第五章Trigonometry三角尽管三角学在ACT数学考试中所占比例不足7%，只有4或5道题，但这个知识点涉及面却很广。ACT数学考试试题可能会来自下列知识点中的一个。•①Angles角；•②TrigonometricFunctions三角函数；•③TrigonometricIdentities三角恒等式；•④GraphsofTrigonometricFunctions三角函数图像；•⑤RightTriangleTrigonometry直角三角函数；•⑥TriangleProblems三角形问题。第一节Angles角•一、Radians弧度•Anglescanbemeasuresindegreesorinradians(abbreviatedas“rad”).Theanglegivenbyacompleterevolutioncontains360°,whichis2πrad.•Therefore,•1rad=(180/π)°≈57.3°•1°=(π/180)rad≈0.017radThefollowingtablegivesthecorrespondencebetweendegreeandradianmeasuresofsomecommonanglesDegrees0°30°45°60°90°120°135°150°180°270°360°Radians0π/6π/4π/3π/22π/33π/45π/6π3π/22π二、AngleinStandardPosition角的标准坐标位置•Thestandardpositionofanangleoccurswhenweplaceitsvertexattheoriginofacoordinatesystemanditsinitialsideonthepositivex-axis.•Thequadrantthatcontainstheterminalsidedeterminesthequadrantthattheangleliesin.•Inthefigureabove,αrepresentsanangleinQuadrantI,whileβisinQuadrantIII.•Apositiveangleisobtainedbyrotatingtheinitialsidecounterclockwiseuntilitcoincideswiththeterminalside.Likewise,negativeanglesareobtainedbyclockwiserotation.•Inthefigureabove,αispositive,whileβisnegative.•Iftheterminalsideofanangleinstandardpositionisoneoftheaxes,theangleisaquadrantangle.•Forexample,90°(π/2)and-180°(-π)arequadrantangles.Everyangleinstandardpositionhasareferenceangle,whichisthepositiveacuteangleformedbytheterminalsideofthegivenangleandthex-axis.Seeexamplesbelow.第二节TrigonometricFunctions三角函数•Forageneralangleθinstandardposition,weletP(x,y)beanypointontheterminalsideofθandletrbethedistance|OP|asshowninthefigureabove.Thenwedefinethefollowingtrigonometricfunctions:•sinθ=y/rcscθ=r/y•cosθ=x/rsecθ=r/x•tanθ=y/xcotθ=x/y•NoticefromthediagramthatθisinQuadrantII,wherex<0andy>0(risalwayspositive).Therefore,sinθandcscθaretheonlytworatiosthatarepositiveinQuadrantII.Alltheotherratiosarenegative.ThisistrueforallQuadrantIIangles.TrigFunctionsofImportantAngles重要角的三角函数值Angle(θ)Radiansinθcosθtanθ0°001030°π/61/2√3/2√3/345°π/4√2/2√2/2160°π/3√3/21/2√390°π/210UNDEFINED第三节TrigonometricIdentities三角恒等式•Atrigonometricidentityisanequationinvolvingtrigonometricfunctionsthatholdstrueforallangles.Herearesomeofthefamiliaridentitiesthatyoushouldknow.•1.QuotientIdentities•sinθ=1/cscθcosθ=1/secθcotθ=1/tanθ•tanθ=sinθ/cosθ•cotθ=cosθ/sinθ•2.PythagoreanIdentities•Sin²θ+cos²θ=1•1+tan²θ=sec²θ•1+cot²θ=csc²θ•3.Periodicity•Sinceanglesθand2kπ(wherekZ)havetheterminal∈side,wehave•Sin(θ+2kπ)=sinθ•cos(θ+2kπ)=cosθ•4.Symmetry•Sin(-θ)=-sinθ•con(-θ)=cosθ•5.DoubleAngleFormulas•Sin2θ=2sinθcosθ•Cos2θ=cos²θ-sin²θ=2cos²θ-1=1-2sin²θ•6.SumandDifferenceofTwoAngles•Sin(α+β)=sinαcosβ+cosαsinβ•Sin(α-β)=sinαcosβ-cosαsinβ•Cos(α+β)=cosαcosβ-sinαsinβ•cos(α-β)=cosαcosβ+sinαsinβ第四节TheGraphsofTrigonometricFunctions三角函数图像•1.Periodicity周期性•Allofthetrigfunctionsareperiodic,thatis,f(x+p)=f(x)forallxinthedomainoff,meaningthegraphrepeatsitpatternaftersomeintervalinx.•Thesmallestpossiblevalueofpintheexpressionf(x+p)=f(x)iscalledthefundamentalperiodofthefunction,sometimesjustcalledtheperiod.•2.Amplitude幅度•Thesineandcosinefunctionshaveanadditionalproperty,amplitude,whichishalfthedistancefromthecrest(top)tothebottomofawave.Forasineorcosinecurvethathasnotbeenverticallytranslated,theamplitudeissimplythedistancefromthex-axistothecrestofthewave.•Thefollowingarethegraphsofthesixtrigfunctions.Thedomain,range,fundamentalperiod,andamplitude(whereapplicable)aregivenforeachfunction.•Thegraphofy=AsinBxandy=AcosBx•Fundamentalperiod=2π/|B|•Amplitude=|A|•Forexample,thegraphofthefunctiony=4sin3xhasfundamentalperiod2π/3andamplitude4•Thefunctiony=-6cos1/2xhasfundamentalperiod4πandamplitude6.第五节RightTriangleTrigonometry直角三角函数•sinx=b/candsiny=a/c•cosx=a/candcosy=b/c•tanx=b/aandtany=a/b•cotx=a/bandsecy=c/b第六节TriangleProblems三角形问题•1.TheLawofSines正弦定律•sinA/a=sinB/b=sinC/c•2.TheLawofCosines余弦定律•a²=b²+c²-2bccosA•b²=a²+c²-2accosB•c²=a²+b²-2abcosC
