Trigonometry 三角学.ppt

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1、Trigonometry 三角学尽管三角学在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。Angles 角；Trigonometric Functions 三角函数；Trigonometric Identities 三角恒等式；Graphs of Trigonometric Functions 三角函数图像；Right Triangle Trigonometry 直角三角函数；Triangle Problems 三角形问题。第一节 Angles 角一、Radians 弧度Angles can be measures

2、in degrees or in radians(abbreviated as“rad”).The angle given by a complete revolution contains 360,which is 2 rad.Therefore,1 rad=(180/)57.31=(/180)rad 0.017 rad ThefollowingtablegivesthecorrespondencebetweendegreeandradianmeasuresofsomecommonanglesDegrees030456090120135150180270360Radians0/6/4/3/2

3、2/33/45/63/22二、AngleinStandardPosition角的标准坐标位置Thestandard positionofanangleoccurswhenweplaceitsvertexattheoriginofacoordinatesystemanditsinitial sideonthepositivex-axis.Thequadrantthatcontainstheterminalsidedeterminesthequadrantthattheangleliesin.Inthefigureabove,representsanangleinQuadrantI,whileis

4、inQuadrantIII.Apositive angle isobtainedbyrotatingtheinitialsidecounterclockwiseuntilitcoincideswiththeterminalside.Likewise,negative anglesareobtainedbyclockwiserotation.Inthefigureabove,ispositive,whileisnegative.Iftheterminalsideofanangleinstandardpositionisoneoftheaxes,theangleisaquadrantangle.F

5、orexample,90(/2)and-180(-)arequadrantangles.Everyangleinstandardpositionhasareference angle,whichisthepositiveacuteangleformedbytheterminalsideofthegivenangleandthex-axis.Seeexamplesbelow.第二节TrigonometricFunctions三角函数Forageneralangleinstandardposition,weletP(x,y)beanypointontheterminalsideofandletrb

6、ethedistance|OP|asshowninthefigureabove.Thenwedefinethefollowingtrigonometricfunctions:sin=y/rcsc=r/ycos=x/rsec=r/xtan=y/xcot=x/yNotice from the diagram that is in Quadrant II,where x0(r is always positive).Therefore,sin and csc are the only two ratios that are positive in Quadrant II.All the other

7、ratios are negative.This is true for all Quadrant II angles.TrigFunctionsofImportantAngles重要角的三角函数值Angle()Radiansincostan0001030/61/23/23/345/42/22/2160/33/21/2390/210UNDEFINED第三节 Trigonometric Identities 三角恒等式Atrigonometricidentityisanequationinvolvingtrigonometricfunctionsthatholdstrueforallangles

8、.Herearesomeofthefamiliaridentitiesthatyoushouldknow.1.Quotient Identitiessin=1/csccos=1/seccot=1/tantan=sin/coscot=cos/sin2.Pythagorean IdentitiesSin+cos=11+tan=sec1+cot=csc3.PeriodicitySinceanglesand2k(wherekZ)havetheterminalside,wehaveSin(+2k)=sincos(+2k)=cos4.SymmetrySin(-)=-sincon(-)=cos5.Doubl

9、e Angle FormulasSin2=2sincosCos2=cos-sin=2cos-1=1-2sin6.Sum and Difference of Two AnglesSin(+)=sincos+cossinSin(-)=sincos-cossinCos(+)=coscos-sinsincos(-)=coscos+sinsin第四节TheGraphsofTrigonometricFunctions三角函数图像1.Periodicity 周期性周期性Allofthetrigfunctionsareperiodic,thatis,f(x+p)=f(x)forallxinthedomaino

10、ff,meaningthegraphrepeatsitpatternaftersomeintervalinx.Thesmallestpossiblevalueofpintheexpressionf(x+p)=f(x)iscalledthefundamental periodofthefunction,sometimesjustcalledtheperiod.2.Amplitude 幅度幅度Thesineandcosinefunctionshaveanadditionalproperty,amplitude,whichishalfthedistancefromthecrest(top)tothe

11、bottomofawave.Forasineorcosinecurvethathasnotbeenverticallytranslated,theamplitudeissimplythedistancefromthex-axistothecrestofthewave.Thefollowingarethegraphsofthesixtrigfunctions.Thedomain,range,fundamentalperiod,andamplitude(whereapplicable)aregivenforeachfunction.Thegraphofy=AsinBxandy=AcosBxFund

12、amentalperiod=2/|B|Amplitude=|A|Forexample,thegraphofthefunction y=4sin3x hasfundamentalperiod2/3andamplitude4Thefunctiony=-6cos1/2 x hasfundamentalperiod4andamplitude6.第五节RightTriangleTrigonometry直角三角函数sinx=b/candsiny=a/ccosx=a/candcosy=b/ctanx=b/aandtany=a/bcotx=a/bandsecy=c/b第六节TriangleProblems三角形问题1.The Law of Sines 正弦定律sinA/a=sinB/b=sinC/c2.The Law of Cosines 余弦定律a=b+c-2bc cosAb=a+c-2ac cosBc=a+b-2ab cosC 谢谢观赏谢谢观赏