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1、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。 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 b
2、e measures 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 ThefollowingtablegivesthecorrespondencebetweendegreeandradianmeasuresofsomecommonanglesDegrees0304560901201351501802
3、70360Radians0/6/4/3/22/33/45/63/22二、AngleinStandardPosition角的标准坐标位置Thestandard positionofanangleoccurswhenweplaceitsvertexattheoriginofacoordinatesystemanditsinitial sideonthepositivex-axis.Thequadrantthatcontainstheterminalsidedeterminesthequadrantthattheangleliesin.Inthefigureabove,representsanang
4、leinQuadrantI,whileisinQuadrantIII.Apositive angle isobtainedbyrotatingtheinitialsidecounterclockwiseuntilitcoincideswiththeterminalside.Likewise,negative anglesareobtainedbyclockwiserotation.Inthefigureabove,ispositive,whileisnegative.Iftheterminalsideofanangleinstandardpositionisoneoftheaxes,thean
5、gleisaquadrantangle.Forexample,90(/2)and-180(-)arequadrantangles.Everyangleinstandardpositionhasareference angle,whichisthepositiveacuteangleformedbytheterminalsideofthegivenangleandthex-axis.Seeexamplesbelow.第二节TrigonometricFunctions三角函数Forageneralangleinstandardposition,weletP(x,y)beanypointonthet
6、erminalsideofandletrbethedistance|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 Q
7、uadrant II. All the other 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Atrigonometricidentityisanequationinvolvingtrigonometricfunc
8、tionsthatholdstrueforallangles.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. Sy
9、mmetrySin(-)=-sincon(-)=cos5. Double 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,
10、thatis,f (x+p)=f (x)forallxinthedomainoff,meaningthegraphrepeatsitpatternaftersomeintervalinx.Thesmallestpossiblevalueofpintheexpressionf (x+p)=f (x)iscalledthefundamental periodofthefunction,sometimesjustcalledtheperiod.2. Amplitude 幅度幅度Thesineandcosinefunctionshaveanadditionalproperty,amplitude,wh
11、ichishalfthedistancefromthecrest(top)tothebottomofawave.Forasineorcosinecurvethathasnotbeenverticallytranslated,theamplitudeissimplythedistancefromthex-axistothecrestofthewave.Thefollowingarethegraphsofthesixtrigfunctions.Thedomain,range,fundamentalperiod,andamplitude(whereapplicable)aregivenforeach
12、function.Thegraphofy=AsinBxandy=AcosBxFundamentalperiod=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
