% % 6'23" %to%dd%% % % (B)%%Convert%128 % WITHOUT%A%CALCULATOR% % % WITHOUT%A%CALCULATOR% (C)%Convert% ' %to%dd%% %
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1 G Lessn1.1:Angles,Degrees,andArcs DegreeMeasurefAngles Anangleisfrmedbyrtatingarayarunditsendpintsthatthe sideftheangleremainsfixedand thesecndray,calledthe side,rtatesarundthecmmnendpint,calledthevertex,inaplaneuntil itreachesitsterminalpsitin. Symbls:α =, β =,γ =,θ =,φ = Cunterclckwisertatin= Clckwisertatin= Cterminalangles: ClassifyingAngles:AcuteB RightB ObtuseB StraightB Acirclecntains360,therefre,1 = revlutin= 2psitiveanglesarecmplementaryif 2psitiveanglesaresupplementaryif Adegreecanbedividedintsmallerpartsfraccuracyintwways: (1)Decimaldegree(DD)ex:36.25 (2)DegreeAMinuteASecnd(DMS)ex: 5 12'32" =5degrees,12minutes,32secnds Nte:InDMS,eachdegreeindividedint60equalminutes,andeachminuteisdividedint60equalsecnds. EXAMPLE1 (A)Cnvert tdms (B)Cnvert tdms (C)Cnvert tdms (D)WITHOUTACALCULATOR,cnvert 43.5 tdms EXAMPLE2 (A)Cnvert12 6'23" tdd (B)Cnvert128 42'8" tdd WITHOUTACALCULATOR WITHOUTACALCULATOR (C)Cnvert 32 30' tdd (D)Cnvert 32 45' tdd
2 AnglesandArcs GivenanarcRQfacirclewithcenterP,theangleRPQiscalledthe thatis by thearcrq.likewise,arcrqissubtendedbytheanglerpq. ArcLength UsethegivencircletLABEL:θ isthecentralangle,sisthearcsubtendedby theangle. Usethefllwingprprtintslvefrθ,srCwhennecessary: θ s = wherec=circumference 360 C (recall C = πd = 2πr,dntrund π ) Example3 Hwlngisanarcsubtendedbyacentralanglef 6.23 nacirclewithradius10cm?includea labeledcircleandprprtinbefreslving. Example4 Hwlargeisthecentralanglefasubtendedarcwhselengthis4.427mnacirclewithradius5 meters?includealabeledcircleandprprtinbefreslving. SectrArea UsethegivencircletLABEL:θ isthecentralangle,aistheareafthe sectrdefinedbythesubtendedarcandradiifthecentralangle. Usethefllwingprprtintslvefrθ,A,rC,swhennecessary: A πr = θ r A πr = s 2 C wherec=circumference Example5 Findtheareafasectrwithacentralanglef andaradiusf20yd.includealabeled circleandprprtinbefreslving.
3 Lessn1.2:SimilarTriangles Euclid stherem:if2trianglesaresimilar,thentheircrrespndingsidesareprprtinal. a b c = = ex: 2 = 1 = 3 a' b' c' ac a c b b Recallfrmgemetry: Twtrianglesaresimilariftwanglesfnetrianglehavethesamemeasureastwanglesfthethertriangle. Iftwrighttrianglesaresimilar,thenanacuteangleinnerighttrianglemusthavethesamemeasureasanacuteangle inthether.(nte:thesumfthemeasuresfthethreeinteriranglesfanytriangleis180.) Explre/Discuss1:Drawandidentifypairsftrianglesfrmpage15thataresimilarandexplainwhy. Example1 Draw&labeladiagram Atreecastsashadwf31feetatthesametimea5ftverticalplecastsashadwf0.56feet.Hwhighisthetree? EXAMPLE2 UsethedrawingstfindthevaluesfBCandAB B ACTUALSIZE SCALEMODEL B 1.76in C A C A 500ft 3in
4 Lessn1.3:TrignmetricRatisandRightTriangles TrignmetricRatis Therearesixtrigratis:sine,csine,tangent,csecant,secant,&ctangent(Recall:SOHCAHTOA) Theirabbreviatins,respectively,are:sin,cs,tan,csc,sec,ct TRIGFUNCTION RECIPROCALFUNCTIONS sinθ = ppsite hyptenuse hyp pp csθ = adjacent hyptenuse hyptenuse cscθ = ppsite = 1 sinθ hyptenuse secθ = = 1 adjacent csθ tanθ = ppsite adjacent adjacent ctθ = ppsite = 1 tanθ adj *Frcscθ,secθ,andctθ yumustusethesin/cs/tanfunctinscmbinedwiththe 1 x r x 1 functinnthe calculatr.donotusethe sin 1,cs 1, tan 1 functinsnthecalculatrfrthereciprcalfunctins. Example1:Evaluate.(Makesureyurcalculatrisindegreemde.) (A) cs38.27 (B) sin 37 44' (C) ct (D) csc 77 53' *Whenyuareslvingfrθ,yuusethe sin 1,cs 1, tan 1 buttns.* (Theirnamesarearcsine,arccsine,&arctangent.They und thesine,csine&tangentfunctins) Prblemswilllklikethefllwing: (1) θ = sin (equivalentt θ = arcsin.7214 ) r (2) csθ =.3174 (takearccsfbthsides) θ = cs 1 csθ ( ) = cs θ = Example2 Findeachacuteangle,θ ttheaccuracyindicated. (A) θ = arccs.0367 tthenearest1 (B) θ = arcsin.0367 tthenearest10 (C) θ = sin tthenearest10 (D) θ = tan tthenearest1 (E) tanθ = t2decimalplaces (F) csθ =.7335 t3decimalplaces (G) secθ = 1.6 tthenearestthusandth (H) ctθ = tthenearestdegree
5 SlvingRightTriangles Methd#1:Given1sidelength&1acuteangle Methd#2:Given2sidelengths Step1:Usesin,cs,rtanfthegivenangle Step1:Usearcsin,arccs,rarctanfthe tfind2missingsidelengths. givensidelengthstfindanacuteangle. Step2:Slvefrthethercmplementaryangle. Step3:UsethePythagreanTheremtslve Frthe3 rd sidelength. Step2:Slvefrthethercmplementaryangle. Example3:Usethefigurettherighttslveeachrighttriangle. (A)α = 62 18'36", b = 14.2 (B)b = 1.6, c = 6.5 b' a' c'
6 Lessn1.4:RightTriangleApplicatins Strategiesfrslvingrighttriangleapplicatinprblems: 1.AlwaysdrawawellBlabeledrighttrianglefthegiveninfrmatin. 2.Determinethevariable(s)finterestnthedrawing. 3.Usetrigfunctins,Pythagreantherem,rareafrmulastslve. (Nte:AreafaTriangle=½base*height) AnglefElevatin: AnglefDepressin: Example1:Onawindyday,Mr.MntrellatakeshisfamilyutttheparkfrafunBfilleddayfkiteflying.Asthekite fliesverhead,mr.mntrellaiscncernedthenearbycalifrniaoaktreespseathreatttheblissfulday.mr.mntrella knwstheaveragecalifrniaoakgrwsfrm50 120feetandhiskitehas200feetfstring.Whatistheminimumangle felevatinthatmr.mntrellamustmaintainwithhiskitetkeepclearfthetrees? Example2:Ahelicpterhvering1000metersveradistressedshipintheceannticesarescuevesseltthewest apprachingthedistressedship.thehelicpterpiltdeterminestheanglefdepressinttherescuevesseltbe apprximately15 45.Iftherescuevesselistravelingataspeedf50kmperhur,hwlngwillittaketherescuevessel treachthedistressedship? Example3:AtriangularpltismarkedbypstsA,B,andC.ThedistancebetweenAandBis200feet.Thesurveyr measurestheanglebacandtheangleabcanddiscverstheanglesmeasure40 and25,respectively.determinethe areafthetriangularpltfland.
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