trigonometry.pdf

of 4
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Information Report
Category:

Documents

Published:

Views: 6 | Pages: 4

Extension: PDF | Download: 0

Share
Description
Download trigonometry.pdf
Tags
Transcript
  Trigonometry(trig)meansmeasurementoftriangles;itisusuallystudiedas measurementsofsidesandanglesoftrianglesandaspointsonaunitcircle;this studyguideisbasicallyseparatedintothesetwomainsections:trigwithtrianglesand trigwithaunitcircle;intrigonometry,themeasuresofanglesareusuallyrepresented bylettersfromtheGreekalphabet;theGreekletters 8. a,v, and 13 willbeused throughoutthisstudyguidetorepresentanglemeasures   I TRIGWITHTRI NGLES A. RightTriangle I.A righttriangleis a trianglewith exactlyone90°(right)angle 2.Thehypotenuseisthelongestside ofarighttriangle,andisalways locatedoppositetheright(90°)angle 3.Thetwoshortersidesofarighttri- anglearebothcalledlegs 4.ThePythagoreanTheorem(Ieg'+leg' = hypotenuse'ora'+ b' = c' wherea and b areleglengthsandcisthehypotenuselength)maybeusedtofindthe lengthofanythirdsideofarighttrianglewhenanytwosidelengthsareknown a.Whenthetwoleglengths areknown,squarethe lengthofeachleg,add thesetwosquarestogether andsquareroottheresult- ingsum;forexample: a 2 + b I = c 2 9 2 + 16 2 = c 2 81+256=c 2 337 = c 2 ..Jill = c 18.36 = c=hypotenuse b.Whenthelengthofthe hypotenuseandeitherleg areknown,squarethe lengthofthehypotenuse, squarethelengthoftheleg, subtractthesetwosquares, andsquareroottheresult- ingdifference;forexample: a 2 + b 2 = c 2 8 2 + b 2 = ]02 64+ b 2 =100 b 2 =100-64 b 2 = 36 b =..J36 leg= b =65.Specialrighttrianglesexistthatareusedsooftenthattherelationshipsofthe sidelengthsshouldbememorized a.30 -60°-90 triangles havesidelengthswith ratiosof I: -J3 :2; thatis. thelongestlegisalways -J3 timesthelengthofthe shortestleg,andthe hypotenuseisalways 2 timestheshortestleg. Thisrelationshipcanhe usedtofindallside lengthswhengivenonly onesidelength;forexample: c=hypotenuse=2x b = longestleg = x...f3 a =shortestleg= x Ifa=5thenc=2ã5=10and b = 5ã '-13'  8.7 Ifb =8then a =8 + '-13' 4.6and c = 2ã8 + '-13'  9.2 c a a=]eg=x~ .45°_45°_90°triangleshaveside lengthswithratiosof I:1:.,f2; thatis, b =leg= x b thetwolegshavethesamelength(ifc=hypotenuse= x{2 ctwoanglesofarighttriangleare Ifa =7then b =7equal,thenthetwolegsareequal), andthehypotenuseis -J2 timestheandc=7ã {2' 9.9lengthofeitherleg;forexample:Ifc=12then a =12 + {2'  8.5 I andb=85B.RightTnangleTrigonometry I Thetrigonometl'ic(trig)functionsofananglearerelated 10 theranosofthe SIdesofarighttnangle I 2.Thetrigfunctionsaredefined in thefollowingmannerwhere8standsfor eitheroftheacute(lessthan ~OO) anglesintherighttriangle;thesedefinitions shouldbememorized: sine El = sin El oppositelegcosecant El = csc 8 =hYllot~nute hypotenuseoppositeeg adjacentleg hypotenusecosine 8 = cas 8 secant 8 = see 8 =d' I hypotenusea jacent egoppositelegadjacentlegtangent 8 = tan 8 = ~d~  --t]-cotangent 8 = cot 8 = osite leg ajacen egopp I  NOTE:Thelegoftherighttrianglewhichisconsideredeithertheoppositelegorthe adjacentlegchangesdepending 011 whichoftheacuteangles is beingevaluatedin thetrigfunction) 3.Theoppositelegofarighttriangleisthelegwhichdoesnottouchthevertex oftheanglethatisnamedinthetrigfunction 4.Theadjacentlegofarighttriangleistheleg A ----- -----,-,C whichdoestouchthevertexoftheanglethatis namedinthetrigfunction;forexample: Whenevaluatingthetrigfunctionsforangle A inthisrighttriangle,leg y istheoppositeleg forangle A becauseitdoesnottouchpoint A; however,leg x istheadjacentlegforangle A becauseitdoestouchpointA;thehypotenuse is sidez Inthesame-righttriangle,legxistheopposite legforangleBbecauseitdoesnottouchpointB;however,leg y istheadja- centlegforangleBbecauseitdoestouchpointB  NOTICE:Thelegthai is theoppositelegforangleAisthesamelegthat is theadja-cent legforangleB,and thelegthatis the adjacent legfor angle A is the same legthat is theoppositelegforangleB;thehypotenuseis /leverconsidered  IS theoppositeside 1101  astheadjacentsidebecauseit is notaleg) 5.Sincetrigfunctionsareratios,andratioscanbewrittenasdecimalnumbers, trigfunctionsareeitherconvertedtodecimalnumbersorleftasradical expressionsinlowestterms(forexample, 4 or.866);forexample,intheright trianglesabove,if:- r B z; = 9,y =7,and x =  Y32 =4-,J2then sin A = cas B = L= L .7778   cas A = sin B =~=4--/2 == .6285   9 tall A =   =7[,_ == 1.2374 x. 4'12 tanB=~=4-,J2 == .8081 Y7 6.Usingthetrigfunctiondecimalnumbervaluestofindoruseanglemeasures requireseitheratrigfunctionchartoracalculatorwithtrigfunctionoptions; forexample,ifyouhavefoundthatsin a = .7778then,byusingeitheratrig chartorcalculatorthemeasureofanale a  isabout51 o b tdegrees sine  St tane · · ããã ã ã ã , ã ·· -, ã 51°00' .7771.62931.23510.7790,62711.24220.7808.62481.25030.7826.62251.25740.7844.6202 1.26550.7862 .61801.27252°00'.7880,61571.280 ]0 .7898.61341.288 ã ã · ã ã · ã ã ã · ã ã Ifusingacalculator,followthecalculatordirections C.TriangleTrigApplications Therearetwobasicwaysinwhichtrigfunctionsareusedwithtriangles:tofind anglemeasuresandtofindsidelengths 1. RightTriangles a.FindingAcuteAngleMeasures Tofindthetwoacuteanglemeasures A whengiventwoSIdesofarighttriangle, II~ IS easiest tofindthelengthofthe third side a  first:forexample,inthefollowingrig htb 20tnangle,ifyouknowthelengthofanytwo sides,thenyoumayusethePythagorean~ Theorem(leg-+leg'=hypotenuse')toC8B findthelengthofthethirdside Oncethethreesidelengthsarefound(itisnotnecessarytofindthethreeside lengthsinordertofindtheanglemeasures,butitiseasier),thenusethetrigfunctions tofindthedegreemeasureofoneacuteangle;usingthesamerighttriangleabove,the measureof a canbefoundusinganyofthetrigfunctions,sojustpickoneofthem; forexample: Themeasureofthesecondacuteanglemay befoundbysimplysubtractingthemeasureof theacuteanglejustfoundfrom90°becausethe sumofthethreeanglesofanytriangleis180° 8 2 + b 2 = 20 2 64+ b 2 =400 b 2 = 336 b = ..J336 == 18.33 sino. =~ =.4000 so a  =23°30' 20 f3 =90°-23° )0  =66°30'  ~ ~~~~~~ b.FindingSideLengths Tofindthesidelengthsofarighttrianglewhengivenonlyonesidelengthandone acme angie. first,subtractthe givenacuteanglemeasurefrom90 becausethesumof thethreeanglesofanytriangleis180°;second,usethetrigfunctionstofindthelength ofanothersideofthetriangle;forexample: A~ = 90°- 35° = 55°sonowweuseeither acuteanglewithalligfunction a 14 Lseatrigchartor orcalculatorto getthisdecimal value. B®~ a = 14(.5736) a = 8.0304 Usethe cos )50, sin 55°, or cos35°, butnotthetangent functionbecause neitherleglength isgiven Oncetwo sidesoftherighttriangleareknown,thePythagoreanTheoremcanbe used 10 fin.theIenerhofthethirdside c. ApplyingSampleSituations i, Definition:Theangleofelevationistheangleformedbyahorizontal line(eitherrealorimagined)andthelineofsightlookingupfromthe horizontal:forexample: Problem:Annastood4,800feetfromarocketlaunchingpad;she measuredtheangleofelevationas73°whentherocketwasatitshigh- estpoinr:ifAnnameasuredtheangleofelevationfromaheightof5.5 feet.findthegreatestheightthattherocketreached -3' is theangleofele:ti~l>./ f'1 tan 73° rr 3.2709 \lmas hecla./h1 tohereyesis ,... I 5.S1eeL./ 15700.32 ft  \rl + 5.50 ft ~-SOL J 9rizontalview -L-l 15705.82feet 2;. c L,bd Iã -i800 hã Iaovegroun h 4800 h 4800 hAnna ii. Definition:Theangleofdepressionistheangleformedbyahorizon- talline(eitherrealorimagined)andthelineofsightlookingdown fromthehorizontal;forexample: Problem: A CoastGuardcrewwasflyingarescuemissioninaheli- copter:amemberofthecrewspotted a boatintrouble;thiscrewmem- berwaslookingdownatabouta25°angleofdepression;ifthe helicopterwasabout300feetabovewaterlevel,howfardidtheheli- copterhavetotraveltobeabovetheboat? OOri2~'i(-'I': _d_-\25° ~I u tan 25° = ____---I  g --------I =-  A  .4663 1 d 300 d 300 d 643.36 ft 15 U isme angleofdepression. 2. ObliqueTriangles Obliquetrianglesdonotcontainarightangle;therefore,anytrianglethatisnota rightmangleisanobliquetriangle a.Acute TrianglesAnyacutetriangle(trianglewithallacute angles I can be sepratedintotworighttriangles o·. onszructinsalinesegmentfromoneofthe verticesamij)~rpendicul'iirtothesideopposite the Y TIe:\.: furexample.6..ABCcanbeformed intorishtriangles A13D andBCDbydrawing BDperpendiculartosideA<:'AC ..en me trigfunctiondefinitionsforrighttrianglescanbeappliedasdiscussed aOOye  ã.YOTE:Another OPtion tor solvingacutetrianglesis 10 leavethetrianglesastheyare  acme)andtoapplv thelawofcosinesorthelaw ofsines, bothofwhicharedis- cussedatthetop ofthe next column  b.ObtuseTrianzles .- ny obtusetrianglec'irianglewithexactly B oneobtuseanelercanbeconverted  to arightrrianslebvconstructinzalinesegmentfrom one ofthe verticesandperpendiculartothe linecontainingthesideoppositethevertex; L forexample.in.lABC, LC isobtuse;extend L.. L side AC__ hendraw BD perpendiculartothe A---- ..L D extension:theresult 1S rightMBD.C Thenthetrigfunctiondefinitionsforrighttrianglescanbeappliedasdiscussedabove {SOTE: .~nOiher option for solvingobtusetrianglesistoleavethetriangles,asthey are tobtuse) andtoapplyeitherthelawofcosinesorthelawofsines,both ofwhicn _..1:..__ .J_~ ~1.r~L -__1.._.ã~.\ c.LawofCosines I  Thelawofcosinesstatesthatina triangle ABC: a'=b 2 +c -2bccos0: b 2 = a 2 + c 2- 2accos~ c1 = a2 + b 2- 2abcos v B a v A b ii. Whentoapplythelawofcosines Thelawofcosinesmaybeusedeitherwhenallthreesidelengthsofthetriangle areknown(SSS),orwhenonlytwosidelengthsandthemeasureoftheangle formedbythesetwosidesareknown(SAS,thatis,twosidesandtheincludedangle) d.LawofSines I  The lawofsinesstatesthatin: L'1ABC (asindicatedinMBCaboveinthelawof sill  J  cosines): ii.Whentoapplythelawofsines: Thelawofsinesmaybeusedeitherwhenonesidelengthandtwoanglemeasuresare known(SAA,thatis.oneoftheanglesmustbeoppositetheside)orwhentwoside lengthsandoneanglemeasureareknown(SSA,thatis,theanglemustbeoppositeone ofthetwosides) iii,Caution Whenusingthelawofsines,occasionallytherewillbenosolution;thisisbecause notallcombinationsofanglemeasuresandsidelengthsactuallyformtriangles; rememberthatthethirdsideofanytrianglemusthavealengthlongerthanthediffer- enceoftheothertwosidesandshorterthanthesumoftheseothertwosides b sin~ a c sin v TRIGWITH UNIT IR LE A. Circles I. Definitions a. A circleisthesetofpointsinaplanethatareequidistant(thesame distance)fromonepoint,thecenterofthecircle(whichisnotactuallya pointonthecircle,butonlythecenter)b. A radius(r)isalinesegmentwhoseendpointsareapointonthecircleand thecenterofthecirclec. A chordisalinesegmentwhoseendpointsarebothpointsonthecircle;all otherpoints 011 thechordarepointsintheinteriorofthecircle d. A diameter(d)isachordthatcontainsthecenterofthecircle ~ ~B DC AD isa chord. Ae  isachordandadiameter. EB is a radius.e.Thecircumference (C) ofacircleisthedistancearoundthecircle,andmay befoundbyusingtheformulaC=redwhere1 isapproximatelyequalto3.14 f.Thearea(A)of a circleisthenumberofsquareunitsthatareneededtocover theinteriorofthecircle,andmaybefoundbyusingtheformulaA=m.:  g.Thearcofacircleisthesetofallpointsonthecirclebetweenanytwo pointsonthecircle;aminorarcmeasureslessthan180°;asemicircleisan arcthatmeasuresexactly180°;amajorarcmeasuresmorethan180 0 B. CentralAngles I.A centralangleisananglewhosevertexisthecenterofacircleandwhose sidescontainpointsonthecircle 2.A centralanglehasthesamedegreemeasureasthecirculararcit A intercepts(thearclocated.intheangleinterior);additionally,an 0 rchasthesamedegreemeas.ureasthecentralanglethatinter- B ceptsit;forexample, LABC interceptsarc AC andtheirdegree measureisequal C 3.Degreesa.Onedegreeis 1/360  ofthe360 0 containedillacompletecircle;adegreemay besubdividedinto60minutes(written60');aminutemaybesubdivided into60seconds(written60 ) b.Thedegreemeasureofanangleisthedegreemeasureoftheintercepted circulararcofthecircleforwhichitisacentralangle 4.Radians a.Oneradianisthemeasureofacentralanglethatinterceptsanarcequalin lengthtotheradiusofthecircle b.Theradianmeasureofacentralangleis A theratioofthecirculararclengthtothe ( JS s radiusofthecircle.Rememberthe dis-BC4ABC =- radians tancearoundacircleis nd; forexample: rr 5.DegreeandRadianConversions a. A semicirclehasadegreemeasureof180'andalengthequaltohalfthecircle, .51 dornr;theradianmeasureistheratiobetweenthecirculararclengthand theradius;therefore,theradianmeasureof a semicircleis m/r=rt; so: i. 180°=1tradians ii. I adian = 180/ n iii. 1°= nit so radiansb.Degreeandradianconversionscanbeaccomplishedusingthesepropor- tionsorequations: i. radianmeasureoftheangle nradians 11. theradianmeasureofanangle ~reemeasureoftheangle 180°1 (degreemeasureoftheangle) 180°180 0 (radianmeasureoftheangle') 1t ii. thedegreemeasureofanangle   £J[j{j 1. Thedomainofthesinefunctionisthesetofrealnumbers;therangeis thesetofrealnumbersbetween -I and],inclusively;i.e., -I~Y~1 lV. Forexample: IfLA = 40° then LA = ni40) = n(2) = 2n radians180 99 c.Seetheradiansanddegreeschartunderthetopicof UnitCircleforthe Measurements ofSpecial Angles C. GeneratedAngles J. Ageneratedangle(anothertypeofangleoftenusedintrigonometry)isa centralanglewiththevertexplacedattheoriginofthecoordinateplane,and oneofthetwosidesplacedandkeptonthepositivex-axis,whilethesecond sideisrotatedineitheraclockwiseorcounterclockwisedirection a.Thesidethatdoesnotrotateiscalledtheinitialside b.Thesidethatdoesrotateiscalledtheterminalside c.Negativeanglesareformedwhentheterminalsiderotatesclockwise d.Positiveanglesareformedwhentheterminalsiderotatescounterclockwise x-axis initialside D. UnitCircle1.Theunitcircleis a circlewhosecenteristheorigin(0,0)oftherectangularcoordi- nateplaneandwhoseradiusisequaltoexactlyoneunit(radius= I anddiameter= 2) 2. Theequationoftheunitcircleisx=+ v= I3. Apoint, P, isontheunitcircleifandonlyifthedistancefromthecenterofthe circletothepointisequaltotheradiusofexactlyoneunit 4.Theunitcircleissymmetricwithrespecttothe x-axis,the y-axis, andtheorigin;therefore,if y-axis pointP=(a,b)isontheunitcircle,thenthese pointsarealsoontheunitcircle(-a,b),(-a,-b), and(a,-b);forexample: x-axis (a,-b) 5.Thedistancebetweenanytwopointsonthe rectangularcoordinateplanemaybefoundbyusingtheformula:  (Xl- x2 l + (YI- Yz 2 wherethe points are (XI'YI) and (x2'Y2)' 6. Thelengthofanarcoftheunitcircleisbasedonthecircumference, xd = 1t2 =17(;because,d=2 7.PointsontheUnitCircle a.Pointscanbelabeledusingtheappropriateorderpair, (x.y) b. Pointscanalsobelabeledusingthecirculararclengthdeterminedbythe generatedanglewhoseterminalsidecontainsthepoint, (x.y); forexample: Remember theradiusof a unitcircle is one;r = 1c.Constructingarighttrianglebydrawingaperpendiculartothex-axis,and determiningthesidelengthsofthetriangleresultsinthefollowingunit circletrigfunctiondefinitions 8.Unitcircletrigfunctiondefinitions(seethediagramabove): \'i11ere t radianssill t sin a = y a = degrees cas t= cas a= X tan t=tall a= L; x l 0 X csce+csco. =. . y see t= see a =. . x ,YOTE:Thesefunctionsarereciprocals: sinandcsc;csc a = '/,ill a casandsee;see a =  /eosa tanandcot;cot a = i/ lalla 9.Frequentlyusedanglesandtrigfunctionsareindi- catedinthefollowingchart x cot t= cot a. =~; y1'O a = degree t= radians 0=undefined (L 30 qj 6 <)0 110 t i 0 .1L.1L.1L 2rr 4 0 2 1 v2 ss. ss. 2 2 2 \ 2 1·1   TT 1  .i3 0 - J}  ll. Boththedomainandtherangeofthecosinefunctionarethesameas thedomainandtherangeofthesinefunction III  Thedomainofthetangentfunctionisthesetofallrealnumbers exceptthosevalueswherethefunctionisundefinedandgoesoff asymptotically,suchas± rrl2, ± 3rr/2,... Therangeisthesetofallrealnumbers;thedashedlinesaretheverti- calasymptotes t:- axis -2n I I I I 11. Periodsofthefunctions a.Afunction, f, isperiodicifthereisapositivenumber z- f suchthat j  t+v)=f t) forall tin thedomainofthefunction;thismayalsobestated using inplaceoftheevalue.Thesmallestvalueof o iscalledtheperiod ofthefunction;thatis,thesmallestvalueatwhichafunctionbeginsto repeatitsrangevalues,andthusrepeatitsgraphingpattern,istheperiodof thefunction b.Theperiodofthesinefunction,j(t)=sill t  is 2n: because sin t+2n:)=sin t c.Theperiodofthecosinefunction, f  t)=COSt, is-also 2n: because cos t+21t)=cost d.Theperiodofthetangent function,j t)=tan f; is rt because tonte+ixv=tan t  NOTE:TheseperiodsCOilbeobservedinthegraphscfthefunctionsasindicatedabove) e.Theperiodof a function, f t)=sin Be, is 2rr/B; theeffectofthevalueof B isthatitstretchesthegraphouthorizontallywhenO<B< I andshrinksthe graphhorizontallywhenB> I v .y = sill x: where B = I   = sill 2x; whereB = 2  NOTE:Theredsectionofthegraphindicatesoneperiodofy=sinx;andtheblue sectionisaileperiodofy=sin b) 12. Amplitudea.Theamplitudeofatrigfunction, y=j t)=Asint or y=j t)=Acost canbedefinedasIA1 NoticethatwhenIA1>1,themaximumandtheminimumvaluesofyequalA,so thegraphgetstaller;likewise,whenIA1<],themaximumandtheminimumvalues ofyequalA,sothegraphgetsshorter;inthefunction f t)=Arant, thevalueofAdoesaffectthecurveofthetangentgraph,butnotthemaximumandtheminimumvalues 210  l 14  270300 1 31; .\30 36  ,rr4rr J.zL .2IL JJ1.  lrr 2rr 4  3 2 0 4 e;- X ~\ 1. ·V.l -1 -\.; - ;2 -t V = -l.Sill X , , T T --0/2 -t 1 sz: .:il 2 T T  2 2 2 ss. ..J3 0 -.,J3 -1 ~ \ \ Y ns 150t80 10.Trigfunctiongraphs a.Graphingthevaluesof.l~etrigf~mc~i~~s~ind c~led.inthechartabove)on b.Theamplitudemaybeconsidered 10 behalfthedifferencebetweenthe _:_:_ãã_~_.l.ã.1...:.  L__    II--_~ tias J-_C 8shit to {heleftbecause C=TC,andB=: . mncncnsare denoted by arc orby -I exponents:without ilifo__:.tb imrsesofthetrigfunctionswouldnotbe 'eh-: n icthattherangeof each inversefunctionisa ~~h]oif'l':;c= ;:~= orrespondingtrigfunction _tn~,.~,j;:e ~ ~~fiflfl.. -i-: t Q  theinejunction while(sill xr  = / ;IIX. -siB<'- nction:DOSOTconfusethese because): tan+:   (tall,,)-1 R<\ IGE -it ::; s it Y - 22 0 ::; v ::; it -11: 11: - < v < 2 2 -11I~~;~;~~~:g;~~::~~~~=- ement. Y·. in herangewhosetrig fix.zrexample.arcsin.J4 = theanglewhose I~ - ;;-=--=-=--= r  V  .'-----~ - ---- : y = arctan x = tan:« 'f:::IC lI~,·~l d eationships,butno =, < or>;they ualsign:theymaybesolvedto hniques.suchasfactoring,may :-I=0 t: = -1 soin the interval0::; t;::; 2n: III tb domain:theymaybeprovenor =:_ ntitiincludeworkingtheleftsideof ri_:side:workingtherightsideuntilit '4'1 side-untiltheyareidentical to REC1PROCALS cset=J... sillt 1 sect=- cost cott=J... tallt COFUNCTIONS cos(~- t;) = sillt 2 sill(~-t) = eos t: 2 tall ( £- t;) = cot t: 2 BASICIDENTITIES tallt=s ...t COSt; sinl-e + cos « = I or Sill 2 t = 1- cos «tmzlt + 1 = secte or tan « = sec e- I cot « + 1 = csc « or cot « = csc-e- 1 COS(r±t) = cas s cosessin «sin t ADDITION I SUBTRACTIONFORMULAS sint«): t) = sillrCOSt±cas «sin t: tants):t) = fall r+ fall t 1 + tan.stan i: NEGATIVES sin(-t) = -sin t: cas( -t) = costtall  -t) = -tant HALF-ANGLEFORMULAS .t +~ /l1- =_---   cas . = ± -V 1 + cast   fall . = ± -V I  ast 21 + COSt DOUBLE-ANGLEFORMULAS sill2t = 2sintcost cos2t = cosse-sin « cosl» = 1- 2sill1teos2t=2eos 2 t- 1 fall2t= Ltan« 1- tan e sin 2 t= 1- cos2t 2 cos e» 1 + cos2t 2 PRODUCT-SUMFORMULAS stnrcose» sint» + t;  + stntr- t) 2 coss sine» sints « t) -sin(r- t;  2 cosscose COS(r + t) + COS(r-t) 2 sin, sine=COS(r-t)-COS(r + t) 2 sinr vsine»2Sil1(r;t)cosE r; t)sinr-sine»2eos( f; t)sinE r; t)C+t)cot) osr+COSt = 2eos  Z cos   /.C+ t). C  t)OSr-COSt = -_S/11  z  S/11  z  < - J6 0 n~ads & ~ãã ~eofjitlesat C   u y com ISBN 13:978 15722261 4 ISBN 1 :15722261 2 J~ IJtlWill I Illl~I lllliII1lli1i1llll -.--,a -:;=P(\S~ '. andis nOI --ã.;-.-~~':'-  toitscondensed -- , ~itherBcr'Cbarts.its -_ãããã-~1:~ liablefortheuseor CustomerHotline   1 8 23 9522 Wewelcomeyourfeedbacksowecanmaintainandexceedyourexpectations. U.S.S4.95/CAN.$7.50 6
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks