Lesson 18.3: Triangle Trigonometry ( ) : OBTUSE ANGLES


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1 Lesson 1.3: Tringle Trigonometry We now extend te teory of rigt tringle trigonometry to nonrigt or olique tringles. Of te six omponents wi form tringle, tree sides nd tree ngles, te possiilities for omintion of ny tree of te omponents re two ngles nd side 1. two ngles nd side inluded y te ngles (S). two ngles nd side not inluded y te ngles (S) two sides nd n ngle 3. two sides nd n ngle opposite one of te sides (SS) 4. two sides nd n ngle inluded y te sides (SS) 5. tree sides (SSS) 6. tree ngles () Geometrilly, te first five of tese ses will determine tringle of speifi dimensions. Tree ngles will not. (onsider te possiilities for similr tringles.) In oter words, given ny tree omponents of tringle, WITH TH XPTION of tree ngles (), we n solve te tringle. In te first tree ses listed S, S, SS we n use te Lw of Sines to solve te tringle. In ses 4 nd 5 SS nd SSS we n use te Lw of osines. However, efore we n proeed wit tis we need to onsider te se of te otuse ngled tringle : OTUS NGLS Te study of te trigonometri rtios of ny ngle is eyond our sope t present. It is suffiient for our purposes to note two fts: (1) te sine of n ute ngle is te sme s te sine of its supplement. Tt is: Tis n e quikly verified using lultor. ( ) sinθ sin 10 θ xmple 1: sin 0 sin sin 5.3 sin sin 0 sin 10 0 Tis n led us to possile miguity in te se of olique tringles. If we do not know tt θ is n ute ngle, or tt te tringle is n utengled tringle, tere my e possiilities for solution. xmple : xmple 3: If we find for ertin ngle in tringle, tt sin 0.5, ten migt e 30 or migt e te supplement, 150, sine sin 30 sin Te ngle ould e eiter 30 or 150. If we find for ertin ngle sin Ten: sin or Te ngle ould e eiter 7 or 173.
2 () te osine of n ute ngle is te sme s te negtive of te osine of its supplement. Tt is: os ( 10 θ ) osθ Tis n lso e verified using lultor. xmple 4: os os os os os0 1 os10 1 We do not need to worry out miguity in te se of te osine rtio: if te osine is positive te ngle is ute, nd if te osine is negtive te ngle is otuse. xmple 5: If os Ten sine te rtio is positive, we know te ngle is ute. os In te se of te osine rtio, if te osine is negtive te lultor will give us n otuse ngle s n nswer. (Tis is not te se wit te sine or tngent rtios.) xmple 6: If os Ten sine te rtio is negtive, we know te ngle is otuse. os 1 10 ( 0. 15) Note tt tis is te supplement of in xmple 5, 7. xerise Find θ to te nerest degree given te following () sinθ () osθ () sinθ 0.56 (d) osθ (e) sinθ 3 (f) osθ 1 (g) sinθ () osθ 1. Given te ngles θ elow, find (i) Find te supplements (10  θ) (ii) sinθ nd sin(10  θ), to 4 deiml ples, nd verify tt tey re equl (iii) osθ nd os(10  θ), nd verify tt tey ve equl mgnitude ut opposite signs () 1 () 69 () 115 (d) 135 pter 3: Tringle Trigonometry Pge
3 1.3.: TH LW OF SINS onsider tringle. Drop perpendiulr line D of lengt from te vertex to te side. (Digrm 1.) y rigt tringle trigonometry: sin sin nd sin sin Hene: sin sin Dividing ot sides y sin sin sin sin results in: Now drop perpendiulr line of lengt k from te vertex to te side. (Digrm.) D Digrm 1 y rigt tringle trigonometry: Hene: Sine: k sin k sin sin sin sin sin sin nd sin nd sin k sin k sin sin k Digrm ten: sin sin sin Te Lw of Sines sttes tt in ny tringle, sin sin sin n lterntive sttement of te Lw of Sines rises from te ft tt if onvenient to use wen we wis to determine te mesure of n ngle: ten x y y x nd is In ny tringle, sin sin sin pter 3: Tringle Trigonometry Pge 3
4 xmple 1: (S) Given tt 66, 5, nd, solve. Find sides orret to te nerest tent. First, drw digrm leling te dt given nd te unknown vlues. To find, sine we know,, nd we use sin sin sin 5 sin 66 sin 5 sin Using te ngle sum of tringle, To find, sine we know,, nd we use sin sin sin 56 sin 66 sin 56 sin xmple : (S) prent is 3.5 meters long strigt pt from er ild, wo is flying kite diretly ove te pt. Te ngle of elevtion of te kite from te ild is 74., wile te ngle of elevtion of te kite from te prent is 7.. How long is te kite string? First, drw digrm leling te dt given nd te unknown vlues. Suppose te ild is stnding t point, te prent t point, nd te kite is flying t point. Given tt D 7., Ten To find, sine we know,, nd we use sin sin 3. 5 sin101. sin sin101. sin D pter 3: Tringle Trigonometry Pge 4
5 xmple 3: (SS) Given tt 45, 5.7, nd 5.1, nd solve. To find, sine we know, nd we use sin sin sin sin sin 45 sin 5. 7 sin sin We lso need to onsider te supplement: or However sine is smller tn, ten must e smller tn. 39 Using te ngle sum of tringle, we n find To find, sine we know, nd now we use: sin sin 5. 7 Note tt tis is sensile nswer. Sine sin96 sin 45 is te lrgest ngle of te tringle, is 5. 7sin96 te longest side of te tringle. sin Te miguous se : Sometimes te dt given is not suffiient to identify unique tringle, s we see in tis next exmple: xmple 4: (SS) Given tt 6., 13.5 nd 5, nd solve. To find, sine we know, nd, we use sin sin sin sin sin 5 sin 6. sin sin or We do not ve enoug informtion out tis tringle to eliminte one of te possile two solutions, nd so te dt gives two possile tringles: pter 3: Tringle Trigonometry Pge 5
6 Tringle 1: If To find : sin sin 6. sin9 sin 5 6. sin9 sin Tus is te longest side of tis tringle. Sine is te lrgest ngle in te tringle, tis result is onsistent. Tringle : If To find : sin sin 6. sin3 sin 5 6. sin3 sin Tus < < wi is onsistent wit < <. ngles formed y Prllel Lines To omplete some of te following exmples we need to use some fts out te ngles formed wen trnsversl uts pir of prllel lines. Wen te trnsversl l uts prllel lines m nd n : v x y w l m pirs of orresponding ngles re equl x t, y s, u v, r w pirs of lternte interior ngles re equl t w nd s v. u t r s n pirs of lternte exterior ngles re equl u y nd r x pirs of ointerior ngles re supplementry t + v 10 nd s + w 10 pter 3: Tringle Trigonometry Pge 6
7 xmple 5: pilot, flying over strigt igwy, determines te ngles of depression from mileposts on opposite sides of te irplne: te ngle of depression to one milepost is, wile te ngle of depression to milepost 4 miles from te first is 5. Find te ltitude of te pilot to te nerest 100 feet. We need to find. 5 D 4 miles 5 To do so we first need to determine (or ) y te Lw of Sines. Ten we n use rigt tringle trigonometry to find. Sine is n lternte ngle wit te ngle of depression of 5 we know 5 nd likewise sine is n lternte ngle wit te ngle of depression of we know To find, we use te strigt ngle: We now ve two ngles of te tringle, nd, nd one side,. To find : sin sin 4 sin5 sin93 4 sin5 sin mi ft 143 ft We n now use te rigt tringle D to find : sin sin sin 15 Terefore te irplne s n ltitude of pproximtely 100 ft. pter 3: Tringle Trigonometry Pge 7
8 xerise Solve te tringle (pproximte to te nerest wole), given: () 67, 0 nd () 49, 45, 1 (), 16 nd 4 (d) 9, 14 nd 170 (e) 36, 30 nd 40 (f) 10, 14 nd 0. Find te vlue of te side leled x or te ngle leled θ orret to te nerest tent of unit. () () () 55 7 x 3.4 x x (d) (e) (f) θ θ θ 1 3. ot ir lloon is kept t onstnt ltitude y two ropes nored t two points on te ground. One rope is 150 feet long nd mkes n ngle of 6 wit te ground. How long is te seond rope, to te nerest ten feet, if it mkes n ngle of 75 wit te ground? 150 ft x ft tree lens t n ngle of 7 from te vertil (digrm ove.) t point 0 feet from te se of te tree te ngle of elevtion to te top of te tree is 67. Find te eigt of te tree to te nerest foot. 7 x 0 ft n oserver in te street notes tt te ngle of elevtion to te top of te uilding is 44 (digrm elow.). He wlks 100 feet diretly towrds te uilding nd finds tt te ngle of elevtion is 56. Find te eigt of te uilding to te nerest foot. x ft ft 56 pter 3: Tringle Trigonometry Pge
9 6. wt tower is 40 meters tll, loted t te top of ill (digrm ove.) From te top of te tower, te ngle of depression to rngers ut t te se of te ill is 4. From te se of te tower, te ngle of depression to te ut is 34. Find te distne, to te nerest meter, from te ut to te se of te tower x m 40 m 7. To find te widt of river surveyor on one side of te river mesures te ngle to tree on te opposite side of te river from two lotions 100 ft prt (digrm elow.) pproximte te widt of te river, orret to te nerest foot. d pilot is flying over strigt freewy. Te ngle of depression to two mile posts, miles prt, on eiter side of te plne re determined to e 14 nd 16. Find te ltitude of te plne to te nerest undred feet. Questions 9 nd 10 refer to te digrm elow: 9. Find D to te nerest degree. 10. Find D to te nerest degree. 50 D pter 3: Tringle Trigonometry Pge 9
10 1.3.3: TH LW OF OSINS We now onsider te se were we know eiter two sides of te tringle nd te ngle inluded y tose sides (SS) or we know tree sides of te tringle (SSS.) In tese two ses we pply te Lw of osines: Te Lw of osines sttes tt in ny tringle, + os Proof of te Lw of osines: in ny tringle, + os In te tringle, drop perpendiulr line D of lengt from te vertex to te side. y Pytgors Teorem, sine tringle D is rigt tringle: ( x) + (1) nd sine tringle D is rigt tringle: rerrngin g, x + x Sustituting for in qution (1), ( x) () ( x) + ( x ) x + x + + x + x (3)  x D x y rigt tringle trigonometry: rerrngin g, x os x os (4) Sustituting for x in qution (3), + x + ( os) ten: + os In similr wy it n e sown tt: + os nd + os pter 3: Tringle Trigonometry Pge 10
11 Wen using te Lw of osines to find n ngle it is ndy to ve rerrnged for of tis formul: + os os + + os y te Lw of osines, in ny tringle, os + Likewise, + os nd os + xmple 1: (SS) Solve te tringle given tt 6,.0 nd 9.0 We know two sides, nd, nd te inluded ngle + os os To find te seond ngle, eiter te Sine Lw or osine Lw ould e used: os os (. ) os To find te tird ngle we n use te ngle sum of tringle: xmple : To find te widt of smll lke, surveyor tkes te mesurements sown: te distne from te fixed point to points on eiter side of te lke, nd, nd te ngle etween tese lines of mesurement. Find te widt of te lke to te nerest tent of mile. We know two sides, nd, nd te inluded ngle. + os (. )(. 5) os Te lke is pproximtely 1.9 miles in widt.. mi mi pter 3: Tringle Trigonometry Pge 11
12 NOT on lultors: Rter tn writing down intermedite nswers, it is more effiient to prepre te nswer for one step lultor use. In tis exmple, (. )(. 5) os 4. 5 (. )(. 5) os 4. 5 (. +.5 x X. X.5 X os 4.5 ) x 1.9 xmple 3: (SSS) Find te ngles (to te nerest degree) of tringle wi s sides of m, 3 m nd 4 m. Drw tringle, lel te verties s unknowns, nd sides wit te know vlues. To find, use te formul os + os os 9 ( 1) To find we need to use te formul for os : os os + ( 11) Finlly, y te ngle sum of tringle, pter 3: Tringle Trigonometry Pge 1
13 erings Tere re numer of different onventions in nvigtion for desriing te diretion in wi ody is eding. Te metod we will disuss uses te ute ngle mesured est or west from due nort or sout. Tis is est explined y exmples: N eding in n ngle of est from due Nort. Te ourse mkes n ngle of, in lokwise diretion, wit Nort. N N 75 W eding in n ngle of 75 west from due Nort. Te ourse mkes n ngle of 75, in ounterlokwise diretion, wit Nort. 75 N W W S S S40.5 eding in n ngle of 40.5 est from due Sout. Te ourse mkes n ngle of 40.5, in ounterlokwise diretion, wit sout. N S17 W eding in n ngle of 17 west from due Sout. Te ourse mkes n ngle of 17, in lokwise diretion, wit sout. N W W S S xmple 4: 34 N W S One te first dy of ike, group of ikers set out in diretion of S34 nd wlk for 0 miles in strigt line (more or less) efore setting up mp. On te seond dy tey ike for 4 miles due nort. Find te distne nd ering tey must ike on dy 3. Drw digrm nd lel ll known ngles nd distnes wit teir vlues. If we lel te strting point s, te first mp s nd te seond mp s, ten te distne tt te group must ike k to strt is. 0 mi 34 5 mi Te ngle is lternte nd equl to 34 ie 34. Hene we know n ngle nd te two sides tt inlude it nd n use te osine Lw to find. + os os miles Tey will need to ike 14 miles to get k to teir strting point. pter 3: Tringle Trigonometry Pge 13
14 To find te ering on wi tey must ike, we need to find te ngle. + os ( ) 1 os 53 0 mi Sine is ute it is onsistent wit ering of S53 W 5 mi 34 xmple 5: Two ots leve port t te sme time. Te first is trveling t n verge speed of 15 mp, on ering of N 30 from te port. Te seond is trveling t n verge speed of 1 mp, on ering of N 35. How fr prt (to te nerest tent of mile) re te two ots fter ours? Drw tringle, lel te verties s unknowns, nd sides wit te know vlues. fter ours te ot on ourse of N 30 s trveled 30 miles, nd te ot on ourse of N 35 s trveled 36 miles. Te ngle etween teir ourses is 5, nd te distne etween te ots is d miles. N OT OT OT OT 15 mp 35 1 mp 30 mi 36 mi 30 5 PORT PORT Sine we ve tringle in wi we know two sides nd te inluded ngle we n use te Lw of osines, were is te distne etween te two ots: os ( )( 36) 30 os5 Te ots re pproximtely 6.6 miles prt. pter 3: Tringle Trigonometry Pge 14
15 xmple6: pilot leves irport nd flies 310 miles on ering of N70 W to irport. Se ten flies 50 miles on ering of N10 from irport to irport. Find () te distne (to te nerest 10 miles) nd () te ering (nerest degree) se must tke to fly k to irport. Drw tringle, lel te verties s unknowns, nd sides wit te know vlues. W N y te teory of prllel lines, te ngles leled x nd y re ointerior nd tus supplementry. y 70, ten x 110. Terefore S x y 310 () First we need to find te distne. Sine we ve two sides nd te inluded ngle, we n pply te Lw of osines: os ( )( 500) 300 os miles Terefore se must fly pproximtely 650 miles k to irport. () To find te ering on wi se must fly we need to find te ngle. Sine we know n ngle,, nd two sides, () nd (), of te tringle so we n use te Lw of Sines: sin sin sin sin sin100 sin sin or tringle n ve only one otuse ngle, terefore nd Te ngle formed t point etween due nort nd te line is y te teory of prllel lines, tis must e equl to te ngle formed t point etween due Sout nd te line (sine tey re lternte ngles.) Terefore te ering te pilot must ed on to return to irport is S 1. pter 3: Tringle Trigonometry Pge
16 xerise Solve te tringle (pproximte to te nerest wole), given: () 67, nd () 0, 1, 1 () 10, 16 nd 4 (d) 9, 16 nd 170. Find te vlue of te side leled x or te ngle leled θ orret to te nerest tent of unit. () () 4 () (d) x 14 x 35 θ 5 6 θ Te sides of prllelogrm re 0 m nd 16 m nd te longest digonl is 3 m. Find () te ngles of te prllelogrm to te nerest degree () te lengt of te sortest digonl to te nerest m. 4. Two trees (t points nd ) re on te opposite side of lke from n oserver (t point.) Te oserver notes tt te ngles etween te lines of sigt to te trees is 35, te distne from te oservtion point to one tree () is 35 m nd to te oter tree () is 6 m. Find te distne etween te two trees to te nerest meter. 35 m D 35 6 m 5. Two sips leve port t te sme time. One trvels t speed of 30 mp in diretion N40 W, wile te oter trvels t speed of 5 mp in diretion S5 W. How fr, to te nerest mile, re te ots prt fter one our? 6. irport is 30 miles from irport on ering of S40. pilot wises to fly from to, ut to void storm must first fly due st t speed of 10 mp for n our, nd ten from tis point (ll it ) turns to fly to. Find te distne, to te nerest mile, nd te ering, to te nerest degree, tt te pilot must fly to irport W N S 7. ot is nored off long strigt soreline wi runs due est nd west. Te ering of te ot from two ottges on te sore 1 miles prt re S41. nd S5. W. How fr is te ot from e ottge, to te nerest tent of mile? mi 5. pter 3: Tringle Trigonometry Pge 16
17 1.3.4: TH R RUL Te re Rule sttes tt for ny tringle, re 1 sin Proof of te re Rule: In te tringle, drop perpendiulr line D of lengt from te vertex to te side. y rigt tringle trigonometry: sin rerrnging: sin D Te re of te tringle is : 1 Sustituting for : xmple 1: 1 1 (sin) re 1 sin Find te ext re of regulr otgon, if te mesure of te distne etween opposite verties is 4 units. Te otgon is omposed of ongruent isoseles tringles, wi e ve n pex ngle mesuring Te equl sides of e tringle mesure 1 units Using te re Rule te re of one tringle will e: Te re of tringles is re 1 sin sin re 36 Terefore te re of te otgon is sq. units pter 3: Tringle Trigonometry Pge 17
18 xmple : Find te re of tringle wi s sides of 3 m, 5 m nd 6 m, to te nerest tent. Drw tringle, lel te verties s unknowns, nd sides wit te know vlues. To find te re we need to ve one ngle of te tringle. To find, use te Lw of osines wit os + + os os ( 13) Now we n use te re Rule: re 1 sin sin30 1 Terefore te re of te tringle is pproximtely 7.5 units Heron s Formul ltoug it is eyond our sope to prove t tis time, Heron s Formul gives us metod to find te re of tringle given te lengt of te tree sides. Te re of tringle is re s( s )( s )( s ) were s 1 ( + + ) is te semiperimeter of te tringle. xmple : Find re of tringle wi s sides of 3 m, 5 m nd 6 m, to te nerest tent. Drw tringle, lel te verties s unknowns, nd sides wit te know vlues. Ten s 1 7 ( ) nd re s 7 7 ( s )( s )( s ) ( 7 5)( 7 6)( 7 3) ( )( 1)( 4) units pter 3: Tringle Trigonometry Pge 1
19 xerise Use te re Rule to find te re of te tringle (pproximte to te nerest wole), given: () 67, nd () 0, 45, 1 () 10, 16 nd 4 (d) 9, 16 nd 170. Use Herons Formul in 1() nd 1(d) to verify your nswer 3. Using te results you found in xerise 3.3 question 3, find te re of te prllelogrm wit sides of 0 m nd 16 m D 4. To estimte te re of smll lke, students tke te following mesurements: F 50 m 10 m D 60 m D 100 m F 0 m F 10 m 90 D D Find te re of te lke to te nerest undred squre meters. pter 3: Tringle Trigonometry Pge 19
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