Radians/Arc+Length+++! Converting++Between++Radians++and++Degrees+

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1 Radians/ArcLength ConvertingBetweenRadiansandDegrees Anglemeasurementcanbeexpressedinboth & Dependingonthecircumstance,itmaybenecessarytoconvertbetweenthetwounits ofangularmeasurement. Since2#=360,thefollowingequationscanbedetermined: ConvertingfromRadians'toDegrees' ConvertingfromDegrees'toRadians' ExactValue ApproximateValue Ex1:Convert todegrees Ex2:Convert240 toradians Ex3:Covert0.85radtodegrees.Roundtothenearesttenth. GeneralFormofCoterminalAngles Anygivenanglehasaninfinitenumberofanglescoterminalwithit,sinceeachtimeyoumake onfullrotationformtheterminalarm,youarrivebackatthesameterminalarm.angles coterminalwithanyanglecanbedescribedusingtheexpressions: Ex1: a) Expresstheanglescoterminalwith110 ingeneralform.identifytheanglescoterminal thatsatisfythedomaink720 <720 For'DegreesFor'Radians ± 360 ± 2

2 b)expresstheanglescoterminalwith ingeneralform.identifytheanglescoterminal with 8 inthedomain 4 < 4 3 CalculatingArcLength Therelationshipbetweentheradianmeasureofanangle,thearclength,andthe radiusofacircleisgivenbytheformula: Where: ristheradius ismeasuredinradians ArcLength= Ex1:Rosemarieistakingacourseinindustrialengineering.Foranassignmentsheis designingtheinterfaceofadvdplayer.inherplan,sheincludesadecorativearcbelow theon/offbutton.thearchascentralangle130 inacirclewithradius6.7mm. Determinethelengthofthearctothenearesttenthofamillimeter. Homework #2,4,6,7[abe],8[ad],11[aceg],12[bc],16

3 4.2$StandardPosition/Co$terminalAngles StandardPosition: AnangleinstandardpositionisanangleonaCartesianplanwithitsvertexatthe originandonearm,calledtheinitialarm,alongthepositivex$$$axis.allanglesare measurebetweentheinitialarmandtherotatingarmcalledtheterminalarm. Whentheterminalarmrotatescounter clockwise:positiveangle Whentheterminalarmrotates clockwise:negativeangle Ex1)Determinethequadrantinwhichtheangle terminates. Ex2)DeterminethequadrantinwhichtheEx3)Determinethequadrantinwhichthe angle terminates.angle# terminates. CoterminalAngles: Sinceanglesinstandardpositioncanbepositive,negative,orhavemultiplerotations aroundthecartesianplane,therewillbecasesinwhichangleswillhaveterminalarmsin thesameposition.

4 Ex1)Findonepositiveandonenegativeanglethatiscoterminalwith100.Then,finda coterminalanglethatrotatesatleastonetime. Positive Negative Atleastonepositiverotation

5 TrigonometricRatiosofAnglesinStandardPosition: Ex1)Findthevaluesofcsc,sec,andcotforanangleterminatingatpointP(3,7) Ex2)Findtheexactvaluesofsin,cos,andtanforanangleterminatingatpoint P($2,5)

6 Lesson&3(&The&Unit&Circle& AcircleontheCartesianplanewithitscenterat theoriginandaradiusof1unitiscalledthe: YoucanderivetheequationoftheunitcirclebyapplyingthePythagoreantheorem. LetP(x,y)representanypointontheunitcircle: Toindicatethethreesidesofthetriangle,adottedverticallineisdrawnfrompointPtothexCaxis.The legsoftherighttrianglearelabeledasxandy. ApplythePythagoreantheoremtotherighttriangle givestheequationoftheunitcircle,whichis: Usingthesidelengthsofxand&yandthehypotenuseof1,theprimarytrigratioscanbe definedasfollows: sin = 1 = cos = 1 = tan = Thereciprocaltrigonometricratioscanbedefinedas: csc = 1 sec = 1 cot =

7 DefiningSpecialTrianglesintheUnitCircle Together,therearetwospecialtrianglesthatmakeuptheentireunitcircle.Thefirstoneisthe45C45C90triangle withhypotenuse1,andthesecondisthe30c60c90trianglewithhypotenuse sin cos Anglesintheunit circlecorrespondto anglesinthesespecial triangles Thesetrianglescanbeplaced intoall4quadrantsoftheunit Circle.But,tofullycomprehend thebehaviorofthiscirclewe reallyonlyneedtounderstand thebehaviorinqi,andthen applyittoqii,qiii,andqiv. Infact:Almostalloftheangles intheunitcirclearemultiples of30 ( 6 )

8 TheUnitCircle Usingthedatacollectedfromthespecialtriangleswecanfillintheunitcircle

9 Findtheexactvaluesof#, #, # ontheunitcircleforthefollowinganglemeasurements Ex1) 4 3 rad Ex2)330 Findtheexactvaluesof, #, # ontheunitcircleforthefollowinganglemeasurements Ex1) radex2) 120

10 Homework: 1)sin135 A:[ ] 2)csc750 A:[2] 3)sec 5 3 A:[2] 3)cos( 7 4 )A:[ ] Page186:#1[a,c],4,5,10,15

11 Homework:Page202#1,2,3,4[bd],6,7,9,10[abd],12,13,17[a] Page211#1,2,3[ad],4[ace],5[abdf],6,19,20 MoreTrigonometricRatiosandIntroductiontoTrigonometricEquations Warm6upProblem:Determinetheexactvalueofthefollowingtrigratios a)sin( 300 ) b)sec 6 ApproximateValuesforTrigonometricRatios Youcandetermineapproximatevaluesforsine,cosine,andtangentusingyourgraphingcalculator. MAKESUREYOUAREINTHECORRECTMODEWHENEVALUATINGTRIGRATIOSONYOURCALCULATORS. [RADIANORDEGREE] Somedevicescancomputenegativeangles,howeveryoushoulduseyourknowledgeofreferenceangles andcoterminalanglestocomputenegativeangles. Whencomputingreciprocaltrigratios[secant,cosecant,cotangent]onacalculator,makesuretousethe correctreciprocalrelationship. Ex1)Determinetheapproximatevalueforeachtrigonometricratio.Giveyouranswertofour decimalplaces. a)tan 7 5 b)csc( 70 ) Howcanyoufindthemeasureoftheanglewhenthevalueofthetrigonometricratioisgiven?Usetheinverse trigonometricfunctionkeysonyourdevices. Ex:sin 30 = 0.5 sin 0.5 = 30 sin istheabbreviationforthe inverseofsine Donnot confusethiswith(sin 30 ) whichmeans #,orcsc 30 Thesekeysonlyreturnoneanswer,whenthereareusuallytwo angleswiththesamevalueinafullrotation.youwillneedto applyknowledgeofreferenceanglesandcoterminalangles todetermineallsolutions

12 Determininganglesgiventhetrigonometricratio Ex2:Determinethemeasuresofallanglesthatsatisfythefollowingratios a) sin θ = 0.879inthedomain0 < 2.Giveanswerstothenearesttenthofaradian b)cos θ = 0.366inthedomain0 θ < 360.Giveexactanswers. b)sec = inthedomain 2 < 2.Giveexactanswers.

13 Calculatingtrigonometricratiosforpointsnotontheunitcircle Ex3)ThepointA(64,3)liesontheterminalarmofanangleinstandardposition.Whatistheexact valueofeachtrigonometricratiofor? IntroductiontoTrigonometricEquations Lookattheequationcos = 1,0 < 2.Whataretheexactmeasuresofθ? 2 Howisthisequationrelatedto2 cos 1 = 0? Whatifthedomaingivenwas0 < 360? How$do$you$know$ whether$to$give$ your$answers$in$ degrees$or$radians?$

14 IntervalNotation Thenotation 0, representstheintervalfrom0to,andisanotherwayofwriting0 θ 0, meansthesameas: θ 0, meansthesameas: Ex1:Solveeachtrigonometricequationinthespecifieddomain. a)5 sin θ 2 = 1 3 sin θ,0 < 2b)3 csc 6 = 0,0 < 360 SolvinganEquationbyFactoring Ex2:Solvefor:tan θ 5 tan θ 4 = 0, 0 θ < 2

15 GeneralSolutionofaTrigonometricEquation Ex3: a)solveforxovertheinterval0 θ < 2if:sin 1 = 0 b)determinethegeneralsolutionforsin 1 = 0overtherealnumbersifxismeasuredinradians

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