College algebra and trigonometry

649 Pages·1986·26.275 MB·English

by  Bernard Kolman, Arnold Shapiro

COLLEGAEL GEBRA
ANDT Rl,GONOMETRY
SECONDE DITION
BernarKdo lman
DrexUenli versity
ArnoldS hapiro
TempUlnei versity
ACADEMPIRCE SCSO LLEDGIEV ISION
HarcoBurratcJ eo vanovPiucbhl,i shers
OrlanSdaDoni egSoaF nr ainsccNoe wY ork
LondToonr onMtoon trSeyadln Teoyk ySoa Poa ulo
Tot hem emoryo fo urf athers
Copy©r i1g89h6bt yA cadePmrieIcsn sc,.
Alrli grhetvssee dr.
Nop aorftt h piusb limcaabyte ri eoopndr ucoerd
transimnain tfoytr emod rb ya nmye anesl,ec torro nic
mechainnicclapulhd,oi tnogrc eodcpioyrno,gra , n y
inforsmtaotraianorgdnere ti esvyasltw eimt,h out
peirmssiinwor ni tfrionmtg hp eu blisher.
AcadePmrieIcsn sc,.
OrlaFnldoor3,i2 d8a8 7
UniKtiendg eddoimpt uibolnbi ys hed
AcademiIcn (cPL.ro en)sLd sto,dn .
241O2v8aR lo aLdo,n dNoWln7 DX
ISB0N1-:-2 1479035 -
LibroafCr oyne gsCrsa taClaorNgdu mb8e57r-2:3 88
Prinitnteh Uden iStteadot fAe mse rica
Tot hem emoryo fo urf athers
Copy©r i1g89h6bt yA cadePmrieIcsn sc,.
Alrli grhetvssee dr.
Nop aorftt h piusb limcaabyte ri eoopndr ucoerd
transimnain tfoytr emod rb ya nmye anesl,ec torro nic
mechainnicclapulhd,oi tnogrc eodcpioyrno,gra , n y
inforsmtaotraianorgdnere ti esvyasltw eimt,h out
peirmssiinwor ni tfrionmtg hp eu blisher.
AcadePmrieIcsn sc,.
OrlaFnldoor3,i2 d8a8 7
UniKtiendg eddoimpt uibolnbi ys hed
AcademiIcn (cPL.ro en)sLd sto,dn .
241O2v8aR lo aLdo,n dNoWln7 DX
ISB0N1-:-2 1479035 -
LibroafCr oyne gsCrsa taClaorNgdu mb8e57r-2:3 88
Prinitnteh Uden iStteadot fAe mse rica
PREFACE
Weh avbee evne rpyl eastehwdei dpberyse aacdc epotfta hnfeci er sto fe dition
thbioso. I kth abse eens pecgiraaltlityfof y iinondtg h teerxa tdso pting
somoefo uirn noviadte.Fia ovserex amtphlee ,ie natrrloyod fuc cotmipolne x
numbpeerrms,i ttthsieon lgu otfai noqynu aderqautaihtcai nsoo nbw,e come
thaec ceapptopeardc h.
Thoeb jecotfti hevidesis at rieo n
toi mprtohcvehe a potnte rrisg onboym:e try
•
a.e litmiitnnhg"ae w rapfpuitnnicgot no" paef ramspitatec roi tfnh goe p en­
insge ctions;
b. interaordtluhcyceo i nncgoe fap ntg umleaars( uirbneo trha diaannds
degsrt)eoa e votihadeb rsuwpitfrt ocmh tthoae poprlyi cation;
c.g uidtihsnetg u dteoun scte a lactuofolrrds e mtienritnhvega loufte hse
trigonfuonmcettiroincs ;
•troe sptoosn udg gefrsotmii onrnsustc taonsrdts u dtehniutmssp ,r otvhien g
exposainttdhi eeox na mples;
•t eol niamttihene e edt hsfeto urtd oep unrtc ahs atsued yb gypu riodveai, d ing
sectiinto hbnea cokft hbeo otkh caotn twaoirnkse sdo-louuttsi eolnesc tteod
ReviEexwe rcises;
•t eon saucrceu ortfah cAeny s wesresc btyiu otni lciozmipnpugrtgo erraw mist h
ratiaorniatlht move etrtiihcfae yn swers;
• etnol itvhbeeon ob kyi ntroFdeuactiuonrfieg sn tetrobe oststht udaenndt
inst(rsupecaetg oweris ct ohl moarr kailnotgnh eged e g)s;
• atdotd o ptihcaasrt e cutrarueagntmht atln syyc ho(ofeloxsra mlpeilaner,
progra)m.m ing
Weh avree tatihnseeu dp poretlievmete hnatts b ehcaovtmehhe e a mlalrk
oft hsiesr ies:
SPLITS CREENMSa nayl gepbrroaciecad rudere essc rwiibttehhade i odfa " spslcirtet ehna"t
dispsliamayunsle tobuosttlhhsye t eopafsn a lgoarniadwt ohrmk eedx-aomuptl e.
PROGRESSC HECKSA tc aurleslfeyl epcltaepcdre osb,sl iemmtisotl haorws oer kietndh t ee hxatv e
beenis ner(tweiadtn hs w)e ertnsoa tbhlseet utdtoee nstot r huhinesdr e rstanding
oft hmea tejruidsaetls cribed.
ix
X PREFACE
WARNINGS Toh eelmlpii nmaitsec oncaenppdrvt eeibnoatnmd sa themhaatbiwicetha sal,v e
insertedWa rnnuim(neigrsnao dtubiesycdt hsey msbhooliw tnnh m ea rgtihna)t
* poionuttt h ien coprrraeccmttoi scte s commoinnhl oym efwaoonurdnk d
exapma pe.r s
END-OF-CHAPTER
MATERIAL Evecrhya pctoenrta as iumnmasr iyn,c luding
and wiatphp roppargrieeae fetnrec es;
Terms Symbols
tos trtehcseos n cepts;
KeIyd efaoRsre view
top rvoiadded itpiroancatli ce;
ReviEexwe rcises
top rovsiedlef -eavnardle uiacntefoimroenn t.
ProgTreesstss
ANSWERS Thaen swteoar lRsle v iEexwe rcainsPedrs o grTuesstsas p pietnah rbe a cokf
thbeo .o k
[mlEXE RCISAbESun dcaanrtle,ygf rualded perxoevrpicrdiaesci etntsi h cmeee chaannidc al
conceapstpueoacfalt l sg. eE bxrearrceiuqsierasic nagl culator are indicated by
thsey mbsohloi wtnnh mea rngA.in srwtse oo dd-nuemxbeerracepidps aeetas r
thbea ockft h beo .oA knswteeorv se n-nuemxbeerarcpeipdsie etnash rI en s truc­
torM'asn uaTlh.Ie n struMcatnoura'ls/ BTaeniskats v aitltoah ibenl set ornu ctor
reeqsut.
ACKNOWLEDGMENTS
Wet hatnhkfe ol lowifornt gh erierv oieftw hm ea nusacnrtdih pehtie rl pful
commePnrtosfe:Ws asrorBreu nra cBthr evCaormdm uCnoiltlyMe rgW.ey ,a tt
CasPeryo,fe RsiscrohdFra . DaeStTa acrr aCmieCtnoytl olP ergoefe,As rstohru r
P.D ualtlD i aVballoCl oelyl Pergoefe,Ms asroFkrr eeamtCa anl ifoSrntiaat e
UniversittoyPn-r,Fo ufeFlsrlsaeoGnrruk n naiMtpa coamnbOd a klCaonmd­
muniCtoyl lPergoefe,Ms iscohrLa eevlia ttVa inl laUnniovvea,Pr rseoisftsyor
CatherOilnsaeet Sn UL Y.N- BuflfaoP,r ofeWsisolrlP ieaamar tcK eio lrge
CollPergoefe,Gs esoorPrge ega lItao rwS at aUtnei vePrrsoifetSsyus,eQo yur a n
atG oldWeensC to llege,J aPmrSeotsfer seasetCot lrea rc kCaommamsu nity
CollPergoefe,Js ascoTkre nnaitsT oenxL aust hCeorlalnPe rgoefe,Ks asyo r
WeiastKs a nsCaosm muCnoiltlyae ngPder ,o feJsosWeoi rt koawtKs ekein e
StaCtoel .l ege
We woaullsdo tlhiaTknoekd R dti ome mra nJda Rni chwahraods s,st iiend
chectkhimena gn usacnrgdia pltla enJydas c,q uSehlaipwnihereo on ,t edraetda
into csoomfptufowtrfiae nrrde ri onogot pfso lyneoqmuiaaatlnii don nvseo rfs es
matrices.
Filnyal,g roauttreh fuatlnoo ku ser d itWoersslL,ea yw taonTnde Bdu chholz,
too uprr ocdtuieodni tAonrCnso ,l oawniJdce kn nKiefieatrnht d,ot hset oafff
AcadePmriefcso sr otnhgeociiornm gm ittmoee xncte l.l ence
xi
TO THE STUDENT
Thbioso wkaw sr ifotrty eonu g.i vyIeotesu v eproys scihbaltneosc uec ceed-if
youus iept r operly.
Wew oullidtk hoea vet hyioonmfuka theamasac thiaclsl geanmgei-nnbogut t
asas pecstpaotTrohtrwi. is sleha tdoosu p rr imarRyea tdrh uitsel xet:bw oiotkh
pencainldp apehra ndEyv. enreywi d eorat echniisiq lsulteur baytf eludl y
workeedx-aomuptl esr.e tahdAt ese xcyator,ue fofulllloywe xtahmaepn lde s
thdeont hPer ogrCehsesc kTsh.ke e tyos uccienams ast cho urwsoer kiisn g
proebmlasn,td hP er ogress aCrthehe cetkrpose r oviimdmee dpiraatcweti itche
thmea teyroihuaa lv el ejaurnsetd .
Youirnr sutcwtioalrsl s hiogmwneo rfrko mt heex tesneslievocefet x ieornc ises
thfoaltl oewascs he cittnih obeno .oD kot haes signmreengtuslt ahrolryo,u ghly,
andi ndependBeynd toliyn.g o fpl rbooltesym oswu,i dlelv etlhnoeep c essary
skiilanll sg eabnrydao ,cu orn fiwdieglnrlco ew .a lSbgirenatcieecc hnainqdu es
concbeupitlspd r eovnri eosu,uys lo ctuas na'fftot rosd k ainpoy ft hweo rk.
Toh eplerpv eonertml iiniamtper hoapbeairntt dhos e ylopau v otiheder rtohrast
wes eeea cshe measstw eegr r apdaep ,ew rehs avien tersWaprenrisnegds
throutghbheoo .ouT kth Wea rnipngosio nuctto mmon aenerdmr pohratssh iez e
prompeetrh od.
Theirisem porretvaminaettw e arttih eaenl od fe acchha pTtheTeerr m.sa nd
Symboslhso aulbllefd a imlbiyta hrte i ymoeru e atcehhmI .yf o uunrd erstanding
ofat eromrs ymbiohsla , zu ysteh pea greee fretnofic netd h pel aicnteh t ee xt
wheiriteis n treodGd.oub cacaknr de tadhd eei fnition.
Iitps o sstibobe lceos moe inwviottlhhdve ee tdoa tfie lcsh ntihqautle osys oeu
traocftkh b er oacdoentrcsT e.hple i osKfte yI deasR efvoirae ttwh eeno def a ch
chapwtihelerlyl opfou c uosnt hper inicdiepaasl.
ThRee viEexwe rcaittseh esen odfe acchha pctabeneur s eapsda orfyt o ur
preaptaifrooenrx aminnsTa.ht seie occtoivoener aicenhxg e ricisin sdeis coa ted
thaintfe, e dyeodcu,a g nob actkor esttuhmdeay t ae.lr yiIofgu e stt uocnak
probsleeiemtf ,h p er obtlheiamgst i vyiondugi ffiocraus litmypi orlbalrie sm
numbeicrnoe ldio nrd,i tchaaatw tio nrgk esdo-louautptip oeinatn rh bsea octfkh e
bo.oY koaur teh reena tdtoyr P yro grTeesstAs .Y owui slolpo inn pyooiwunerta k
spoatncsda gnob acfokr fruervtaihnemedwor re ex erictnih soaesrasee sO .n ly
thsehno yuolpudr octeoe dP roTgesrte ss
Web eltihetavhteee v entuali Bsn."t puadyyoiffn"g imasani t mhpermoavteidc s
abitloti atcyk le prporbailcynteo imufcsira oe lifl nd teTrot ehseatntt.d h ,bi oso k
plascpeesec mipahloa ns iwspo rbrolde ,wm hsircehc seunrtvs ehyoaswro ef ten
troubtloes stoumdSeei nntacsle.g eibsr bata hsleia cn goufta hgmeea themati­
catle chnuisqeiudnve i sr taulfallil ed,lts yhm ea stoefar lyg eibwsre walo lr th
youerff ort.
xiii
THEF OUNDATIOONFS
ALGEBRA
Noo nweo udledb taht2ae t + 42,o rt= h 5a t+ 35 .T=h se3i g+n ioffi cance
thset ate"m2+e n2t4 " =l iients h ree cogtnhiiatittits or nu e wthheet her
objects undaeprp oldraei ns,stc csruasodsrcli aeorFsnus r.at trhhesee tr a,t ement
5 + 3 '=i' n 3d itc+ha tat5hoteer s do eafrd diitisimm oant earnitdah plir,si n­
''
cipiltser uea npfyao oirfir n te.g ers
Thessiem ple ieulxslattmrhpafeltu eens d atmaeosnafktl a glet bor aa:b stract
thopsreot pieterhsaa ptp tloany u mbseyrs tOefcm o.u rtshpeer, or ptedieepse nd
ont htey opfen umbweecr hso toods eewa ilt Whew. i tlhle rbeefogrwiein at h
discuosfts hrieeo anlnu m bsesyrt eamn idtp sr orpteiseisnm,cu eco hfo uwro rk
ina lgewbiirlnalv otlhnviuesm bseyrs. tW eewm itlhlie nnd iacc aoetrsrep on­
denbceet wtehreeenn a ulm baenrtdshp eo ionnatr sen aulm bleiarnn ewd i ll give
ag raipcpharle seonftt ahctioisero srnp oned.e nc
Three maionftdh ceihrsa pidtsee vro ttaore edv oifseo wm fuen damentals
ofa lgetbhmreea a:n ainnudgs o efv ariaalbgleeebsxr;pa riecas nspdio olnysn o­
mifoarlm fasc;t oarniodnp ge;r awtiirtoahnt sie oxnparleo sras ligoenfrbsar ca­ic
tions.
1.1 Wew inlelte udos tehn eo taatntideo rnm ionsfo elfrtoosgmt y i tmtoei mAes e.t
THE REAL NUMBER issi mapc loyl leocfot bijoeoncrn tusm bwehrisac,rh ce a ltlheeeld e meonrt s
SYSTEM membeorfts hsee Tth.ee l emoefans teastr w er iwttietbnhr iansc ote hsta hte
notation
SETS
A={ 45,,6 }
teulstl hsta htse eA tc onsoifts hnteus m b4e,5r ,as n dT h6se.e t
B ={ExxFoonrH,do ,n eywell}
consoifts hnteas m oefts h etsherc eoer pioornWaseat. l wsroi4 t Ae,w hiwceh
e
reaasd" 4i asm emboeftr hs eeA t. S"i imlaFrolryBd, i r se aasd" Foirasd
e
memboeftr hs eeB t," anIdB Mfl. iBr se aasd" IBiMns o atm emboeftr hs ee t
B."
PAGE 1
THE
2 FOUNADTIONS ALGEBRA
OF
lefv eerlye moefn tAa i ss eaatm lesmobo efar Bs,et thA e inas s ubseotf
B.F oerx amtphlseeeo ,tfa lrlo biiasns su bosfe tst eohtfea lbli rds.
EXAMPLE 1
Thsee Ctc onsoifs nttahsme oe fs c aoliwlnh sod seen ominaltetishos5an 0ns
cenWtesm .a wyr iCt iens ento taatsfoi lolno ws: are
C ={pennyni,c ,dk iemlqeu,art er}
Wese et hdaitmE eCb uhta dlofl f.lC .aF ru rthtehsree, t = {nicdkie}mli e,as
suebtos fC. H
PROG CHECK
ThRESSse e tc onsoifts hvteos w ienl sEt nhgela ilsphh abet.
V
(a)W ritiens e ntot tiaon.
(b) Itsh lVe e tkt meaer m beorf
(c)I tsh lee tut meaer m beorfV ?
(d)L itshtse u bsofeh tasv Vfoi? unergl ements.
V
ANSWERS
(a)= {a,e ,i, o ,u } (b)N o (cY) es
(d{)V,a e ,i, o ,}{ ,ei, o ,u },{ ,ai, o ,u ,}{ ,ae ,o ,u },{ ,ae ,i, u }
THEREAL NUMBER Sinmcueco hf woourirkan l gdeebrawali tsth he reasly s,ntw uee'mm 1bb1ee gri n
SYSTEM wiatr he voifte hwce o mpoosfit thniiuosmn bs eyrs. t em
Thneu mb1e2,r,s3 ,. .,u .s edc ofuon,rfot rimnt ghseeo tfn atural
numbeIrwsf.e h aodn tlhye nsuem bteour sste os hotwhp er oefarnietbd y a
comp,aw newy ouhladvn eow atyoi nditchatathtcee o mphaanndyo profit or
haadl osTsoi .n dincopa rtoewf ei ti nt,ra ondfodur lc oes ws0een se etdo
intrnoedguacnteui mvb.eTe hrnesu mbers
. .-.2 ,1 012,,. .
' - , ,
fortmh eo fisn ette. gT ehr,use svenrayt nuurmabliae snr i nt,ea gntedhr seeo tf
natnuurmablie ssre ste onb e a s ubosfte hte o fis nette. g ers
Whewnet rtyod ivtiwdaoep pelqeusaa lmlofoynu gpr e owpelf ei nnod
numbientr hs eeo tfi ntetghewaritsel xlp rheoswmsa nayp pelaecpshe rson
should negeetdtoi . n trWtoehd seue ctre a otifon nuamlb ewrhs,i chn umar­e
betrhsca atbn ew riattsare ant itowi ono tfe, g ers
p_ wiqtn ho etq utaol zero
q
Examopflr eansta inlou mbaerres
2 7 -3
0 -4
3 5 4
Byw ritainin ngt ne ignte hrfoe r mni lw,e s eteh at ienvteieragsry e a rt ional
numbTehrde.e cinmuamlb1 e.ir3as l asr oa tinounma,bsl ei rn1c.=e3
M.
THER EAL NUMBSVSTEERM 3
1.1
Weh avneos we tehnr fuened amennutmabsley rs tenmast:nu urtmahblee r
ssyt,e mstyhseot ifen mt resag,net dhr ea tinounmabsley rs tEeamcl.has tyesrt em
incltuhpdere esv siyosuotsrse yms taenmedsa ,ci mhso rceo mipclatthetadhn e
onbee fo.rH eowevtherera ,t inounmabsley rs itsset mii nlald efqosuroa pthei s­
ticuastoeefmsd a thteimscaisn,tch eee rxei st tnhuaamrtnbe oer tra st ,it ohnaatl
is, numberbsew rtihatatstte hr neac toaifno nit onwtto e. Tg heersse are called
irratniuomnbaelIr ctsa. b nes hotwhnta htne u mbat ehrsa att ia saf iiess
·
suacn hu mbTehrne.u mbewrh iictshh r ea ottifho ce i mrfecrueonfac=c ei2 rcle
'7T,
toi tdsi amieastl essruo,ca nh u mber.
Thdee cifomramol fa r atinounmabwlei erli lt theerrm ,ia nsa te
3
= 0.75
4
orw iflolra rm e pienpagate trn,ta s
2 1
06.66. .. 0.090.9 09� = 0 .4128571
3 = II=
Remaryk,tah dbeel cifomramol fa ni rrantuibmoennrea vlefo rr masr eapteing
parnt.te
Threa tiaonindara trli nounmablte orgsefo trhmte hrree anlu mbseyrs tem
(Fieg1 )u.r
Reanlu mbers
Ratinounmable rs
lmtional
nubmers
AGURE 1
PROPERTIES OFTH E REAL Witrshepe cttot hoepe ratoifao dndsia tnimdou nl tiptlhirece anatulim obenr,
NUMBERS( Opitonal) syshtaepsmr operttihareea stf undatmaoe lngtaeTlbhl reea t.at ,be ,r sw ill
and c
denroetnaeul m sbe.r
Closure PropertyT hseu omfa a nbd,d enobtyae +d b ,i asr enaulm ber.
I.
PropertyT hper uocdotfa a nbd,d enobtyae ·bd o rab i, asr enaulm ber.
2.
Wes atyh tahtse eo tfr enaulm biecsrl so sweidrt ehs pteotc hotep aetrioofn s
addiatnimdou nl tipsliinctcahseteu iamon tndh, pe r odoufc trt ewnaoul m bers
are raelnasulom bers.