8.1. The Sine Law. Investigate. Tools
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1 8.1 Te Sine Lw Mimi 50 ermud Tringle ermud 1600 km Sn Jun 74 Puerto Rio Te ermud Tringle, in te nort tlnti Oen, is te lotion of severl unexplined plne nd sip disppernes. Vrious teories ve een suggested to explin tis mysterious penomenon, su s voltile weter ptterns, mgneti ompss vrition, umn error, nd even lien dution. Suppose you took ruise round te ermud Tringle. How ould you determine its perimeter from te given informtion? Investigte Tools protrtor nd ruler OR omputer wit Te Geometer s Sketpd Tenology Tip If you re using Te Geometer s Sketpd, you n ess te sine funtion from te Funtions drop-down menu on te lultor. Tenology Tip Insted of onstruting new tringles to ompre results, simply move one of te verties of to exmine oter ute tringles. Te lulted mesures will nge utomtilly. How re te side lengts nd sines of ngles relted in n ute tringle? 1. ) Drw n ute. ) Mesure te side lengts nd te ngles. ) lulte te sine of e ngle. sin 2. ) lulte te rtios,,,,, nd. sin sin sin ) Do you notie ny reltionsips etween te rtios? Explin. 3. ) lulte e rtio. sin sin ) ompre tese results nd explin wt you notie. 4. Repet steps 1 to 3 for two different tringles. re te results te sme? Explin. 5. Reflet Summrize te reltionsip of te sides nd sines of ngles in n ute tringle. 396 MHR pter 8
2 Te reltionsip you investigted is lled te sine lw. To sow wy te sine lw olds true, drw n ute tringle nd dd n ltitude,, from one of te verties. Te ltitude splits into two smller rigt tringles, X nd X. X sine lw te reltionsip etween te sides nd teir opposite ngles in ny ute : = = sin sin Find n expression for. Fous on X: Multiply ot sides y. () Fous on X: sin Multiply ot sides y. (sin ) Set te two expressions for equl. () (sin ) ( ) ( sin ) Divide ot sides y. ( sin ) ( sin ) Divide ot sides y sin. sin ( sin ) sin Tis proess n e repeted using different ltitude. omining te results gives te sine lw. Y sin sin Even toug tere re tree prts to tis eqution, you only use two prts t time. Te oie of wi two to use depends on wt informtion is given. You n pply te sine lw to find n unknown side lengt of n ute tringle if you know two ngles nd one of te side lengts. 8.1 Te Sine Lw MHR 397
3 Exmple 1 Find Side Lengt Using te Sine Lw iyle pt forms 66 ngle wit one lok of nl. t distne of 2.5 km long te iyle pt, te ngle seprting tis lok from te next lok is 52. How fr prt re te two loks, to te nerest tent of kilometre? Solution Lok 1 Lok km 52 Drw simplified digrm sowing te given informtion. L km p 52 P K efore you n use te sine lw to find lengt p, you need to find te mesure of K. K Te sum of te interior ngles in tringle is Now, pply te sine lw. p k sin P sin K p 2.5 sin 52 sin 62 p sin 52 sin 52 Sustitute te given informtion. sin Multiply ot sides y sin 52. sin 62 p sin 52 p sin 62 Te distne etween te two loks is 2.2 km, to te nerest tent of kilometre. 398 MHR pter 8
4 You n lso use te sine lw to find n unknown ngle if you know two sides nd n ngle opposite one of te known sides. Sometimes te sine lw ppers in different form. Rell te result otined erlier for. () (sin ) ( ) ( sin ) Divide ot sides y. omining te results gives n lterntive form of te sine lw. sin sin sin Eiter form n e used to find missing side or ngle tis lterntive form is little esier to use wen finding n unknown ngle. X Liter onnetions proportion is sttement tt sys tt two rtios re equl. For exmple, 2 = Te proportion still olds true if you invert te numertor nd denomintor of ot rtios. 3 = Tis is wy eiter form of te sine lw n e pplied. Exmple 2 Find n ngle Using te Sine Lw In ute TUV, TU 11.1 m, UV 5.8 m, nd T 31. Find te mesure of V, to te nerest degree. Solution Drw digrm nd lel te given informtion. V pply te sine lw. sin V v sin T t m sin V sin 31 Sustitute te given informtion sin V sin Multiply ot sides y sin 31 sin V sin V V sin 1 ( ) Use te inverse sine opertion. V 80 T 5.8 m U Tenology Tip ns in te lultor disply indites te lst nswer. To use te lst nswer in lultion, press n for [NS]. Te mesure of V is out Te Sine Lw MHR 399
5 Exmple 3 Perimeter of te ermud Tringle Use te informtion given on te digrm to determine te perimeter of te ermud Tringle, to te nerest undred kilometres. M km 74 S Solution Drw simplified digrm sowing te given informtion. First, find Now, pply te sine lw twie to find sides m nd s. m sin M sin m 1600 sin 50 sin 56 m sin 50 m sin 56 M km 74 S Metod 1: Use Side s sin S s sin 74 sin 1600 sin 56 s sin 74 s sin 56 Metod 2: Use Side m s sin S s sin 74 m sin M 1478 sin 50 s sin 74 s sin 50 To find te perimeter, dd te sides. P m s Terefore, te perimeter of te ermud Tringle is pproximtely 4900 km. 400 MHR pter 8
6 Key onepts In n ute, te sine lw sttes tt sin sin Te sine lw n e used to find n unknown side if two ngles nd side re known n unknown ngle if two sides nd te ngle opposite one of te known sides re known Te sine lw n lso e written in te form sin sin ommunite Your Understnding 1 ) Wy does te sine lw ve tree prts to its eqution? ) Do you use ll tree prts t one? How n you tell wi prts to use? 2 Wi of te following tringles n e solved using te sine lw? For tose tt n, explin ow. For tose tt nnot, explin wy not. K 5.3 m m 52 J m L Q mm R 16 mm P V 38 km 40 km U T 27 km 3 4 How mny piees of informtion out tringle s side lengts nd ngles re needed in order to solve it using te sine lw? Wt possile omintions will work? Tere re two forms of te sine lw, relting sides nd nd nd : nd sin sin Explin wen it is esier to pply e form. rete exmples to support your explntion. 8.1 Te Sine Lw MHR 401
7 Prtise For elp wit questions 1 nd 2, see Exmple Find te lengt of te indited side in e tringle, to te nerest unit. ) ) F 6. Solve e tringle. Round nswers to te nerest unit, if neessry. ) D 34 m =? m 2. Find te lengt of te indited side in e tringle, to te nerest tent of unit. ) ) D e =? m D E ) 73 H 12 m 29 m Q W 2.5 m For elp wit questions 3 nd 4, see Exmple Find te mesure of te indited ngle in e tringle, to te nerest degree. ) Y ) X 62? mm =? Z mm 4. Drw digrm nd lel te given informtion. Ten, find te mesure of te indited ngle in e tringle, to te nerest degree. ) In ute GHK, G 47, 5 m, nd g 4 m. Find H. ) In ute RST, S 72, t 1.5 m, nd s 1.8 m. Find T. E R 2.2 m mm? e =? m F K P 11 m 7. Drw digrm nd lel te given informtion. Ten, solve e tringle. Round nswers to te nerest unit, if neessry. ) In ute KR, k 15 mm, r 13 mm, nd K 68. ) In ute UJF, j 23 km, U 57, nd F Use Tenology ek your nswers to question 7 using dynmi geometry softwre. onnet nd pply 9. sign is supported y pole nd le, s sown. Te le is tted to te wll 2.2 m ove te se of te pole. le 61 R 46 For elp wit questions 5 to 7, see Exmple Solve e tringle. Round nswers to te nerest unit, if neessry. ) P ) F L 18 m 70 M 402 MHR pter 8 E m D WELOME pole 74 ) Find te lengt of te pole, to te nerest tent of metre. ) Find te lengt of te le, to te nerest tent of metre.
8 10. smll ommeril plne nd jet irliner re 7.5 km from e oter, t te sme ltitude. From n oservtion tower, te two irrft re seprted y n ngle of 68. If te jet irliner is 5.2 km from te oservtion tower, ow fr is te ommeril plne from te oservtion tower, to te nerest tent of kilometre? plne 7.5 km jet 13. pter Prolem You nd your prtner re oserving n irrft from two oservtion deks, loted 5.0 km prt. From your point of view, te irrft is t n ngle of elevtion of 70. From your prtner s point of view, te ngle of elevtion is 55. Determine te ltitude of te irrft, to te nerest tent of kilometre km km telepone pole is supported y n 18-m guy wire tt mkes n ngle of 50 wit te orizontl ground. 14-m guy wire is to e fstened on te oter side of te pole for reinforement. ot wires tt to te pole t its top. Round your nswers to te nerest unit, if neessry. ) Wt ngle sould te seond wire mke wit te ground? ) How tll is te pole? ) How fr is te se of e wire from te se of te pole? d) ould you solve tis prolem witout using te sine lw? Explin. 12. ridge ross vlley is 150 m in lengt. Te vlley wlls mke ngles of 60 nd 54 wit te ridge tt spns it, s sown. How deep is te vlley, to te nerest metre? 14. In isoseles, 15 m, 11 m, nd n tis tringle e solved using te sine lw? If so, solve it nd explin your resoning. If not, explin wy not. 15. To mesure te eigt of wter tower, Kelly wlks 0.15 km from te se of te tower long n inlined pt to point P. From P, te top of te tower ppers t n ngle of elevtion of 12. P 12 8 Find te eigt of te wter tower, to te nerest metre Te Sine Lw MHR 403
9 16. You n find te re of tringle if you know ll tree of its side lengts y pplying Heron s formul, 2s(s )(s )(s ), were,, nd re te side lengts, nd 1 s ( ). 2 ) Use Heron s formul to find te re of te ermud Tringle. ) ek tis result y estimting te eigt of te tringle nd pplying te formul for te re of tringle,. 2 Did You Know? Heron ws mtemtiin, engineer, nd inventor orn in lexndri, Greee, during te first entury. mong is omplisments ws te invention of te first stem engine, lled n elopile. Te priniple Heron used in te elopile is similr to tt used in tody s jet engines. 17. Refer to question 16. Ln s grden is in te spe of n ute tringle. Two of te verties ve ngles of 55 nd 58. Te side joining tese verties is 6.2 m in lengt. Wt is te re of tis grden, to te nerest tent of squre metre? 18. ) rete prolem involving te sine lw for wi te nswer is 15 m. ) Solve te prolem. ) Trde wit prtner nd solve e oter s prolem. ek your solutions. 19. Does te sine lw work if you reple sines wit osines or tngents? Investigte severl tringles nd explin weter tere is su reltionsip. Extend 20. ) Sow tt te sine lw simplifies to te sine rtio wen one of te ngles of tringle is 90. ) Would you use te sine lw to solve rigt tringles? Explin. 21. Refer to question 16. Sow tt Heron s formul n lso e written s follows: 1 2( )( )( )( ) Mt ontest Frist, Mielle, Dniel, nd Vpv e soot two rrows t te trget sown. E ring s different point vlue. Frist soots one rrow in ring nd one in ring. Her sore is 17. Mielle soots one in ring nd one in ring. Her sore is 12. Dniel soots one in ring nd one in ring. His sore is 15. Vpv soots two rrows into ring. Vpv s sore is D 18 E Mt ontest Wen e of tree different numers is dded to te verge of te oter two, te results re 105, 106, nd 125. Wt is te verge of te tree originl numers? D 57 E MHR pter 8
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