TRIGONOMETRY:+2.1++Degrees+&+Radians+ Definitions: 1degree/ 1radian s s FORMULA: θ = radians;wheres=arclength,r=radius r θ r IMPLICATIONOFFORMULA:Ifs=rthen θ =1radian EXAMPLE1:Whatistheradianmeasureofacentralanglesubtendedbyanarcof32cminacircleofradius8cm.? THINK!!Whatistheradianmeasureofanangleof180?Of90?Of60?Of45?Of30? ConvertingRadians Degrees Deg Rad:multiplyby π 180 o Rad Deg:multiplyby 180o π EXAMPLE2a:Findthedegreemeasureof1.5radiansinexactformandindecimalformto4places. EXAMPLE2b:Findtheradianmeasureof 120o inexactformandindecimalformto4places.
AnglesinStandardPosition:vertexisatorigin,initialsideispositionedalongthex/axis( 0 o ) LABEL: α ispositive(counterclockwise) β isnegative(clockwise) 0 o EXAMPLE3:Sketchthefollowinganglesinstandardposition: π (A) 6 (B) 495 o Coterminalangleshavethesameinitial&terminalsides. Therefore,themeasuresofcoterminalanglesdifferbyintegermultiplesof or. EXAMPLE4:Whichofthefollowingpairsofanglesarecoterminal?Createalabeledsketchofeachangleonthesame axis. (A) α = 90, β = 90 (B) α = 750, β = 30 π 25π (C) α =, β = 3π 7π (D) α =, β = 6 6 4 4
FORMULAtofindarclength: s rθ = wheres=arclength,r=radius,θ =centralanglein+radians EXAMPLE5:Inacircleofradius6ft,findthearclengthsubtendedbyacentralangleof: (A) 1.7 θ = radians (B) 40 θ = o CIRCLE:AREAOFASECTOR part whole : A πr 2 = θ 2π NowsolvethisequationforA: EXAMPLE6:Inacircleofradius7in,findtheareaofthesectorwithcentralangle: (A)0.1332radians (B)110 o
2.3+ +Trigonometric+Functions:+Unit+Circle+Approach+ Algebra2Review:Thegraphof a 2 + b 2 = 1 isacirclewithcenterat(0,0)andradius=1 ThisisthedefinitionofaUNITCIRCLE P(a,b) WITHINAUNITCIRCLE:a=cosx,andb=sinxorP(a,b)=P(cosx,sinx) EXPLAINWHY: (1,0) Whatisthedomain&rangeforsine(y=sinx)&cosine(y=cosx)? DOMAIN: RANGE: Definitions: isanangleindegrees. isanangleinradians. representsarandompointontheterminalsideofangleθ oranglex. isthedistancefromtheorigintopointp. Referencetriangle: Referenceangle: Definethe6trigfunctions: Fillin+or valuesforthefunctionsinrelationtothe4quadrants sinθ = cscθ = = sin sin cosθ = secθ = = cos cos tanθ = cotθ = = tan tan sin sin tanθ = cotθ = = cos cos tan tan
EXAMPLE1:Findtheexactvaluesofeachofthe6trigfunctionsfortheanglexwithterminalsidecontainingP(/6,8). Besuretosketchthetriangleonthecoordinateplane. EXAMPLE2:Findtheexactvaluesofeachofthe6trigfunctionsfortheangleθ withterminalsidecontainingp(/4,/3). Besuretosketchthetriangleonthecoordinateplane. EXAMPLE3:Findtheexactvalueofeachoftheother5trigfunctionsfortheanglex(withoutfindingx)giventhatthe terminalsideofxliesinquadrantiand sin x = 5 13 EXAMPLE4:Findtheexactvalueofeachoftheother5trigfunctionsfortheangleθ (withoutfindingθ )giventhatthe terminalsideofθ liesinquadrantiiand tanθ = 3 4 EXAMPLE5:Useacalculatorandevaluateto4decimalplaces(thisiswherethecalculatormodematters) (A) cos 303.73 = (B) sec( 2.805) = (C) tan 83 29 ( ) = (D) sin12 = (E) csc100 52 43" = (F) cot 9 =
TRIGONOMETRY:+2.5+Exact+Values+and+Properties+of+Trigonometric+Functions+ Quadrantalangle: Example1:Evaluateeachfunctionatthegivenquadrantalangle.Sketcheachangle. (A)sin3π/2 (B)sec(/π) (C)tan90 (D)cot(/270 ) DevelopingtheUnitCircle:EvaluatingTrigFunctionsofMultiplesofπ/4 Drawthereferencetrianglewithareferenceangleof45 ineachquadrant. Recordthesineandcosinevaluesatthegivenangle. QuadrantII(135 ) QuadrantI(45 ) QuadrantIII(225 ) QuadrantIV(315 ) Example2:Evaluateeachfunctionatthegivenangle.Giveexactanswersonly. (A)cos(5π/4) (B)tan(3π/4) (C)csc(45 ) (D)sec(/π/4) DevelopingtheUnitCircle:EvaluatingTrigFunctionsofMultiplesofπ/6 Drawtworeferencetriangleswithreferenceanglesof30 and60 ineachquadrant. Recordthesineandcosinevaluesatthegivenangle. QuadrantII(120 &150 ) QuadrantI(30 &60 ) QuadrantIII(210 &240 ) QuadrantIV(300 &330 ) Example3:Evaluateeachfunctionatthegivenangle.Giveexactanswersonly. (A)cot(5π/6) (B)csc(330 ) (C)sin(2π/3) (D)tan(4π/3)
FindingSpecialAngles Usingthegivenratioforeachtrigfunction,determinetheleastpositiveθindegreeandradianmeasure. Supposesinθ= 3.Drawareferencetriangleinthefirstquadrantwithsideoppositereferenceangle 3 and 2 hypotenuse2.observethatthisisaspecial30 /60 /90 triangle: Example4:Findtheleastpositiveangleforsecθ= 2. PeriodicFunctions: Note:Boththesineandcosinefunctionhaveaperiodof2π.Tangentandcotangentfunctionshaveaperiodof. LetQ(a,b)bethepointontheunitcirclethatliesontheterminalsideofananglehaving Radianmeasurex.Then,sincethereare2πradiansintheonecompleterotation,thepoint Q(a,b)liesontheterminalsideofx+2π. b=sinx=sin(x+2π) a=cosx=cos(x+2π) Example5:Ifsinx=0.7714,whatisthevalueofeachofthefollowing? (A)sin(x+2π) (B)sin(x 2π) (C)sin(x+14π) (D)sin(x 26π) FundamentalIdentities: ReciprocalIdentities TangentIdentities Odd/EvenIdentities PythagoreanIdentities! csc! = tan! = sin(!) = sin! sin!! + cos!! = 1! sec! = cot! = cos(!) = cos!! cot! = tan(!) = tan! Example6:Simplifyeachexpressiontoonetrigonometricfunctionusingthefundamentalidentities. (A)!!!"#!!!"#!! (B)tan(!) cos!(!)(c)!(!)!"!!