngth. Howvr, as w hav sn, ach paralll of latitud has a diffrnt circumfrnc. Sinc ths lins ar all rprsntd as th sam lngth on th map, th scal of th vrtical lins must b changing, thus making thm diffrnt distancs apart. Lt us now considr th scals rquird. t th Equator w hav a latitud lin of lngth 2π. Lt us say that th scal w us is S km/cm. t latitud 60 N w hav a latitud lin with lngth 2π cos 60. Thrfor, th scal of this lin must b S cos 60. In turn, this mans that th vrtical scal would also b S cos 60 km/cm at this point. Qustions 1 Find th distanc around ach of th following parallls of latitud, assuming th radius of th Earth to b 1 unit. Giv your answrs corrct to two dcimal placs. (a) 0 (b) 20 N (c) 40 N (d) 60 N () 80 N 2 What do you know about th distanc around th 20 S paralll of latitud? 3 Whr would you b if th distanc around th paralll of latitud of your position was givn as ach of th following valus? Giv your answrs to th narst dgr. (a) 6.12 (b) 5.17 (c) 4.96 (d) 4.25 () 2.25 Rsarch Prpar a visual prsntation of som of th othr map projctions. B prpard to point out th good and bad faturs of ach projction that you choos. r trigonomtry 313

35 Th dirction of point from point B can b prssd in two ways: Th convntional baring of is N 36 E of B. Th tru baring of from B is 036. N 36 Convntional barings ar dirctions givn in trms of turning a numbr of dgrs ast or wst of north or south. N 36 E is th dirction 36 ast of north. W B S E 0 Tru barings giv th angl in dgrs in a clockwis dirction from north. convntional baring of SE (S 45 E) has a tru baring of 135. This can also b writtn as 135 T, whr T indicats a tru baring B SE (S45 E) Tru barings rquir thr digits for th whol numbr of dgrs. Thrfor, a tru baring of 48 is writtn as Hinmann MTHS ZONE 10

36 workd ampl 17 Find th convntional baring and tru baring of point B from point in ach. (a) N (b) N 58 B B 13 Stps (a) From south, turn 58 towards ast. Moving clockwis from north covrs (180 58). (b) From north, turn (90 13) towards wst. Moving clockwis from north covrs ( ). Solutions (a) Convntional baring is S 58 E. Tru baring is 122. (b) Convntional baring is N 77 W. Tru baring is 283. workd ampl 18 yacht sails 64 km on a tru baring of 215. How far south is th yacht from its original position? Giv th answr corrct to two dcimal placs. Stps 1. Draw a diagram. Lt b th distanc rquird. Not that = 35. Solution N original position of yacht 64 km 35 final position of yacht 2. Dcid which trigonomtric ratio to us. djacnt and hypotnus, so us cosin. 3. Form an quation using cos θ = ---. cos 35 = H Mak th subjct. = 64 cos Entr th information into th calculator. Scrn shows Round th answr to two dcimal placs km 7. Stat th answr. Th yacht is about km south from its original position. 6 trigonomtry 315

37 workd ampl 19 On a hiking trip, Sarah walks 8.3 km wst and thn 5.4 km north. What ar hr tru baring and convntional baring from hr original position, to th narst dgr? Stps 1. Draw a diagram. Lt θ b th angl w want to find. Solution N 5.4 km θ 8.3 km 2. Dcid which trigonomtric ratio to us. Opposit and adjacnt, so us tangnt. O Form an quation using tan θ = ---. tan θ = Find θ. θ= tan Entr th information into your calculator. Scrn shows Round th answr to th narst dgr. θ Convrt θ to tru and convntional barings. It may b hlpful to draw anothr diagram. Tru baring = = 303 Convntional baring: N = = = N57 W 8. Stat th answr. Tru baring = 303 Convntional baring = N57 W rcis 6.9 Barings Skills 1 Writ down (i) th convntional baring and (ii) th tru baring of ach of th following points from O. (a) N (b) N (c) N Workd Eampl O O 73 O 32 Worksht C Hinmann MTHS ZONE 10

38 N (d) () (f) N N O 53 O O 2 ship sails for 150 km on a baring of N 40 E. How far north of its starting point is it, to th narst kilomtr? N Workd Eampl km 3 hlicoptr travlling on a tru baring of 098 is now 75 km ast of its original position. How far has th hlicoptr travlld, to two dcimal placs? 4 tai drivr travls 630 m south and thn 410 m ast. To th narst dgr, what is th baring of th tai from its starting position? Workd Eampl 19 5 plan flis on a tru baring of 280 at 600 km/h aftr taking off at 1300 hours. To th narst kilomtr, how far north of its starting position is it at 1530 hours? 6 (a) Th convntional baring of point B from point, as shown in th diagram, is: N N35 E B S55 E C N35 W D S35 E 35 E S55 W N km B (b) Th baring of point D from point C, as shown in th diagram, is: 076 T B 256 T C 194 T D 284 T E 166 T D 14 N C pplications 7 hikr walks 5 km on a tru baring of 025 from camp sit to camp sit B. ll distancs ar roundd to two dcimal placs. (a) How far ast has sh walkd? Sh thn continus for anothr 6 km on a tru baring of 078 from camp sit B to camp sit C. (b) How far ast of camp B is sh now? (c) How far ast of camp is sh now? N 25 B N 78 C 6 trigonomtry 317

39 8 n ant walks 16 cm on a baring of S 62 W and thn turns NW and walks anothr 23 cm. How far wst is th ant from its starting point, to two dcimal placs? 9 Dsign a sailing cours around fiv buoys lablld, B, C, D and E. On ach lg of th cours, writ down th tru baring to b takn by th sailors compting in th rac. Rmmbr that th cours finishs at th starting point. nalysis 10 crw row thir boat from point to point B, a distanc of 200 m on a baring of S 55 E. Thy thn row du north a furthr 440 m to point C, as shown in th diagram. (a) Find, to th narst mtr: (i) th distanc from B to Z (ii) th distanc from to Z. (b) Find th baring of C from, to th narst dgr. Finally th crw row back from C to. (c) What is th total distanc rowd by th crw, to th narst mtr? 11 (a) From his hom, Minh jogs 2.7 km NE and thn 7.4 km SE. To th narst dgr, find his baring from his starting position. (b) If Minh thn jogs 6.9 km wst, how far south is h from his hom (to two dcimal placs)? N m Homwork 6.2 C Z B Sing th light 1 ship lavs its dock, which is du wst of a lighthous, and travls NE at a constant spd of 15 km/h. (a) To th narst minut, whn is th ship du north of th lighthous? (b) t this tim, how far is th ship from th lighthous? (c) How far has th ship travlld? 2 Rwork ths answrs if th ship s baring is N 20 E. 318 Hinmann MTHS ZONE 10

40 workd ampl 20 Find th valus of and y (to two dcimal placs). 24 y B C D Stps 1. To find, us triangl BC. Solution B C O 2. Form an quation using sin θ = ---. sin 41 = H Mak th subjct. = 24 sin Entr th information into your calculator. Scrn shows Lav this valu on th scrn. 5. To find y, us triangl DC. y 19 C O 6. Form an quation using sin θ = ---. sin 19 = - H y 7. Mak y th subjct. y sin 19 = Not that is still on th scrn. y = sin y = sin19 8. Solv for y by dividing by sin 19. Prss: Scrn shows sin 1 9 =. 9. Round answrs to two dcimal placs y D 6 trigonomtry 319

41 workd ampl 21 Tha sights two swimmrs (both on th sam baring from hr) at angls of dprssion of and from hr vantag point on th cliff. If hr y lvl is 40 m abov th watr lvl, find th distanc btwn th two swimmrs (to two dcimal placs). Stps 1. Draw a diagram. Not that altrnat angls allow us to find th angls within th triangl. Solution m To find distanc BC, us triangl BC. B C y D 40 m B C O Form an quation using tan θ = ---. tan = Mak th subjct. tan = = tan Entr th information into your calculator. Scrn shows Prss th following kys: 4 0 tan 5 5 D M S 1 0 D M S = 6. Stor th valu for in th mmory of your calculator. Prss STO M+. 7. To find th distanc BD, us triangl BD. 40 m O Form an quation using tan θ = ---. tan = y B y D 320 Hinmann MTHS ZONE 10

42 9. Mak y th subjct. y tan = y = tan Entr th information into your calculator. Scrn shows Lav this valu on th scrn. 11. Calculat CD by finding y. RCL M+ = Prss. Scrn shows Round th answr to two dcimal placs. CD m 13. Stat th answr. Th distanc btwn th swimmrs is about m. It is inappropriat mathmatics to round off in th middl of a qustion as it introducs an approimation into your solutions. If this occurs too arly, inaccuracis may b compoundd throughout th whol solution procss. Us th calculator s mmory ky to stor valus, if ncssary, and thn round off onc only for your answr. rcis 6.10 Skills Mid two-dimnsional problms 1 Find th valu of ach pronumral, corrct to two dcimal placs. (a) (b) y y Workd Eampl 20 (c) y (d) y () (f) y 49 y 6 trigonomtry 321

43 2 Find th valus of th pronumrals. Round your answrs to on dcimal plac. (a) (b) 8 Workd Eampl a 21 k (c) m 72 (d) a slid in a playground has a vrtical support and a laddr as shown in th diagram. Th slid is 2.85 m long and maks an angl of with th vrtical support. If th bas of th laddr is 80 cm away from th m support, th lngth of th laddr, to two dcimal placs, is: 80 cm 1.73 m B 0.37 m C 0.42 m D 1.53 m E 2.14 m laddr pplications 4 Whn Liz is 70 m away from a building sh notics a stuntman at an angl of lvation of 58 from hr horizontal lin of sight. H is climbing th sid of th building. What will b th angl of lvation, to th narst dgr, whn th stuntman has climbd a furthr 20 m? 20 m 5 pilot viws two lights positiond in a lin on th runway at angls of dprssion of 18 and 22. If th plan is 300 m abov th runway, find th distanc btwn th two lights. nswr corrct to two dcimal placs m 322 Hinmann MTHS ZONE 10

44 6 4.5 m laddr laning against a wall maks an angl of 64 5 with th ground. If th laddr slips 83 cm down th wall, what angl dos it now mak with th ground, in dgrs, minuts and sconds? 4.5 m 64 5 nalysis 7 n nvlop has a lngth of 12 cm and a width of 8 cm. Th dg of th back flap maks an angl of 35 with th top of th nvlop. Find th distanc btwn th bottom of th flap and th bottom of th nvlop. (: Draw a vrtical lin from th bottom of th flap to th top of th nvlop.) Round your answr to two dcimal placs. 8 chord 15 mm long forms an angl of 126 at th cntr of a circl. Find th lngth of th diamtr of th circl corrct to two dcimal placs cm 15 mm cm 9 cabl car is supportd by two wirs attachd to a horizontal cabl. On of th wirs is 5.4 m long and maks an angl of with th horizontal cabl. Th othr wir maks an angl of with th cabl. How far apart along th cabl ar th wirs, corrct to two dcimal placs? 6 trigonomtry 323

45 n altrnativ formula for th ara of a triangl You will nd: calculator, rulr and protractor. Rcall th formula for th ara of a triangl studid in prvious yars: h 1 = -- bh 2 To us this formula, th bas and hight of th triangl must b known. Sinc th bas is on of th b sids of th triangl, it is usually givn, but th sam cannot b said of th hight. In many cass, sids and on or mor angls ar known or can b found asily. formula involving trigonomtry that may b usd in such instancs is: 1 = -- ab sin θ 2 a whr a and b ar th lngths of two sids, and θ is th angl btwn thm. θ 1 By taking appropriat masurmnts, find th ara b 1 of ach triangl blow using th formula = -- bh. (a) 2 (b) a h a h b b (c) (d) h a a h b b 2 By taking appropriat masurmnts, find th ara of ach triangl in Qustion 1 using 1 th formula = -- ab sin θ. Do your answrs agr with thos obtaind using th othr 2 formula? 1 3 Can you s how th formula = -- 1 ab sin θ was dvlopd from = -- bh? Eplain this 2 2 dvlopmnt using a diagram and trigonomtry. 324 Hinmann MTHS ZONE 10

46 To solv a thr-dimnsional problm using sin, cosin or tangnt (or Pythagoras Thorm), w nd to look for right-angld triangls. Th problm nds first to b drawn to appar thr-dimnsional. This may mak it mor difficult to locat th right-angld triangls as a right angl will not always b drawn as actly 90 on your pag. Considr this cub: B D C W can find right-angld triangls such as EH and CDH quit asily. F G E Within th cub w can locat right-angld triangls such as EG and DBF. (Can you find othrs?) B H C B C D D F G F G E lways rdraw th right-angld triangl you wish to solv as a sparat diagram. H E B H D E G F You may nd to solv mor than on right-angld triangl in th thr-dimnsional diagram to obtain th unknown you rquir. 6 trigonomtry 325

47 workd ampl 22 right pyramid BCDE stands on a squar bas of dg lngth 10 cm. Th sloping dgs ar ach of lngth 15 cm. F is th prpndicular hight. (a) Find th angl that a sloping dg maks with th bas of th pyramid (to th narst dgr). (b) Hnc find th hight of th pyramid, corrct to two dcimal placs. B C 10 cm F E 15 cm 10 cm D Stps (a) 1. Lt θ b th angl rquird. Choos a triangl to work with. W could us any of th four possibl triangls. 2. Rdraw this triangl. Solutions (a) Us DF to find θ. 15 F θ D 3. W nd to find DF bfor w can find θ. Us BDE to find BD and DF. 4. Rdraw this triangl. D 10 B 10 E 5. Us Pythagoras Thorm to find BD. (BD) 2 = (BD) 2 = 200 BD = 200 Scrn shows DF is half of BD. Divid th scrn valu Scrn shows by 2. Stor this valu in mmory. 7. Using DF, dcid which trigonomtric ratio to us. djacnt and hypotnus ar providd, so us cosin. DF 8. Form an quation. cos θ = Find θ. θ= cos 1 DF Substitut th stord valu for DF. θ= cos Entr th information into your calculator. Scrn shows Stor th valu for θ in mmory. 326 Hinmann MTHS ZONE 10

48 12. Round th answr to th narst dgr. θ Stat th answr. sloping dg maks an angl of about 62 with th bas of th pyramid. F (b) 1. Using triangl DF again, form an (b) sin θ = quation for F. 2. Mak F th subjct. F = 15 sin θ 3. Substitut th stord valu for θ. F = 15 sin Entr th information into your calculator. Scrn shows Round th answr to two dcimal placs. F Stat th answr. Th hight of th pyramid is about cm. To solv a thr-dimnsional problm: 1 Draw a diagram that rprsnts th problm. 2 Idntify right-angld triangls that contain th unknown you want or that will hlp you find it. 3 Rdraw ths right-angld triangls. 4 Solv ths triangls using trigonomtry. Tutorial rcis 6.11 Skills 1 right pyramid BCDE stands on a squar bas of dg lngth 8 cm. Th sloping dgs ar ach 14 cm long. F is th prpndicular hight. (a) Find th angl that a sloping dg maks with th bas of th pyramid (to th narst dgr). (b) Hnc find th hight of th pyramid to two dcimal placs. Solving thr-dimnsional problms B C F 8 cm E 14 cm D Workd Eampl 22 2 (a) To find th angl that plan BFE maks with plan BCD, w could calculat: FCB B BFC C DB D ED E DE D (b) To find th lngth F, w could us th triangl: BC B EF C BCF D BD E DEF E B F C 6 trigonomtry 327

49 pplications For Qustions 3 to 11, unlss statd othrwis, round off angls in answrs corrct to on dcimal plac and round off othr answrs corrct to two dcimal placs. 3 tnt has th dimnsions shown in th diagram. Find th ara of th groundsht ndd to fit insid th tnt. 1.2 m m 4 PQRST is a squar-basd pyramid T with a bas of sid lngth 14 mm and a hight of 6 mm. Th point X on th bas is vrtically blow T at th ap, and point Y is midway along PQ. (a) Find th angl TXY. S R (b) What angl dos th sloping fac PQT mak with th bas? X 14 mm (c) Find th lngth of TY. Y P 14 mm (d) Find th lngth of a sloping Q dg of th pyramid. () Find th angl that a sloping dg maks with th bas of th pyramid. 5 hill BCD riss at an angl of 24 to th horizontal. Th hill is 60 m high. B C (a) If you walkd straight up th hill along B, what distanc would you travl? (b) What horizontal distanc would 24 F E you covr walking along B? D (c) Your frind walks along a track from to C, taking 30 minuts to rach th top. Horizontally, th distanc sh covrs is 510 m. How fast dos sh walk in m/min? (d) Find th angl of th track C to th horizontal. () Gradint rprsnts th stpnss of a slop and can b calculatd ris by using th formula or vrtical distanc. Calculat th run horizontal distanc gradints of B and C, and hnc dtrmin which track is stpr. 6 cylindrical pic of hollow pip is 47 cm long and has both nds covrd with plastic. tiny hol is mad in th middl of on nd and a pic of wir is thradd through. Th wir dos not bnd. If th pip has a diamtr of 12 cm, find: 47 cm (a) th angl th wir maks with th plastic if it touchs th opposit nd whr th plastic mts th pip (b) th lngth of wir within th pip. 6 mm 60 m 328 Hinmann MTHS ZONE 10

50 7 Val is 50 m du north of a communications towr. Th top of th towr is at an angl of lvation of 41 from th ground at Val s obsrvation point. From Con s obsrvation point, du ast of th towr, th angl of lvation of th top of th towr is 29. Find: (a) th hight of th towr (b) th distanc from Con to th foot of th towr (c) th distanc btwn Val and Con (d) Con s baring from Val s obsrvation point. 8 Th hight of a vrtical con is 4.72 cm, and its smi-vrtical angl is Find: (a) th slant hight of th con (b) th diamtr of th con cm 50 m Val 41 N 29 Con E nalysis 9 passngr on a cruis ship is positiond 42 m abov th watr lvl and is admiring a tropical island in th distanc. On nd of th island is at a tru baring of 028 and has an angl of dprssion of 11, whil th othr nd is at a tru baring of 118 and has an angl of dprssion of 19. How long is th island? 10 survyor nds to find th hight of an inaccssibl cliff (surroundd by rocks) and taks masurmnts from two diffrnt positions in th watr, as shown in th diagram. Thr is a flagpol on top of th cliff. (a) If th flagpol is known to b 4.5 m tall, find th hight of th cliff. (b) What is th distanc from th top of th flagpol to th survyor at position K? K m 4.5 m flagpol cliff 15 nimation M 11 Liam wants to construct a rctangular dsktop with a diagonal lngth of 70 cm. (a) Whn th dsktop is tiltd at 15 to th horizontal, it is raisd 8 cm. Find th ara of th dsktop, to th narst squar cntimtr. (b) Find at last two diffrnt dimnsions of th dsktop whn th hight is still 8 cm and th diagonal lngth 70 cm but th angl is changd. Considr angls lss than 30 only. Qustions Homwork trigonomtry 329

51 DIY summary Copy and complt th following using words and phrass from th list whr appropriat to writ a summary for this chaptr. word or phras may b usd mor than onc. 1 n angl can b masurd in dgrs. Instad of a dcimal part to th dgr, and ar usd. 2 Th ratio is whn th adjacnt sid is dividd by th hypotnus. 3 Th ratio is whn th opposit sid is dividd by th hypotnus. 4 Th ratio is whn th opposit sid is dividd by th adjacnt sid. 5 Whn a dirction is givn with a masurmnt ast or wst from north or south, this is a. 6 always has thr digits and shows th numbr of dgrs from north in a clockwis dirction. 7 Th is a circl that has a radius of 1 unit. It is dividd into four. 8 Draw a unit circl and labl th quadrants as first, scond, third and fourth. Stat which trigonomtric ratios ar positiv in ach quadrant. 9 Th angl from th horizontal as you look up is calld an. Th angl from th horizontal as you look down is calld an. VELS prsonal larning activity angl of dprssion angl of lvation convntional baring cosin minut quadrant scond sin tangnt tru baring unit circl 1 Draw a right-angld triangl with lablld sids and us it to show th trigonomtric ratios: cosin, sin and tangnt. 2 Draw a flowchart to show how to solv a problm whr you nd to find th valu of an angl. 3 Draw anothr flowchart to show how to find th lngth of a sid in a right-angld triangl. 4 Writ down a valu for an angl in ach of th four quadrants. Using ths angls, plain how to find th sin, cosin and tangnt of angls in ths quadrants. 5 Draw diagrams to clarly show an angl of lvation and an angl of dprssion. Writ down a way that you ar going to rmmbr thm. 330 Hinmann MTHS ZONE 10