1300 MATHS FORMULA.pdf

1300 MATHS FORMULA.pdf
1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë=oÉëÉêîÉÇK=
i = qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK=
ii Preface = = = = qÜáë= Ü~åÇÄççâ= áë= ~= ÅçãéäÉíÉ= ÇÉëâíçé= êÉÑÉêÉåÅÉ= Ñçê= ëíì- ÇÉåíë= ~åÇ= ÉåÖáåÉÉêëK= fí= Ü~ë= ÉîÉêóíÜáåÖ= Ñêçã= ÜáÖÜ= ëÅÜççä= ã~íÜ=íç=ã~íÜ=Ñçê=~Çî~åÅÉÇ=ìåÇÉêÖê~Çì~íÉë=áå=ÉåÖáåÉÉêáåÖI= ÉÅçåçãáÅëI=éÜóëáÅ~ä=ëÅáÉåÅÉëI=~åÇ=ã~íÜÉã~íáÅëK=qÜÉ=ÉÄççâ= Åçåí~áåë= ÜìåÇêÉÇë= çÑ= Ñçêãìä~ëI= í~ÄäÉëI= ~åÇ= ÑáÖìêÉë= Ñêçã= kìãÄÉê=pÉíëI=^äÖÉÄê~I=dÉçãÉíêóI=qêáÖçåçãÉíêóI=j~íêáÅÉë= ~åÇ= aÉíÉêãáå~åíëI= sÉÅíçêëI= ^å~äóíáÅ= dÉçãÉíêóI= `~äÅìäìëI= aáÑÑÉêÉåíá~ä=bèì~íáçåëI=pÉêáÉëI=~åÇ=mêçÄ~Äáäáíó=qÜÉçêóK== qÜÉ= ëíêìÅíìêÉÇ= í~ÄäÉ= çÑ= ÅçåíÉåíëI= äáåâëI= ~åÇ= ä~óçìí= ã~âÉ= ÑáåÇáåÖ= íÜÉ= êÉäÉî~åí= áåÑçêã~íáçå= èìáÅâ= ~åÇ= é~áåäÉëëI= ëç= áí= Å~å=ÄÉ=ìëÉÇ=~ë=~å=ÉîÉêóÇ~ó=çåäáåÉ=êÉÑÉêÉåÅÉ=ÖìáÇÉK=== = =
iii Contents = = = = 1 krj_bo=pbqp= NKN= pÉí=fÇÉåíáíáÉë==1= NKO= pÉíë=çÑ=kìãÄÉêë==5= NKP= _~ëáÅ=fÇÉåíáíáÉë==7= NKQ= `çãéäÉñ=kìãÄÉêë==8= = 2 îdb_o^= OKN= c~ÅíçêáåÖ=cçêãìä~ë==12= OKO= mêçÇìÅí=cçêãìä~ë==13= OKP= mçïÉêë==14= OKQ= oççíë==15= OKR= içÖ~êáíÜãë==16= OKS= bèì~íáçåë==18= OKT= fåÉèì~äáíáÉë==19= OKU= `çãéçìåÇ=fåíÉêÉëí=cçêãìä~ë==22= = 3 dbljbqov= PKN= oáÖÜí=qêá~åÖäÉ==24= PKO= fëçëÅÉäÉë=qêá~åÖäÉ==27= PKP= bèìáä~íÉê~ä=qêá~åÖäÉ==28= PKQ= pÅ~äÉåÉ=qêá~åÖäÉ==29= PKR= pèì~êÉ==33= PKS= oÉÅí~åÖäÉ==34= PKT= m~ê~ääÉäçÖê~ã==35= PKU= oÜçãÄìë==36= PKV= qê~éÉòçáÇ==37= PKNM= fëçëÅÉäÉë=qê~éÉòçáÇ==38= PKNN= fëçëÅÉäÉë=qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==40= PKNO= qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==41=
iv PKNP= háíÉ==42= PKNQ= `óÅäáÅ=nì~Çêáä~íÉê~ä==43= PKNR= q~åÖÉåíá~ä=nì~Çêáä~íÉê~ä==45= PKNS= dÉåÉê~ä=nì~Çêáä~íÉê~ä==46= PKNT= oÉÖìä~ê=eÉñ~Öçå==47= PKNU= oÉÖìä~ê=mçäóÖçå==48= PKNV= `áêÅäÉ==50= PKOM= pÉÅíçê=çÑ=~=`áêÅäÉ==53= PKON= pÉÖãÉåí=çÑ=~=`áêÅäÉ==54= PKOO= `ìÄÉ==55= PKOP= oÉÅí~åÖìä~ê=m~ê~ääÉäÉéáéÉÇ==56= PKOQ= mêáëã==57= PKOR= oÉÖìä~ê=qÉíê~ÜÉÇêçå==58= PKOS= oÉÖìä~ê=móê~ãáÇ==59= PKOT= cêìëíìã=çÑ=~=oÉÖìä~ê=móê~ãáÇ==61= PKOU= oÉÅí~åÖìä~ê=oáÖÜí=tÉÇÖÉ==62= PKOV= mä~íçåáÅ=pçäáÇë==63= PKPM= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê==66= PKPN= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê=ïáíÜ=~å=lÄäáèìÉ=mä~åÉ=c~ÅÉ==68= PKPO= oáÖÜí=`áêÅìä~ê=`çåÉ==69= PKPP= cêìëíìã=çÑ=~=oáÖÜí=`áêÅìä~ê=`çåÉ==70= PKPQ= péÜÉêÉ==72= PKPR= péÜÉêáÅ~ä=`~é==72= PKPS= péÜÉêáÅ~ä=pÉÅíçê==73= PKPT= péÜÉêáÅ~ä=pÉÖãÉåí==74= PKPU= péÜÉêáÅ~ä=tÉÇÖÉ==75= PKPV= bääáéëçáÇ==76= PKQM= `áêÅìä~ê=qçêìë==78= = = 4 qofdlkljbqov= QKN= o~Çá~å=~åÇ=aÉÖêÉÉ=jÉ~ëìêÉë=çÑ=^åÖäÉë==80= QKO= aÉÑáåáíáçåë=~åÇ=dê~éÜë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==81= QKP= páÖåë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==86= QKQ= qêáÖçåçãÉíêáÅ=cìåÅíáçåë=çÑ=`çããçå=^åÖäÉë==87= QKR= jçëí=fãéçêí~åí=cçêãìä~ë==88=
v QKS= oÉÇìÅíáçå=cçêãìä~ë==89= QKT= mÉêáçÇáÅáíó=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90= QKU= oÉä~íáçåë=ÄÉíïÉÉå=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90= QKV= ^ÇÇáíáçå=~åÇ=pìÄíê~Åíáçå=cçêãìä~ë==91= QKNM= açìÄäÉ=^åÖäÉ=cçêãìä~ë==92= QKNN= jìäíáéäÉ=^åÖäÉ=cçêãìä~ë==93= QKNO= e~äÑ=^åÖäÉ=cçêãìä~ë==94= QKNP= e~äÑ=^åÖäÉ=q~åÖÉåí=fÇÉåíáíáÉë==94= QKNQ= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=mêçÇìÅí==95= QKNR= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=pìã==97=== QKNS= mçïÉêë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==98= QKNT= dê~éÜë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==99= QKNU= mêáåÅáé~ä=s~äìÉë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==102= QKNV= oÉä~íáçåë=ÄÉíïÉÉå=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==103= QKOM= qêáÖçåçãÉíêáÅ=bèì~íáçåë==106= QKON= oÉä~íáçåë=íç=eóéÉêÄçäáÅ=cìåÅíáçåë==106= = = 5 j^qof`bp=^ka=abqbojfk^kqp= RKN= aÉíÉêãáå~åíë==107= RKO= mêçéÉêíáÉë=çÑ=aÉíÉêãáå~åíë==109= RKP= j~íêáÅÉë==110= RKQ= léÉê~íáçåë=ïáíÜ=j~íêáÅÉë==111= RKR= póëíÉãë=çÑ=iáåÉ~ê=bèì~íáçåë==114= = = 6 sb`qlop= SKN= sÉÅíçê=`ççêÇáå~íÉë==118= SKO= sÉÅíçê=^ÇÇáíáçå==120= SKP= sÉÅíçê=pìÄíê~Åíáçå==122= SKQ= pÅ~äáåÖ=sÉÅíçêë==122= SKR= pÅ~ä~ê=mêçÇìÅí==123= SKS= sÉÅíçê=mêçÇìÅí==125= SKT= qêáéäÉ=mêçÇìÅí=127= = = 7 ^kîvqf`=dbljbqov= TKN= låÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==130=
vi TKO= qïç=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==131= TKP= píê~áÖÜí=iáåÉ=áå=mä~åÉ==139= TKQ= `áêÅäÉ==149= TKR= bääáéëÉ==152= TKS= eóéÉêÄçä~==154= TKT= m~ê~Äçä~==158= TKU= qÜêÉÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==161= TKV= mä~åÉ==165= TKNM= píê~áÖÜí=iáåÉ=áå=pé~ÅÉ==175= TKNN= nì~ÇêáÅ=pìêÑ~ÅÉë==180= TKNO= péÜÉêÉ==189= = = 8 afccbobkqfî=`î`rirp= UKN= cìåÅíáçåë=~åÇ=qÜÉáê=dê~éÜë==191= UKO= iáãáíë=çÑ=cìåÅíáçåë==208= UKP= aÉÑáåáíáçå=~åÇ=mêçéÉêíáÉë=çÑ=íÜÉ=aÉêáî~íáîÉ==209= UKQ= q~ÄäÉ=çÑ=aÉêáî~íáîÉë==211= UKR= eáÖÜÉê=lêÇÉê=aÉêáî~íáîÉë==215= UKS= ^ééäáÅ~íáçåë=çÑ=aÉêáî~íáîÉ==217= UKT= aáÑÑÉêÉåíá~ä==221= UKU= jìäíáî~êá~ÄäÉ=cìåÅíáçåë==222= UKV= aáÑÑÉêÉåíá~ä=léÉê~íçêë==225= = = 9 fkqbdoî=`î`rirp= VKN= fåÇÉÑáåáíÉ=fåíÉÖê~ä==227= VKO= fåíÉÖê~äë=çÑ=o~íáçå~ä=cìåÅíáçåë==228= VKP= fåíÉÖê~äë=çÑ=fêê~íáçå~ä=cìåÅíáçåë==231= VKQ= fåíÉÖê~äë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==237= VKR= fåíÉÖê~äë=çÑ=eóéÉêÄçäáÅ=cìåÅíáçåë==241= VKS= fåíÉÖê~äë=çÑ=bñéçåÉåíá~ä=~åÇ=içÖ~êáíÜãáÅ=cìåÅíáçåë==242= VKT= oÉÇìÅíáçå=cçêãìä~ë==243= VKU= aÉÑáåáíÉ=fåíÉÖê~ä==247= VKV= fãéêçéÉê=fåíÉÖê~ä==253= VKNM= açìÄäÉ=fåíÉÖê~ä==257= VKNN= qêáéäÉ=fåíÉÖê~ä==269=
vii VKNO= iáåÉ=fåíÉÖê~ä==275= VKNP= pìêÑ~ÅÉ=fåíÉÖê~ä==285= = = 10 afccbobkqfî=bnr^qflkp= NMKN= cáêëí=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==295= NMKO= pÉÅçåÇ=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==298= NMKP= pçãÉ=m~êíá~ä=aáÑÑÉêÉåíá~ä=bèì~íáçåë==302= = = 11 pbofbp= NNKN= ^êáíÜãÉíáÅ=pÉêáÉë==304= NNKO= dÉçãÉíêáÅ=pÉêáÉë==305= NNKP= pçãÉ=cáåáíÉ=pÉêáÉë==305= NNKQ= fåÑáåáíÉ=pÉêáÉë==307= NNKR= mêçéÉêíáÉë=çÑ=`çåîÉêÖÉåí=pÉêáÉë==307= NNKS= `çåîÉêÖÉåÅÉ=qÉëíë==308= NNKT= ^äíÉêå~íáåÖ=pÉêáÉë==310= NNKU= mçïÉê=pÉêáÉë==311= NNKV= aáÑÑÉêÉåíá~íáçå=~åÇ=fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë==312= NNKNM= q~óäçê=~åÇ=j~Åä~ìêáå=pÉêáÉë==313= NNKNN= mçïÉê=pÉêáÉë=bñé~åëáçåë=Ñçê=pçãÉ=cìåÅíáçåë==314= NNKNO= _áåçãá~ä=pÉêáÉë==316= NNKNP= cçìêáÉê=pÉêáÉë==316= = = 12 mol_^_fifqv= NOKN= mÉêãìí~íáçåë=~åÇ=`çãÄáå~íáçåë==318= NOKO= mêçÄ~Äáäáíó=cçêãìä~ë==319= = = = = =
viii = qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK= =
1 Chapter 1 Number Sets = = = = 1.1 Set Identities = pÉíëW=Î=_I=`= råáîÉêë~ä=ëÉíW=f= `çãéäÉãÉåí=W= ^′ = mêçéÉê=ëìÄëÉíW= _ ^ ⊂ == bãéíó=ëÉíW=∅= råáçå=çÑ=ëÉíëW= _ ^ ∪ = fåíÉêëÉÅíáçå=çÑ=ëÉíëW= _ ^ ∩ = aáÑÑÉêÉåÅÉ=çÑ=ëÉíëW= _ y ^ = = = 1. f ^ ⊂ = = 2. ^ ^ ⊂ = = 3. _ ^ = =áÑ= _ ^ ⊂ =~åÇ= ^ _ ⊂ .= = 4. bãéíó=pÉí= ^ ⊂ ∅ = = 5. råáçå=çÑ=pÉíë== { } _ ñ çê ^ ñ ö ñ _ ^ ` ∈ ∈ = ∪ = = =
CHAPTER 1. NUMBER SETS 2 ===== = = Figure 1. = 6. `çããìí~íáîáíó= ^ _ _ ^ ∪ = ∪ = = 7. ^ëëçÅá~íáîáíó= ( ) ( ) ` _ ^ ` _ ^ ∪ ∪ = ∪ ∪ = = 8. fåíÉêëÉÅíáçå=çÑ=pÉíë= { } _ ñ ~åÇ ^ ñ ö ñ _ ^ ` ∈ ∈ = ∪ = = = = ===== = = Figure 2. = 9. `çããìí~íáîáíó= ^ _ _ ^ ∩ = ∩ = = 10. ^ëëçÅá~íáîáíó= ( ) ( ) ` _ ^ ` _ ^ ∩ ∩ = ∩ ∩ = =
CHAPTER 1. NUMBER SETS 3 11. aáëíêáÄìíáîáíó= ( ) ( ) ( ) ` ^ _ ^ ` _ ^ ∪ ∩ ∪ = ∩ ∪ I= ( ) ( ) ( ) ` ^ _ ^ ` _ ^ ∩ ∪ ∩ = ∪ ∩ K= = 12. fÇÉãéçíÉåÅó= ^ ^ ^ = ∩ I== ^ ^ ^ = ∪ = = 13. açãáå~íáçå= ∅ = ∅ ∩ ^ I= f f ^ = ∪ = = 14. fÇÉåíáíó= ^ ^ = ∅ ∪ I== ^ f ^ = ∩ = 15. `çãéäÉãÉåí= { } ^ ñ ö f ñ ^ ∉ ∈ = ′ = 16. `çãéäÉãÉåí=çÑ=fåíÉêëÉÅíáçå=~åÇ=råáçå f ^ ^ = ′ ∪ I== ∅ = ′ ∩ ^ ^ = = 17. aÉ=jçêÖ~å∞ë=i~ïë ( ) _ ^ _ ^ ′ ∩ ′ = ′ ∪ I== ( ) _ ^ _ ^ ′ ∪ ′ = ′ ∩ = = 18. aáÑÑÉêÉåÅÉ=çÑ=pÉíë { } ^ ñ ~åÇ _ ñ ö ñ ^ y _ ` ∉ ∈ = = = =
CHAPTER 1. NUMBER SETS 4 ===== = = Figure 3. = 19. ( ) _ ^ y _ ^ y _ ∩ = = 20. ^ _ ^ y _ ′ ∩ = = 21. ∅ = ^ y ^ = 22. ^ _ y ^ = =áÑ= ∅ = ∩_ ^ . = ===== = = Figure 4. = 23. ( ) ( ) ( ) ` _ y ` ^ ` _ y ^ ∩ ∩ = ∩ 24. ^ y f ^ = ′ 25. `~êíÉëá~å=mêçÇìÅí ( ) { } _ ó ~åÇ ^ ñ ö ó I ñ _ ^ ` ∈ ∈ = × = = =
CHAPTER 1. NUMBER SETS 5 1.2 Sets of Numbers = k~íìê~ä=åìãÄÉêëW=k= tÜçäÉ=åìãÄÉêëW= M k = fåíÉÖÉêëW=w= mçëáíáîÉ=áåíÉÖÉêëW= + w = kÉÖ~íáîÉ=áåíÉÖÉêëW= − w = o~íáçå~ä=åìãÄÉêëW=n= oÉ~ä=åìãÄÉêëW=o== `çãéäÉñ=åìãÄÉêëW=`== = = 26. k~íìê~ä=kìãÄÉêë `çìåíáåÖ=åìãÄÉêëW { } K I P I O I N k = K= 27. tÜçäÉ=kìãÄÉêë `çìåíáåÖ=åìãÄÉêë=~åÇ=òÉêçW= { } K I P I O I N I M kM = K= = 28. fåíÉÖÉêë tÜçäÉ=åìãÄÉêë=~åÇ=íÜÉáê=çééçëáíÉë=~åÇ=òÉêçW= { } K I P I O I N k w = = + I= { } N I O I P I w − − − = − K I= { } { } K K I P I O I N I M I N I O I P I w M w w − − − = ∪ ∪ = + − K= = 29. o~íáçå~ä=kìãÄÉêë oÉéÉ~íáåÖ=çê=íÉêãáå~íáåÖ=ÇÉÅáã~äëW==       ≠ ∈ ∈ = = M Ä ~åÇ w Ä ~åÇ w ~ ~åÇ Ä ~ ñ ö ñ n K= = 30. fêê~íáçå~ä=kìãÄÉêë kçåêÉéÉ~íáåÖ=~åÇ=åçåíÉêãáå~íáåÖ=ÇÉÅáã~äëK =
CHAPTER 1. NUMBER SETS 6 31. oÉ~ä=kìãÄÉêë== råáçå=çÑ=ê~íáçå~ä=~åÇ=áêê~íáçå~ä=åìãÄÉêëW=oK= = 32. `çãéäÉñ=kìãÄÉêë { } o ó ~åÇ o ñ ö áó ñ ` ∈ ∈ + = I== ïÜÉêÉ=á=áë=íÜÉ=áã~Öáå~êó=ìåáíK = 33. ` o n w k ⊂ ⊂ ⊂ ⊂ = = === = = Figure 5. = = = = = =
CHAPTER 1. NUMBER SETS 7 1.3 Basic Identities = oÉ~ä=åìãÄÉêëW=~I=ÄI=Å= = = 34. ^ÇÇáíáîÉ=fÇÉåíáíó= ~ M ~ = + = = 35. ^ÇÇáíáîÉ=fåîÉêëÉ= ( ) M ~ ~ = − + = = 36. `çããìí~íáîÉ=çÑ=^ÇÇáíáçå= ~ Ä Ä ~ + = + = = 37. ^ëëçÅá~íáîÉ=çÑ=^ÇÇáíáçå= ( ) ( ) Å Ä ~ Å Ä ~ + + = + + = = 38. aÉÑáåáíáçå=çÑ=pìÄíê~Åíáçå= ( ) Ä ~ Ä ~ − + = − = = 39. jìäíáéäáÅ~íáîÉ=fÇÉåíáíó= ~ N ~ = ⋅ = = 40. jìäíáéäáÅ~íáîÉ=fåîÉêëÉ= N ~ N ~ = ⋅ I= M ~ ≠ = 41. jìäíáéäáÅ~íáçå=qáãÉë=M M M ~ = ⋅ = 42. `çããìí~íáîÉ=çÑ=jìäíáéäáÅ~íáçå= ~ Ä Ä ~ ⋅ = ⋅ = =
CHAPTER 1. NUMBER SETS 8 43. ^ëëçÅá~íáîÉ=çÑ=jìäíáéäáÅ~íáçå= ( ) ( ) Å Ä ~ Å Ä ~ ⋅ ⋅ = ⋅ ⋅ = 44. aáëíêáÄìíáîÉ=i~ï= ( ) ~Å ~Ä Å Ä ~ + = + = = 45. aÉÑáåáíáçå=çÑ=aáîáëáçå= Ä N ~ Ä ~ ⋅ = = = = = 1.4 Complex Numbers = k~íìê~ä=åìãÄÉêW=å= fã~Öáå~êó=ìåáíW=á= `çãéäÉñ=åìãÄÉêW=ò= oÉ~ä=é~êíW=~I=Å= fã~Öáå~êó=é~êíW=ÄáI=Çá= jçÇìäìë=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=êI= N ê I= O ê = ^êÖìãÉåí=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=ϕ I= N ϕ I= O ϕ = = = á áN = = á áR = = á á N å Q = + = N áO − = = N áS − = = N á O å Q − = + = á áP − = = á áT − = = á á P å Q − = + = 46. N áQ = = N áU = = N á å Q = = = 47. Äá ~ ò + = = = 48. `çãéäÉñ=mä~åÉ= =
CHAPTER 1. NUMBER SETS 9 ===== = = Figure 6. = 49. ( ) ( ) ( ) ( )á Ç Ä Å ~ Çá Å Äá ~ + + + = + + + = = 50. ( ) ( ) ( ) ( )á Ç Ä Å ~ Çá Å Äá ~ − + − = + − + = = 51. ( )( ) ( ) ( )á ÄÅ ~Ç ÄÇ ~Å Çá Å Äá ~ + + − = + + = = 52. á Ç Å ~Ç ÄÅ Ç Å ÄÇ ~Å Çá Å Äá ~ O O O O ⋅ + − + + + = + + = = 53. `çåàìÖ~íÉ=`çãéäÉñ=kìãÄÉêë= Äá ~ Äá ~ ||||||| − = + = = 54. ϕ = Åçë ê ~ I= ϕ = ëáå ê Ä == =
CHAPTER 1. NUMBER SETS 10 = = Figure 7. = 55. mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë= ( ) ϕ + ϕ = + ëáå á Åçë ê Äá ~ = = 56. jçÇìäìë=~åÇ=^êÖìãÉåí=çÑ=~=`çãéäÉñ=kìãÄÉê= fÑ= Äá ~ + =áë=~=ÅçãéäÉñ=åìãÄÉêI=íÜÉå= O O Ä ~ ê + = =EãçÇìäìëFI== ~ Ä ~êÅí~å = ϕ =E~êÖìãÉåíFK= = 57. mêçÇìÅí=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ( ) ( ) O O O N N N O N ëáå á Åçë ê ëáå á Åçë ê ò ò ϕ + ϕ ⋅ ϕ + ϕ = ⋅ = ( ) ( ) [ ] O N O N O N ëáå á Åçë ê ê ϕ + ϕ + ϕ + ϕ = = = 58. `çåàìÖ~íÉ=kìãÄÉêë=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ( ) ( ) ( ) [ ] ϕ − + ϕ − = ϕ + ϕ ëáå á Åçë ê ëáå á Åçë ê | |||||||||| |||||||||| = = 59. fåîÉêëÉ=çÑ=~=`çãéäÉñ=kìãÄÉê=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ( ) ( ) ( ) [ ] ϕ − + ϕ − = ϕ + ϕ ëáå á Åçë ê N ëáå á Åçë ê N =
CHAPTER 1. NUMBER SETS 11 60. nìçíáÉåí=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ( ) ( ) ( ) ( ) [ ] O N O N O N O O O N N N O N ëáå á Åçë ê ê ëáå á Åçë ê ëáå á Åçë ê ò ò ϕ − ϕ + ϕ − ϕ = ϕ + ϕ ϕ + ϕ = = = 61. mçïÉê=çÑ=~=`çãéäÉñ=kìãÄÉê= ( ) [ ] ( ) ( ) [ ] ϕ + ϕ = ϕ + ϕ = å ëáå á å Åçë ê ëáå á Åçë ê ò å å å = = 62. cçêãìä~=±aÉ=jçáîêÉ≤= ( ) ( ) ( ) ϕ + ϕ = ϕ + ϕ å ëáå á å Åçë ëáå á Åçë å = = 63. kíÜ=oççí=çÑ=~=`çãéäÉñ=kìãÄÉê= ( )       π + ϕ + π + ϕ = ϕ + ϕ = å â O ëáå á å â O Åçë ê ëáå á Åçë ê ò å å å I== ïÜÉêÉ== N å I I O I N I M â − = K K== = 64. bìäÉê∞ë=cçêãìä~= ñ ëáå á ñ Åçë Éáñ + = = = =
12 Chapter 2 Algebra = = = = 2.1 Factoring Formulas = oÉ~ä=åìãÄÉêëW=~I=ÄI=Å== k~íìê~ä=åìãÄÉêW=å= = = 65. ( )( ) Ä ~ Ä ~ Ä ~ O O − + = − = = 66. ( )( ) O O P P Ä ~Ä ~ Ä ~ Ä ~ + + − = − = = 67. ( )( ) O O P P Ä ~Ä ~ Ä ~ Ä ~ + − + = + = = 68. ( )( ) ( )( )( ) O O O O O O Q Q Ä ~ Ä ~ Ä ~ Ä ~ Ä ~ Ä ~ + + − = + − = − = = 69. ( )( ) Q P O O P Q R R Ä ~Ä Ä ~ Ä ~ ~ Ä ~ Ä ~ + + + + − = − = = 70. ( )( ) Q P O O P Q R R Ä ~Ä Ä ~ Ä ~ ~ Ä ~ Ä ~ + − + − + = + = = 71. fÑ=å=áë=çÇÇI=íÜÉå= ( )( ) N å O å O P å O å N å å å Ä ~Ä Ä ~ Ä ~ ~ Ä ~ Ä ~ − − − − − + − − + − + = + K K== = 72. fÑ=å=áë=ÉîÉåI=íÜÉå== ( )( ) N å O å O P å O å N å å å Ä ~Ä Ä ~ Ä ~ ~ Ä ~ Ä ~ − − − − − + + + + + − = − K I==
CHAPTER 2. ALGEBRA 13 ( )( ) N å O å O P å O å N å å å Ä ~Ä Ä ~ Ä ~ ~ Ä ~ Ä ~ − − − − − − + − + − + = + K K= = = = 2.2 Product Formulas oÉ~ä=åìãÄÉêëW=~I=ÄI=Å== tÜçäÉ=åìãÄÉêëW=åI=â= = = 73. ( ) O O O Ä ~Ä O ~ Ä ~ + − = − = = 74. ( ) O O O Ä ~Ä O ~ Ä ~ + + = + = = 75. ( ) P O O P P Ä ~Ä P Ä ~ P ~ Ä ~ − + − = − = = 76. ( ) P O O P P Ä ~Ä P Ä ~ P ~ Ä ~ + + + = + = = 77. ( ) Q P O O P Q Q Ä ~Ä Q Ä ~ S Ä ~ Q ~ Ä ~ + − + − = − = = 78. ( ) Q P O O P Q Q Ä ~Ä Q Ä ~ S Ä ~ Q ~ Ä ~ + + + + = + = = 79. _áåçãá~ä=cçêãìä~= ( ) I Ä ` ~Ä ` Ä ~ ` Ä ~ ` ~ ` Ä ~ å å å N å N å å O O å O å N å N å å M å å + + + + + = + − − − − K ïÜÉêÉ= ( )> â å > â > å `â å − = =~êÉ=íÜÉ=Äáåçãá~ä=ÅçÉÑÑáÅáÉåíëK= = 80. ( ) ÄÅ O ~Å O ~Ä O Å Ä ~ Å Ä ~ O O O O + + + + + = + + = = 81. ( ) + + + + + + = + + + + + O O O O O O î ì Å Ä ~ î ì Å Ä ~ K K = ( ) ìî Äî Äì ÄÅ ~î ~ì ~Å ~Ä O + + + + + + + + + + + K K K =
CHAPTER 2. ALGEBRA 14 2.3 Powers = _~ëÉë=EéçëáíáîÉ=êÉ~ä=åìãÄÉêëFW=~I=Ä== mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã= = = 82. å ã å ã ~ ~ ~ + = = = 83. å ã å ã ~ ~ ~ − = = = 84. ( ) ã ã ã Ä ~ ~Ä = = = 85. ã ã ã Ä ~ Ä ~ =       = = 86. ( ) ãå å ã ~ ~ = = = 87. N ~M = I= M ~ ≠ = = 88. N ~N = = = 89. ã ã ~ N ~ = − = = 90. å ã å ã ~ ~ = = = = = = =
CHAPTER 2. ALGEBRA 15 2.4 Roots = _~ëÉëW=~I=Ä== mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã= M Ä I ~ ≥ =Ñçê=ÉîÉå=êççíë=E â O å = I= k â∈ F= = = 91. å å å Ä ~ ~Ä = = = 92. åã å ã ã å Ä ~ Ä ~ = = = 93. å å å Ä ~ Ä ~ = I= M Ä ≠ = = 94. åã å ã åã å åã ã ã å Ä ~ Ä ~ Ä ~ = = I= M Ä ≠ K= = 95. ( ) å ãé é å ã ~ ~ = = = 96. ( ) ~ ~ å å = = = 97. åé ãé å ã ~ ~ = = = 98. å ã å ã ~ ~ = = = 99. ãå ã å ~ ~ = = = 100. ( ) å ã ã å ~ ~ = = =
CHAPTER 2. ALGEBRA 16 101. ~ ~ ~ N å N å å − = I= M ~ ≠ K= = 102. O Ä ~ ~ O Ä ~ ~ Ä ~ O O − − ± − + = ± = = 103. Ä ~ Ä ~ Ä ~ N − = ± m = = = = 2.5 Logarithms = mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=ñI=óI=~I=ÅI=â= k~íìê~ä=åìãÄÉêW=å== = = 104. aÉÑáåáíáçå=çÑ=içÖ~êáíÜã= ñ äçÖ ó ~ = =áÑ=~åÇ=çåäó=áÑ= ó ~ ñ = I= M ~ > I= N ~ ≠ K= = 105. M N äçÖ~ = = = 106. N ~ äçÖ~ = = = 107.    < ∞ + > ∞ − = N ~ áÑ N ~ áÑ M äçÖ~ = = 108. ( ) ó äçÖ ñ äçÖ ñó äçÖ ~ ~ ~ + = = = 109. ó äçÖ ñ äçÖ ó ñ äçÖ ~ ~ ~ − = =
CHAPTER 2. ALGEBRA 17 110. ( ) ñ äçÖ å ñ äçÖ ~ å ~ = = = 111. ñ äçÖ å N ñ äçÖ ~ å ~ = = = 112. Å äçÖ ñ äçÖ ~ äçÖ ñ äçÖ ñ äçÖ ~ Å Å Å ~ ⋅ = = I= M Å > I= N Å ≠ K= = 113. ~ äçÖ N Å äçÖ Å ~ = = = 114. ñ äçÖ~ ~ ñ = = = 115. içÖ~êáíÜã=íç=_~ëÉ=NM= ñ äçÖ ñ äçÖNM = = = 116. k~íìê~ä=içÖ~êáíÜã= ñ äå ñ äçÖÉ = I== ïÜÉêÉ= K TNUOUNUOU K O â N N äáã É â â =       + = ∞ → = = 117. ñ äå QPQOVQ K M ñ äå NM äå N ñ äçÖ = = = = 118. ñ äçÖ PMORUR K O ñ äçÖ É äçÖ N ñ äå = = = = = = = =
CHAPTER 2. ALGEBRA 18 2.6 Equations = oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=éI=èI=ìI=î= pçäìíáçåëW= N ñ I= O ñ I= N ó I= O ó I= P ó = = = 119. iáåÉ~ê=bèì~íáçå=áå=låÉ=s~êá~ÄäÉ= M Ä ~ñ = + I= ~ Ä ñ − = K== = 120. nì~Çê~íáÅ=bèì~íáçå= M Å Äñ ~ñO = + + I= ~ O ~Å Q Ä Ä ñ O O I N − ± − = K= = 121. aáëÅêáãáå~åí= ~Å Q Ä a O − = = = 122. sáÉíÉ∞ë=cçêãìä~ë= fÑ= M è éñ ñO = + + I=íÜÉå==    = − = + è ñ ñ é ñ ñ O N O N K= = 123. M Äñ ~ñO = + I= M ñN = I= ~ Ä ñO − = K= = 124. M Å ~ñO = + I= ~ Å ñ O I N − ± = K= = 125. `ìÄáÅ=bèì~íáçåK=`~êÇ~åç∞ë=cçêãìä~K== M è éó óP = + + I==
CHAPTER 2. ALGEBRA 19 î ì óN + = I= ( ) ( )á î ì O P î ì O N ó P I O + ± + − = I== ïÜÉêÉ== P O O P é O è O è ì       +       + − = I= P O O P é O è O è î       +       − − = K== = = 2.7 Inequalities s~êá~ÄäÉëW=ñI=óI=ò= oÉ~ä=åìãÄÉêëW=    å P O N ~ I I ~ I ~ I ~ Ç I Å I Ä I ~ K I=ãI=å= aÉíÉêãáå~åíëW=aI= ñ a I= ó a I= ò a == = = 126. fåÉèì~äáíáÉëI=fåíÉêî~ä=kçí~íáçåë=~åÇ=dê~éÜë== = fåÉèì~äáíó= fåíÉêî~ä=kçí~íáçå= dê~éÜ= Ä ñ ~ ≤ ≤ = [ ] Ä I ~ = = Ä ñ ~ ≤ < = ( ] Ä I ~ = = Ä ñ ~ < ≤ = [ ) Ä I ~ = = Ä ñ ~ < < = ( ) Ä I ~ = = Ä ñ ≤ < ∞ − I= Ä ñ ≤ = ( ] Ä I ∞ − = = Ä ñ < < ∞ − I= Ä ñ < = ( ) Ä I ∞ − = = ∞ < ≤ ñ ~ I= ~ ñ ≥ = [ ) ∞ I ~ = = ∞ < < ñ ~ I= ~ ñ > = ( ) ∞ I ~ = =
CHAPTER 2. ALGEBRA 20 127. fÑ= Ä ~ > I=íÜÉå= ~ Ä < K= = 128. fÑ= Ä ~ > I=íÜÉå= M Ä ~ > − =çê= M ~ Ä < − K= = 129. fÑ= Ä ~ > I=íÜÉå= Å Ä Å ~ + > + K= = 130. fÑ= Ä ~ > I=íÜÉå= Å Ä Å ~ − > − K= = 131. fÑ= Ä ~ > =~åÇ= Ç Å > I=íÜÉå= Ç Ä Å ~ + > + K= = 132. fÑ= Ä ~ > =~åÇ= Ç Å > I=íÜÉå= Å Ä Ç ~ − > − K= = 133. fÑ= Ä ~ > =~åÇ= M ã > I=íÜÉå= ãÄ ã~ > K= = 134. fÑ= Ä ~ > =~åÇ= M ã > I=íÜÉå= ã Ä ã ~ > K= = 135. fÑ= Ä ~ > =~åÇ= M ã < I=íÜÉå= ãÄ ã~ < K= = 136. fÑ= Ä ~ > =~åÇ= M ã < I=íÜÉå= ã Ä ã ~ < K= = 137. fÑ= Ä ~ M < < =~åÇ= M å > I=íÜÉå= å å Ä ~ < K= = 138. fÑ= Ä ~ M < < =~åÇ= M å < I=íÜÉå= å å Ä ~ > K= = 139. fÑ= Ä ~ M < < I=íÜÉå= å å Ä ~ < K= = 140. O Ä ~ ~Ä + ≤ I== ïÜÉêÉ= M ~ > =I= M Ä > X=~å=Éèì~äáíó=áë=î~äáÇ=çåäó=áÑ= Ä ~ = K== = 141. O ~ N ~ ≥ + I=ïÜÉêÉ= M ~ > X=~å=Éèì~äáíó=í~âÉë=éä~ÅÉ=çåäó=~í= N ~ = K=
CHAPTER 2. ALGEBRA 21 142. å ~ ~ ~ ~ ~ ~ å O N å å O N + + + ≤ K K I=ïÜÉêÉ= M ~ I I ~ I ~ å O N > K K= = 143. fÑ= M Ä ~ñ > + =~åÇ= M ~ > I=íÜÉå= ~ Ä ñ − > K= = 144. fÑ= M Ä ~ñ > + =~åÇ= M ~ < I=íÜÉå= ~ Ä ñ − < K== = 145. M Å Äñ ~ñO > + + = = = M ~ > = M ~ < = = = = M a > = = = N ñ ñ < I= O ñ ñ > = = = = O N ñ ñ ñ < < = = = = M a = = = ñ ñN < I= N ñ ñ > = = = ∅ ∈ ñ = = = = M a< = = = ∞ < < ∞ − ñ = = = = ∅ ∈ ñ = =
CHAPTER 2. ALGEBRA 22 146. Ä ~ Ä ~ + ≤ + = = 147. fÑ= ~ ñ < I=íÜÉå= ~ ñ ~ < < − I=ïÜÉêÉ= M ~ > K= = 148. fÑ= ~ ñ > I=íÜÉå= ~ ñ − < =~åÇ= ~ ñ > I=ïÜÉêÉ= M ~ > K= = 149. fÑ= ~ ñO < I=íÜÉå= ~ ñ < I=ïÜÉêÉ= M ~ > K= = 150. fÑ= ~ ñO > I=íÜÉå= ~ ñ > I=ïÜÉêÉ= M ~ > K= = 151. fÑ= ( ) ( ) M ñ Ö ñ Ñ > I=íÜÉå= ( ) ( ) ( )    ≠ > ⋅ M ñ Ö M ñ Ö ñ Ñ K= = 152. ( ) ( ) M ñ Ö ñ Ñ < I=íÜÉå= ( ) ( ) ( )    ≠ < ⋅ M ñ Ö M ñ Ö ñ Ñ K= = = = 2.8 Compound Interest Formulas = cìíìêÉ=î~äìÉW=^= fåáíá~ä=ÇÉéçëáíW=`= ^ååì~ä=ê~íÉ=çÑ=áåíÉêÉëíW=ê= kìãÄÉê=çÑ=óÉ~êë=áåîÉëíÉÇW=í= kìãÄÉê=çÑ=íáãÉë=ÅçãéçìåÇÉÇ=éÉê=óÉ~êW=å= = = 153. dÉåÉê~ä=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~= åí å ê N ` ^       + = = =
CHAPTER 2. ALGEBRA 23 154. páãéäáÑáÉÇ=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~= fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=çåÅÉ=éÉê=óÉ~êI=íÜÉå=íÜÉ=éêÉîáçìë= Ñçêãìä~=ëáãéäáÑáÉë=íçW= ( )í ê N ` ^ + = K= = 155. `çåíáåìçìë=`çãéçìåÇ=fåíÉêÉëí= fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=Åçåíáåì~ääó=E ∞ → å FI=íÜÉå== êí `É ^ = K= = =
24 Chapter 3 Geometry = = = = 3.1 Right Triangle = iÉÖë=çÑ=~=êáÖÜí=íêá~åÖäÉW=~I=Ä= eóéçíÉåìëÉW=Å= ^äíáíìÇÉW=Ü= jÉÇá~åëW= ~ ã I= Ä ã I= Å ã = ^åÖäÉëW=α Iβ = o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= ^êÉ~W=p= = = = = Figure 8. = 156. ° = β + α VM = =
CHAPTER 3. GEOMETRY 25 157. β = = α Åçë Å ~ ëáå = = 158. β = = α ëáå Å Ä Åçë = = 159. β = = α Åçí Ä ~ í~å = = 160. β = = α í~å ~ Ä Åçí = = 161. β = = α ÉÅ Åçë Ä Å ëÉÅ = = 162. β = = α ëÉÅ ~ Å ÉÅ Åçë = = 163. móíÜ~ÖçêÉ~å=qÜÉçêÉã= O O O Å Ä ~ = + = = 164. ÑÅ ~O = I= ÖÅ ÄO = I== ïÜÉêÉ= Ñ= ~åÇ= Å= ~êÉ= éêçàÉÅíáçåë= çÑ= íÜÉ= äÉÖë= ~= ~åÇ= ÄI= êÉëéÉÅ- íáîÉäóI=çåíç=íÜÉ=ÜóéçíÉåìëÉ=ÅK= = ===== = = Figure 9. =
CHAPTER 3. GEOMETRY 26 165. ÑÖ ÜO = I=== ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK== = 166. Q ~ Ä ã O O O ~ − = I= Q Ä ~ ã O O O Ä − = I=== ïÜÉêÉ= ~ ã =~åÇ= Ä ã =~êÉ=íÜÉ=ãÉÇá~åë=íç=íÜÉ=äÉÖë=~=~åÇ=ÄK== = = = Figure 10. = 167. O Å ãÅ = I== ïÜÉêÉ= Å ã =áë=íÜÉ=ãÉÇá~å=íç=íÜÉ=ÜóéçíÉåìëÉ=ÅK= = 168. Å ã O Å o = = = = 169. Å Ä ~ ~Ä O Å Ä ~ ê + + = − + = = = 170. ÅÜ ~Ä = = = =
CHAPTER 3. GEOMETRY 27 171. O ÅÜ O ~Ä p = = = = = = 3.2 Isosceles Triangle = _~ëÉW=~= iÉÖëW=Ä= _~ëÉ=~åÖäÉW=β = sÉêíÉñ=~åÖäÉW=α = ^äíáíìÇÉ=íç=íÜÉ=Ä~ëÉW=Ü= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 11. = 172. O VM α − ° = β = = 173. Q ~ Ä Ü O O O − = =
CHAPTER 3. GEOMETRY 28 174. Ä O ~ i + = = = 175. α = = ëáå O Ä O ~Ü p O = = = = 3.3 Equilateral Triangle = páÇÉ=çÑ=~=Éèìáä~íÉê~ä=íêá~åÖäÉW=~= ^äíáíìÇÉW=Ü= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 12. = 176. O P ~ Ü = = =
CHAPTER 3. GEOMETRY 29 177. P P ~ Ü P O o = = = = 178. O o S P ~ Ü P N ê = = = = = 179. ~ P i = = = 180. Q P ~ O ~Ü p O = = = = = = 3.4 Scalene Triangle E^=íêá~åÖäÉ=ïáíÜ=åç=íïç=ëáÇÉë=Éèì~äF= = = páÇÉë=çÑ=~=íêá~åÖäÉW=~I=ÄI=Å= pÉãáéÉêáãÉíÉêW= O Å Ä ~ é + + = == ^åÖäÉë=çÑ=~=íêá~åÖäÉW= γ β α I I = ^äíáíìÇÉë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= Å Ä ~ Ü I Ü I Ü = jÉÇá~åë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= Å Ä ~ ã I ã I ã = _áëÉÅíçêë=çÑ=íÜÉ=~åÖäÉë= γ β α I I W= Å Ä ~ í I í I í = o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= ^êÉ~W=p= = =
CHAPTER 3. GEOMETRY 30 ===== = = Figure 13. = 181. ° = γ + β + α NUM = = 182. Å Ä ~ > + I== ~ Å Ä > + I== Ä Å ~ > + K= = 183. Å Ä ~ < − I== ~ Å Ä < − I== Ä Å ~ < − K= = 184. jáÇäáåÉ= O ~ è = I= ~ öö è K= = ===== = = Figure 14. =
CHAPTER 3. GEOMETRY 31 185. i~ï=çÑ=`çëáåÉë= α − + = Åçë ÄÅ O Å Ä ~ O O O I= β − + = Åçë ~Å O Å ~ Ä O O O I= γ − + = Åçë ~Ä O Ä ~ Å O O O K= = 186. i~ï=çÑ=páåÉë= o O ëáå Å ëáå Ä ëáå ~ = γ = β = α I== ïÜÉêÉ=o=áë=íÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉK== = 187. p Q ~ÄÅ Ü O ~Ä Ü O ~Å Ü O ÄÅ ëáå O Å ëáå O Ä ëáå O ~ o Å Ä ~ = = = = γ = β = α = = = 188. ( )( )( ) é Å é Ä é ~ é êO − − − = I== Å Ä ~ Ü N Ü N Ü N ê N + + = K= = 189. ( )( ) ÄÅ Å é Ä é O ëáå − − = α I= ( ) ÄÅ ~ é é O Åçë − = α I= ( )( ) ( ) ~ é é Å é Ä é O í~å − − − = α K= = 190. ( )( )( ) Å é Ä é ~ é é ~ O Ü~ − − − = I= ( )( )( ) Å é Ä é ~ é é Ä O ÜÄ − − − = I= ( )( )( ) Å é Ä é ~ é é Å O ÜÅ − − − = K=
CHAPTER 3. GEOMETRY 32 191. β = γ = ëáå Å ëáå Ä Ü~ I= α = γ = ëáå Å ëáå ~ ÜÄ I= α = β = ëáå Ä ëáå ~ ÜÅ K= = 192. Q ~ O Å Ä ã O O O O ~ − + = I== Q Ä O Å ~ ã O O O O Ä − + = I== Q Å O Ä ~ ã O O O O Å − + = K= = ===== = = Figure 15. = 193. ~ ã P O ^j = I= Ä ã P O _j = I= Å ã P O `j = =EcáÖKNRFK= = 194. ( ) ( )O O ~ Å Ä ~ é ÄÅé Q í + − = I== ( ) ( )O O Ä Å ~ Ä é ~Åé Q í + − = I== ( ) ( )O O Å Ä ~ Å é ~Äé Q í + − = K= =
CHAPTER 3. GEOMETRY 33 195. O ÅÜ O ÄÜ O ~Ü p Å Ä ~ = = = I== O ëáå ÄÅ O ëáå ~Å O ëáå ~Ä p α = β = γ = I== ( )( )( ) Å é Ä é ~ é é p − − − = =EeÉêçå∞ë=cçêãìä~FI= éê p = I== o Q ~ÄÅ p = I= γ β α = ëáå ëáå ëáå o O p O I= O í~å O í~å O í~å é p O γ β α = K= = = = 3.5 Square páÇÉ=çÑ=~=ëèì~êÉW=~= aá~Öçå~äW=Ç= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = Figure 16.
CHAPTER 3. GEOMETRY 34 196. O ~ Ç = == = 197. O O ~ O Ç o = = = = 198. O ~ ê = = = 199. ~ Q i = = = 200. O ~ p = = = = = 3.6 Rectangle = páÇÉë=çÑ=~=êÉÅí~åÖäÉW=~I=Ä= aá~Öçå~äW=Ç= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 17. = 201. O O Ä ~ Ç + = ==
CHAPTER 3. GEOMETRY 35 202. O Ç o = = = 203. ( ) Ä ~ O i + = = = 204. ~Ä p = = = = = 3.7 Parallelogram = páÇÉë=çÑ=~=é~ê~ääÉäçÖê~ãW=~I=Ä= aá~Öçå~äëW= O N Ç I Ç = `çåëÉÅìíáîÉ=~åÖäÉëW= β αI = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ = ^äíáíìÇÉW=Ü== mÉêáãÉíÉêW=i= ^êÉ~W=p= = = ===== = = Figure 18. = 205. ° = β + α NUM = = 206. ( ) O O O O O N Ä ~ O Ç Ç + = + = =
CHAPTER 3. GEOMETRY 36 207. β = α = ëáå Ä ëáå Ä Ü = = 208. ( ) Ä ~ O i + = = = 209. α = = ëáå ~Ä ~Ü p I== ϕ = ëáå Ç Ç O N p O N K= = = = 3.8 Rhombus = páÇÉ=çÑ=~=êÜçãÄìëW=~= aá~Öçå~äëW= O N Ç I Ç = `çåëÉÅìíáîÉ=~åÖäÉëW= β αI = ^äíáíìÇÉW=e= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = ===== = = Figure 19. =
CHAPTER 3. GEOMETRY 37 210. ° = β + α NUM = = 211. O O O O N ~ Q Ç Ç = + = = 212. ~ O Ç Ç ëáå ~ Ü O N = α = = = 213. O ëáå ~ ~ Q Ç Ç O Ü ê O N α = = = = = 214. ~ Q i = = = 215. α = = ëáå ~ ~Ü p O I== O NÇ Ç O N p = K= = = = 3.9 Trapezoid = _~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä= jáÇäáåÉW=è= ^äíáíìÇÉW=Ü= ^êÉ~W=p= = =
CHAPTER 3. GEOMETRY 38 = = Figure 20. = 216. O Ä ~ è + = = = 217. èÜ Ü O Ä ~ p = ⋅ + = = = = = 3.10 Isosceles Trapezoid = _~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä= iÉÖW=Å= jáÇäáåÉW=è= ^äíáíìÇÉW=Ü= aá~Öçå~äW=Ç= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= ^êÉ~W=p= = =
CHAPTER 3. GEOMETRY 39 = = Figure 21. = 218. O Ä ~ è + = = = 219. O Å ~Ä Ç + = = = 220. ( )O O ~ Ä Q N Å Ü − − = = = 221. ( )( ) Ä ~ Å O Ä ~ Å O Å ~Ä Å o O − + + − + = = = 222. èÜ Ü O Ä ~ p = ⋅ + = = = = = = = =
CHAPTER 3. GEOMETRY 40 3.11 Isosceles Trapezoid with Inscribed Circle = _~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä= iÉÖW=Å= jáÇäáåÉW=è= ^äíáíìÇÉW=Ü= aá~Öçå~äW=Ç= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 22. = 223. Å O Ä ~ = + = = 224. Å O Ä ~ è = + = = = 225. O O O Å Ü Ç + = = =
CHAPTER 3. GEOMETRY 41 226. O ~Ä O Ü ê = = = = 227. ~ Ä S Ä ~ U Ä ~ Å Ü Ü O Å ~Ä Å N O Å ê Q ÅÇ Ü O ÅÇ o O O O + + + = + = + = = = = = 228. ( ) Å Q Ä ~ O i = + = = = 229. ( ) O iê ÅÜ èÜ O ~Ä Ä ~ Ü O Ä ~ p = = = + = ⋅ + = == = = = 3.12 Trapezoid with Inscribed Circle = _~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä= i~íÉê~ä=ëáÇÉëW=ÅI=Ç= jáÇäáåÉW=è= ^äíáíìÇÉW=Ü= aá~Öçå~äëW= O N Ç I Ç = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ = o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= ^êÉ~W=p= =
CHAPTER 3. GEOMETRY 42 = = Figure 23. = 230. Ç Å Ä ~ + = + = = 231. O Ç Å O Ä ~ è + = + = = = 232. ( ) ( ) Ç Å O Ä ~ O i + = + = = = 233. èÜ Ü O Ç Å Ü O Ä ~ p = ⋅ + = ⋅ + = I== ϕ = ëáå Ç Ç O N p O N K= = = = 3.13 Kite = páÇÉë=çÑ=~=âáíÉW=~I=Ä= aá~Öçå~äëW= O N Ç I Ç = ^åÖäÉëW= γ β α I I = mÉêáãÉíÉêW=i= ^êÉ~W=p= = =
CHAPTER 3. GEOMETRY 43 = = Figure 24. = 234. ° = γ + β + α PSM O = = 235. ( ) Ä ~ O i + = = = 236. O Ç Ç p O N = = = = = 3.14 Cyclic Quadrilateral páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç= aá~Öçå~äëW= O N Ç I Ç = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ = fåíÉêå~ä=~åÖäÉëW= δ γ β α I I I = o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= pÉãáéÉêáãÉíÉêW=é== ^êÉ~W=p=
CHAPTER 3. GEOMETRY 44 = = Figure 25. = 237. ° = δ + β = γ + α NUM = = 238. míçäÉãó∞ë=qÜÉçêÉã= O NÇ Ç ÄÇ ~Å = + = = 239. Ç Å Ä ~ i + + + = = = 240. ( )( )( ) ( )( )( )( ) Ç é Å é Ä é ~ é ÅÇ ~Ä ÄÅ ~Ç ÄÇ ~Å Q N o − − − − + + + = I== ïÜÉêÉ= O i é = K= = 241. ϕ = ëáå Ç Ç O N p O N I== ( )( )( )( ) Ç é Å é Ä é ~ é p − − − − = I== ïÜÉêÉ= O i é = K= = = =
CHAPTER 3. GEOMETRY 45 3.15 Tangential Quadrilateral = páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç= aá~Öçå~äëW= O N Ç I Ç = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ = o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= pÉãáéÉêáãÉíÉêW=é== ^êÉ~W=p= = = = = Figure 26. = 242. Ç Ä Å ~ + = + = = 243. ( ) ( ) Ç Ä O Å ~ O Ç Å Ä ~ i + = + = + + + = = = 244. ( ) ( ) é O é Ä ~ Ä ~ Ç Ç ê O O O O O N − + − − = I== ïÜÉêÉ= O i é = K== =
CHAPTER 3. GEOMETRY 46 245. ϕ = = ëáå Ç Ç O N éê p O N = = = = 3.16 General Quadrilateral = páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç= aá~Öçå~äëW= O N Ç I Ç = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ = fåíÉêå~ä=~åÖäÉëW= δ γ β α I I I = mÉêáãÉíÉêW=i= ^êÉ~W=p= = = ======= = = Figure 27. = 246. ° = δ + γ + β + α PSM = = 247. Ç Å Ä ~ i + + + = = =
CHAPTER 3. GEOMETRY 47 248. ϕ = ëáå Ç Ç O N p O N = = = = 3.17 Regular Hexagon = páÇÉW=~= fåíÉêå~ä=~åÖäÉW=α = pä~åí=ÜÉáÖÜíW=ã= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= pÉãáéÉêáãÉíÉêW=é== ^êÉ~W=p= = = = = Figure 28. = 249. ° = α NOM = = 250. O P ~ ã ê = = =
CHAPTER 3. GEOMETRY 48 251. ~ o = = = 252. ~ S i = = = 253. O P P ~ éê p O = = I== ïÜÉêÉ= O i é = K= = = = 3.18 Regular Polygon = páÇÉW=~= kìãÄÉê=çÑ=ëáÇÉëW=å= fåíÉêå~ä=~åÖäÉW=α = pä~åí=ÜÉáÖÜíW=ã= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= pÉãáéÉêáãÉíÉêW=é== ^êÉ~W=p= = =
CHAPTER 3. GEOMETRY 49 = = Figure 29. = 254. ° ⋅ − = α NUM O O å = = 255. ° ⋅ − = α NUM O O å = = 256. å ëáå O ~ o π = = = 257. Q ~ o å í~å O ~ ã ê O O − = π = = = = 258. å~ i = = = 259. å O ëáå O åo p O π = I== Q ~ o é éê p O O − = = I==
CHAPTER 3. GEOMETRY 50 ïÜÉêÉ= O i é = K== = = = 3.19 Circle = o~ÇáìëW=o= aá~ãÉíÉêW=Ç= `ÜçêÇW=~= pÉÅ~åí=ëÉÖãÉåíëW=ÉI=Ñ= q~åÖÉåí=ëÉÖãÉåíW=Ö= `Éåíê~ä=~åÖäÉW=α = fåëÅêáÄÉÇ=~åÖäÉW=β = mÉêáãÉíÉêW=i= ^êÉ~W=p= = = 260. O ëáå o O ~ α = = = = = Figure 30. =
CHAPTER 3. GEOMETRY 51 261. O N O N Ä Ä ~ ~ = = = = = Figure 31. = 262. N N ÑÑ ÉÉ = = = ===== = = Figure 32. = 263. N O ÑÑ Ö = = =
CHAPTER 3. GEOMETRY 52 ===== = = Figure 33. = 264. O α = β = = = = Figure 34. = 265. Ç o O i π = π = = = 266. O io Q Ç o p O O = π = π = == =
CHAPTER 3. GEOMETRY 53 3.20 Sector of a Circle = o~Çáìë=çÑ=~=ÅáêÅäÉW=o= ^êÅ=äÉåÖíÜW=ë= `Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ= `Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW=α= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 35. = 267. oñ ë = = = 268. ° α π = NUM o ë = = 269. o O ë i + = = = 270. ° α π = = = PSM o O ñ o O oë p O O == = =
CHAPTER 3. GEOMETRY 54 3.21 Segment of a Circle = o~Çáìë=çÑ=~=ÅáêÅäÉW=o= ^êÅ=äÉåÖíÜW=ë= `ÜçêÇW=~= `Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ= `Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW=α= eÉáÖÜí=çÑ=íÜÉ=ëÉÖãÉåíW=Ü= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 36. = 271. O Ü Üo O O ~ − = = = 272. O O ~ o Q O N o Ü − − = I= o Ü < = = 273. ~ ë i + = = =
CHAPTER 3. GEOMETRY 55 274. ( ) [ ] ( ) ñ ëáå ñ O o ëáå NUM O o Ü o ~ ëo O N p O O − =       α − ° απ = − − = I== Ü~ P O p ≈ K= = = = 3.22 Cube = bÇÖÉW=~== aá~Öçå~äW=Ç= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ëéÜÉêÉW=ê= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = === = = Figure 37. = 275. P ~ Ç = = = 276. O ~ ê = = =
CHAPTER 3. GEOMETRY 56 277. O P ~ o = = = 278. O ~ S p = = = 279. P ~ s = == = = = 3.23 Rectangular Parallelepiped = bÇÖÉëW=~I=ÄI=Å== aá~Öçå~äW=Ç= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = ===== = = Figure 38. = 280. O O O Å Ä ~ Ç + + = = = 281. ( ) ÄÅ ~Å ~Ä O p + + = = = 282. ~ÄÅ s = ==
CHAPTER 3. GEOMETRY 57 3.24 Prism = i~íÉê~ä=ÉÇÖÉW=ä= eÉáÖÜíW=Ü= i~íÉê~ä=~êÉ~W= i p = ^êÉ~=çÑ=Ä~ëÉW= _ p = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = ===== = = Figure 39. = 283. _ i p O p p + = K== = 284. i~íÉê~ä=^êÉ~=çÑ=~=oáÖÜí=mêáëã= ( )ä ~ ~ ~ ~ p å P O N i + + + + = K = = 285. i~íÉê~ä=^êÉ~=çÑ=~å=lÄäáèìÉ=mêáëã= éä pi = I== ïÜÉêÉ=é=áë=íÜÉ=éÉêáãÉíÉê=çÑ=íÜÉ=Åêçëë=ëÉÅíáçåK= =
CHAPTER 3. GEOMETRY 58 286. Ü p s _ = = = 287. `~î~äáÉêáDë=mêáåÅáéäÉ== dáîÉå=íïç=ëçäáÇë=áåÅäìÇÉÇ=ÄÉíïÉÉå=é~ê~ääÉä=éä~åÉëK=fÑ=ÉîÉêó= éä~åÉ=Åêçëë=ëÉÅíáçå=é~ê~ääÉä=íç=íÜÉ=ÖáîÉå=éä~åÉë=Ü~ë=íÜÉ=ë~ãÉ= ~êÉ~=áå=ÄçíÜ=ëçäáÇëI=íÜÉå=íÜÉ=îçäìãÉë=çÑ=íÜÉ=ëçäáÇë=~êÉ=Éèì~äK= = = = 3.25 Regular Tetrahedron = qêá~åÖäÉ=ëáÇÉ=äÉåÖíÜW=~= eÉáÖÜíW=Ü= ^êÉ~=çÑ=Ä~ëÉW= _ p = pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 40. = 288. ~ P O Ü = = =
CHAPTER 3. GEOMETRY 59 289. Q ~ P p O _ = = = 290. O ~ P p = = = 291. O S ~ Ü p P N s P _ = = K== = = = 3.26 Regular Pyramid = páÇÉ=çÑ=Ä~ëÉW=~= i~íÉê~ä=ÉÇÖÉW=Ä= eÉáÖÜíW=Ü= pä~åí=ÜÉáÖÜíW=ã== kìãÄÉê=çÑ=ëáÇÉëW=å== pÉãáéÉêáãÉíÉê=çÑ=Ä~ëÉW=é= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉ=çÑ=Ä~ëÉW=ê= ^êÉ~=çÑ=Ä~ëÉW= _ p = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i p = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = =
CHAPTER 3. GEOMETRY 60 = = Figure 41. = 292. Q ~ Ä ã O O − = = = 293. å ëáå O ~ å ëáå Ä Q Ü O O O π − π = = = 294. éã ~ Ä Q å~ Q N å~ã O N p O O i = − = = = = 295. éê p_ = = = 296. i _ p p p + = = = 297. éêÜ P N Ü p P N s _ = = == = = =
CHAPTER 3. GEOMETRY 61 3.27 Frustum of a Regular Pyramid = _~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW=    å P O N å P O N Ä I I Ä I Ä I Ä ~ I I ~ I ~ I ~ K K = eÉáÖÜíW=Ü= pä~åí=ÜÉáÖÜíW=ã== ^êÉ~=çÑ=Ä~ëÉëW= N p I= O p = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i p = mÉêáãÉíÉê=çÑ=Ä~ëÉëW= N m I= O m = pÅ~äÉ=Ñ~ÅíçêW=â= qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 42. = 298. â ~ Ä ~ Ä ~ Ä ~ Ä ~ Ä å å P P O O N N = = = = = = K = =
CHAPTER 3. GEOMETRY 62 299. O N O â p p = = = 300. ( ) O m m ã p O N i + = = = 301. O N i p p p p + + = = = 302. ( ) O O N N p p p p P Ü s + + = = = 303. [ ] O N O N â â N P Üp ~ Ä ~ Ä N P Üp s + + =               + + = = = = = 3.28 Rectangular Right Wedge = páÇÉë=çÑ=Ä~ëÉW=~I=Ä= qçé=ÉÇÖÉW=Å= eÉáÖÜíW=Ü= i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i p = ^êÉ~=çÑ=Ä~ëÉW= _ p = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = =
CHAPTER 3. GEOMETRY 63 = = Figure 43. = 304. ( ) ( )O O O O i Å ~ Ü Ä Ä Ü Q Å ~ O N p − + + + + = = = 305. ~Ä p_ = = = 306. i _ p p p + = = = 307. ( ) Å ~ O S ÄÜ s + = = = = = 3.29 Platonic Solids = bÇÖÉW=~= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = =
CHAPTER 3. GEOMETRY 64 308. cáîÉ=mä~íçåáÅ=pçäáÇë= qÜÉ= éä~íçåáÅ= ëçäáÇë= ~êÉ= ÅçåîÉñ= éçäóÜÉÇê~= ïáíÜ= Éèìáî~äÉåí= Ñ~ÅÉë=ÅçãéçëÉÇ=çÑ=ÅçåÖêìÉåí=ÅçåîÉñ=êÉÖìä~ê=éçäóÖçåëK== = pçäáÇ= kìãÄÉê= çÑ=sÉêíáÅÉë kìãÄÉê= çÑ=bÇÖÉë= kìãÄÉê= çÑ=c~ÅÉë= pÉÅíáçå= qÉíê~ÜÉÇêçå== Q= S= Q= PKOR= `ìÄÉ= U= NO= S= PKOO= lÅí~ÜÉÇêçå= S= NO= U= PKOT= fÅçë~ÜÉÇêçå= NO= PM= OM= PKOT= açÇÉÅ~ÜÉÇêçå= OM= PM= NO= PKOT= = = Octahedron = = = Figure 44. = 309. S S ~ ê = = = 310. O O ~ o = = =
CHAPTER 3. GEOMETRY 65 311. P ~ O p O = = = 312. P O ~ s P = = = = Icosahedron = = = Figure 45. = 313. ( ) NO R P P ~ ê + = = = 314. ( ) R R O Q ~ o + = = = 315. P ~ R p O = = = 316. ( ) NO R P ~ R s P + = = = =
CHAPTER 3. GEOMETRY 66 Dodecahedron = = = Figure 46. = 317. ( ) O R NN OR NM ~ ê + = = = 318. ( ) Q R N P ~ o + = = = 319. ( ) R O R R ~ P p O + = = = 320. ( ) Q R T NR ~ s P + = = = = = 3.30 Right Circular Cylinder = o~Çáìë=çÑ=Ä~ëÉW=o= aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=
CHAPTER 3. GEOMETRY 67 eÉáÖÜíW=e= i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i p = ^êÉ~=çÑ=Ä~ëÉW= _ p = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = ===== = = Figure 47. = 321. oe O pi π = = = 322. ( )       + π = + π = + = O Ç e Ç o e o O p O p p _ i = = 323. e o e p s O _ π = = = = = =
CHAPTER 3. GEOMETRY 68 3.31 Right Circular Cylinder with an Oblique Plane Face = o~Çáìë=çÑ=Ä~ëÉW=o= qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= N Ü = qÜÉ=ëÜçêíÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= O Ü = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i p = ^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= _ p = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 48. = 324. ( ) O N i Ü Ü o p + π = = = 325. O O N O O _ O Ü Ü o o o p       − + π + π = = =
CHAPTER 3. GEOMETRY 69 326.               − + + + + π = + = O O N O O N _ i O Ü Ü o o Ü Ü o p p p = = 327. ( ) O N O Ü Ü O o s + π = = = = = 3.32 Right Circular Cone o~Çáìë=çÑ=Ä~ëÉW=o= aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç= eÉáÖÜíW=e= pä~åí=ÜÉáÖÜíW=ã= i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i p = ^êÉ~=çÑ=Ä~ëÉW= _ p = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 49.
CHAPTER 3. GEOMETRY 70 328. O O o ã e − = = = 329. O ãÇ oã pi π = π = = = 330. O _ o p π = = = 331. ( )       + π = + π = + = O Ç ã Ç O N o ã o p p p _ i = = 332. e o P N e p P N s O _ π = = = = = = 3.33 Frustum of a Right Circular Cone = o~Çáìë=çÑ=Ä~ëÉëW=oI=ê= eÉáÖÜíW=e= pä~åí=ÜÉáÖÜíW=ã= pÅ~äÉ=Ñ~ÅíçêW=â= ^êÉ~=çÑ=Ä~ëÉëW= N p I= O p = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i p = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = =
CHAPTER 3. GEOMETRY 71 = = Figure 50. = 333. ( )O O ê o ã e − − = = = 334. â ê o = = = 335. O O O N O â ê o p p = = = = 336. ( ) ê o ã pi + π = = = 337. ( ) [ ] ê o ã ê o p p p p O O i O N + + + π = + + = = = 338. ( ) O O N N p p p p P Ü s + + = = = 339. [ ] O N O N â â N P Üp ê o ê o N P Üp s + + =               + + = = = = =
CHAPTER 3. GEOMETRY 72 3.34 Sphere = o~ÇáìëW=o= aá~ãÉíÉêW=Ç= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = Figure 51. = 340. O o Q p π = = = 341. po P N Ç S N e o P Q s P P = π = π = = = = = 3.35 Spherical Cap o~Çáìë=çÑ=ëéÜÉêÉW=o= o~Çáìë=çÑ=Ä~ëÉW=ê= eÉáÖÜíW=Ü= ^êÉ~=çÑ=éä~åÉ=Ñ~ÅÉW= _ p = ^êÉ~=çÑ=ëéÜÉêáÅ~ä=Å~éW= ` p = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s=
CHAPTER 3. GEOMETRY 73 = = Figure 52. = 342. Ü O Ü ê o O O + = = = 343. O _ ê p π = = = 344. ( ) O O ` ê Ü p + π = = = 345. ( ) ( ) O O O ` _ ê oÜ O ê O Ü p p p + π = + π = + = = = 346. ( ) ( ) O O O Ü ê P Ü S Ü o P Ü S s + π = − π = = = = = 3.36 Spherical Sector = o~Çáìë=çÑ=ëéÜÉêÉW=o= o~Çáìë=çÑ=Ä~ëÉ=çÑ=ëéÜÉêáÅ~ä=Å~éW=ê= eÉáÖÜíW=Ü= qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= =
CHAPTER 3. GEOMETRY 74 ====== === = = Figure 53. = 347. ( ) ê Ü O o p + π = = = 348. Ü o P O s O π = = = kçíÉW=qÜÉ=ÖáîÉå=Ñçêãìä~ë=~êÉ=ÅçêêÉÅí=ÄçíÜ=Ñçê=±çéÉå≤=~åÇ= ±ÅäçëÉÇ≤=ëéÜÉêáÅ~ä=ëÉÅíçêK= = = = 3.37 Spherical Segment = o~Çáìë=çÑ=ëéÜÉêÉW=o= o~Çáìë=çÑ=Ä~ëÉëW= N ê I= O ê = eÉáÖÜíW=Ü= ^êÉ~=çÑ=ëéÜÉêáÅ~ä=ëìêÑ~ÅÉW= p p = ^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= N p I= O p = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= =
CHAPTER 3. GEOMETRY 75 ===== = = Figure 54. = 349. oÜ O pp π = = = 350. ( ) O O O N O N p ê ê oÜ O p p p p + + π = + + = = = 351. ( ) O O O O N Ü ê P ê P Ü S N s + + π = = = = = 3.38 Spherical Wedge = o~ÇáìëW=o= aáÜÉÇê~ä=~åÖäÉ=áå=ÇÉÖêÉÉëW=ñ= aáÜÉÇê~ä=~åÖäÉ=áå=ê~Çá~åëW=α= ^êÉ~=çÑ=ëéÜÉêáÅ~ä=äìåÉW= i p = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = =
CHAPTER 3. GEOMETRY 76 = = Figure 55. = 352. ñ o O VM o p O O i = α π = = = 353. ñ o O o VM o o p O O O O + π = α π + π = = = 354. ñ o P O OTM o s P P = α π = = = = = 3.39 Ellipsoid = pÉãá-~ñÉëW=~I=ÄI=Å= sçäìãÉW=s=
CHAPTER 3. GEOMETRY 77 ======= = = Figure 56. = 355. ~ÄÅ P Q s π = = = = = Prolate Spheroid = pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä ~ > F= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 356.       + π = É É ~êÅëáå ~ Ä Ä O p I== ïÜÉêÉ= ~ Ä ~ É O O − = K= = 357. ~ Ä P Q s O π = = =
CHAPTER 3. GEOMETRY 78 Oblate Spheroid = pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä ~ < F= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 358.                   + π = ~ L ÄÉ ~ ÄÉ ~êÅëáåÜ ~ Ä Ä O p I== ïÜÉêÉ= Ä ~ Ä É O O − = K= = 359. ~ Ä P Q s O π = = = = = 3.40 Circular Torus = j~àçê=ê~ÇáìëW=o= jáåçê=ê~ÇáìëW=ê= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= =
CHAPTER 3. GEOMETRY 79 == = Picture 57. = 360. oê Q p O π = = = 361. O O oê O s π = = = =
80 Chapter 4 Trigonometry = = = = ^åÖäÉëW=α I=β = oÉ~ä=åìãÄÉêë=EÅççêÇáå~íÉë=çÑ=~=éçáåíFW=ñI=ó== tÜçäÉ=åìãÄÉêW=â= = = 4.1 Radian and Degree Measures of Angles = 362. ? QR D NT RT NUM ê~Ç N ° ≈ π ° = = = 363. ê~Ç MNTQRP K M ê~Ç NUM N ≈ π = ° = = 364. ê~Ç MMMOVN K M ê~Ç SM NUM D N ≈ ⋅ π = = = 365. ê~Ç MMMMMR K M ê~Ç PSMM NUM ? N ≈ ⋅ π = = = 366. = = ^åÖäÉ= EÇÉÖêÉÉëF= M= PM= QR= SM= VM= NUM= OTM= PSM= ^åÖäÉ= Eê~Çá~åëF= M= S π = Q π = P π = O π = π= O Pπ = π O = = = =
CHAPTER 4. TRIGONOMETRY 81 4.2 Definitions and Graphs of Trigonometric Functions = = = = Figure 58. = 367. ê ó ëáå = α = = 368. ê ñ Åçë = α = = 369. ñ ó í~å = α = = 370. ó ñ Åçí = α = =
CHAPTER 4. TRIGONOMETRY 82 371. ñ ê ëÉÅ = α = = 372. ó ê ÅçëÉÅ = α = = 373. páåÉ=cìåÅíáçå= ñ ëáå ó = I= N ñ ëáå N ≤ ≤ − K= = = Figure 59. = 374. `çëáåÉ=cìåÅíáçå== ñ Åçë ó = I= N ñ Åçë N ≤ ≤ − K=
CHAPTER 4. TRIGONOMETRY 83 = = Figure 60. = 375. q~åÖÉåí=cìåÅíáçå= ñ í~å ó = I= ( ) O N â O ñ π + ≠ I= K ñ í~å ∞ ≤ ≤ ∞ − = = = = Figure 61. =
CHAPTER 4. TRIGONOMETRY 84 376. `çí~åÖÉåí=cìåÅíáçå== ñ Åçí ó = I= π ≠ â ñ I== ∞ ≤ ≤ ∞ − ñ Åçí K= = = = Figure 62. = 377. pÉÅ~åí=cìåÅíáçå= ñ ëÉÅ ó = I= ( ) O N â O ñ π + ≠ K= ==
CHAPTER 4. TRIGONOMETRY 85 = = Figure 63. = 378. `çëÉÅ~åí=cìåÅíáçå== ñ ÉÅ Åçë ó = I= π ≠ â ñ K= = Figure 64.
CHAPTER 4. TRIGONOMETRY 86 4.3. Signs of Trigonometric Functions 379. = = nì~Çê~åí= páå α = `çë α = q~å α = `çí α = pÉÅ α = `çëÉÅ= α = f= H= H= H= H= H= H= ff= H= = = = = H= fff= = = H= H= = = fs= = H= = = H= = = = = 380. = = = Figure 65. = = = = = = = = = =
CHAPTER 4. TRIGONOMETRY 87 4.4 Trigonometric Functions of Common Angles 381. = ° α = ê~Ç α = α ëáå = α Åçë = α í~å = α Åçí α ëÉÅ = α ÅçëÉÅ = M= M= M= N= M= ∞= N= ∞= PM= S π = O N = O P = P N = P = P O = O= QR= Q π = O O = O O = N= N= O = O = SM= P π = O P = O N = P = P N = O= P O = VM= O π = N= M= ∞ = M= ∞ = N= NOM= P Oπ = O P = O N − = P − = P N − O − = P O = NUM= π= M= N − = M= ∞ = N − = ∞ = OTM= O Pπ = N − = M= ∞= M= ∞= N − = PSM= π O = M= N= M= ∞ = N= ∞ = = = = = = = = = = = = = =
CHAPTER 4. TRIGONOMETRY 88 382. = ° α = ê~Ç α = α ëáå = α Åçë = α í~å = α Åçí = NR= NO π = Q O S − = Q O S + = P O− = P O+ = NU= NM π = Q N R − = Q R O NM + R R O R− = R O R+ = PS= R π = Q R O NM − Q N R + = N R R O NM + − R O NM N R − + = RQ= NM Pπ = Q N R + = Q R O NM − R O NM N R − + N R R O NM + − = TO= R Oπ = Q R O NM + Q N R − = R O R+ = R R O R− = TR= NO Rπ = Q O S + = Q O S − = P O+ = P O− = = = = 4.5 Most Important Formulas = 383. N Åçë ëáå O O = α + α = = 384. N í~å ëÉÅ O O = α − α = = 385. N Åçí ÅëÅ O O = α − α = = 386. α α = α Åçë ëáå í~å =
CHAPTER 4. TRIGONOMETRY 89 387. α α = α ëáå Åçë Åçí = = 388. N Åçí í~å = α ⋅ α = = 389. α = α Åçë N ëÉÅ = = 390. α = α ëáå N ÅçëÉÅ = = = = 4.6 Reduction Formulas = 391. = = β = β ëáå = β Åçë = β í~å = β Åçí = α − = α − ëáå = α + Åçë = α − í~å = α − Åçí = α − ° VM = α + Åçë = α + ëáå = α + Åçí = α + í~å = α + ° VM = α + Åçë = α − ëáå = α − Åçí = α − í~å = α − ° NUM α + ëáå = α − Åçë = α − í~å = α − Åçí = α + ° NUM α − ëáå = α − Åçë = α + í~å = α + Åçí = α − ° OTM α − Åçë = α − ëáå = α + Åçí = α + í~å = α + ° OTM α − Åçë = α + ëáå = α − Åçí = α − í~å = α − ° PSM α − ëáå = α + Åçë = α − í~å = α − Åçí = α + ° PSM α + ëáå = α + Åçë = α + í~å = α + Åçí = = = = = = =
CHAPTER 4. TRIGONOMETRY 90 4.7 Periodicity of Trigonometric Functions = 392. ( ) α = π ± α ëáå å O ëáå I=éÉêáçÇ= π O =çê= ° PSM K= = 393. ( ) α = π ± α Åçë å O Åçë I=éÉêáçÇ= π O =çê= ° PSM K= = 394. ( ) α = π ± α í~å å í~å I=éÉêáçÇ=π=çê= ° NUM K= = 395. ( ) α = π ± α Åçí å Åçí I=éÉêáçÇ=π=çê= ° NUM K= = = = 4.8 Relations between Trigonometric Functions = 396. ( ) N Q O Åçë O O Åçë N O N Åçë N ëáå O O −       π − α = α − ± = α − ± = α = = O í~å N O í~å O O α + α = = = 397. ( ) N O Åçë O O Åçë N O N ëáå N Åçë O O − α = α + ± = α − ± = α = = O í~å N O í~å N O O α + α − = = = 398. α α − = α + α = − α ± = α α = α O ëáå O Åçë N O Åçë N O ëáå N ëÉÅ Åçë ëáå í~å O =
CHAPTER 4. TRIGONOMETRY 91 = O í~å N O í~å O O Åçë N O Åçë N O α + α = α + α − ± = = = 399. α − α = α α + = − α ± = α α = α O Åçë N O ëáå O ëáå O Åçë N N ÅëÅ ëáå Åçë Åçí O = = O í~å O O í~å N O Åçë N O Åçë N O α α − = α − α + ± = = = 400. O í~å N O í~å N í~å N Åçë N ëÉÅ O O O α − α + = α + ± = α = α = = 401. O í~å O O í~å N Åçí N ëáå N ÅëÅ O O α α + = α + ± = α = α = = = = 4.9 Addition and Subtraction Formulas = 402. ( ) α β + β α = β + α Åçë ëáå Åçë ëáå ëáå = = 403. ( ) α β − β α = − α Åçë ëáå Åçë ëáå ó ëáå = = 404. ( ) β α − β α = β + α ëáå ëáå Åçë Åçë Åçë = = 405. ( ) β α + β α = β − α ëáå ëáå Åçë Åçë Åçë =
CHAPTER 4. TRIGONOMETRY 92 406. ( ) β α − β + α = β + α í~å í~å N í~å í~å í~å = = 407. ( ) β α + β − α = β − α í~å í~å N í~å í~å í~å = = 408. ( ) β + α β α − = β + α í~å í~å í~å í~å N Åçí = = 409. ( ) β − α β α + = β − α í~å í~å í~å í~å N Åçí = = = = 4.10 Double Angle Formulas = 410. α ⋅ α = α Åçë ëáå O O ëáå = = 411. N Åçë O ëáå O N ëáå Åçë O Åçë O O O O − α = α − = α − α = α = = 412. α − α = α − α = α í~å Åçí O í~å N í~å O O í~å O = = 413. O í~å Åçí Åçí O N Åçí O Åçí O α − α = α − α = α = = = = = = =
CHAPTER 4. TRIGONOMETRY 93 4.11 Multiple Angle Formulas = 414. α − α ⋅ α = α − α = α P O P ëáå ëáå Åçë P ëáå Q ëáå P P ëáå = = 415. α ⋅ α − α ⋅ α = α Åçë ëáå U Åçë ëáå Q Q ëáå P = = 416. α + α − α = α R P ëáå NS ëáå OM ëáå R R ëáå = = 417. α ⋅ α − α = α − α = α O P P ëáå Åçë P Åçë Åçë P Åçë Q P Åçë = = 418. N Åçë U Åçë U Q Åçë O Q + α − α = α = = 419. α + α − α = α Åçë R Åçë OM Åçë NS R Åçë P R = = 420. α − α − α = α O P í~å P N í~å í~å P P í~å = = 421. α + α − α − α = α Q O P í~å í~å S N í~å Q í~å Q Q í~å = = 422. α + α − α + α − α = α Q O P R í~å R í~å NM N í~å R í~å NM í~å R í~å = = 423. N Åçí P Åçí P Åçí P Åçí O P − α α − α = α = = 424. α − α α + α − = α P Q O í~å Q í~å Q í~å í~å S N Q Åçí == =
CHAPTER 4. TRIGONOMETRY 94 425. α + α − α α + α − = α í~å R í~å NM í~å í~å R í~å NM N R Åçí P R Q O = = = = 4.12 Half Angle Formulas = 426. O Åçë N O ëáå α − ± = α = = 427. O Åçë N O Åçë α + ± = α = = 428. α − α = α α − = α + α = α + α − ± = α Åçí ÅëÅ ëáå Åçë N Åçë N ëáå Åçë N Åçë N O í~å = = 429. α + α = α α + = α − α = α − α + ± = α Åçí ÅëÅ ëáå Åçë N Åçë N ëáå Åçë N Åçë N O Åçí = = = = 4.13 Half Angle Tangent Identities = 430. O í~å N O í~å O ëáå O α + α = α = =
CHAPTER 4. TRIGONOMETRY 95 431. O í~å N O í~å N Åçë O O α + α − = α = = 432. O í~å N O í~å O í~å O α − α = α = = 433. O í~å O O í~å N Åçí O α α − = α = = = = 4.14 Transforming of Trigonometric Expressions to Product = 434. O Åçë O ëáå O ëáå ëáå β − α β + α = β + α = = 435. O ëáå O Åçë O ëáå ëáå β − α β + α = β − α = = 436. O Åçë O Åçë O Åçë Åçë β − α β + α = β + α = = 437. O ëáå O ëáå O Åçë Åçë β − α β + α − = β − α = =
CHAPTER 4. TRIGONOMETRY 96 438. ( ) β ⋅ α β + α = β + α Åçë Åçë ëáå í~å í~å = = 439. ( ) β ⋅ α β − α = β − α Åçë Åçë ëáå í~å í~å = = 440. ( ) β ⋅ α α + β = β + α ëáå ëáå ëáå Åçí Åçí = = 441. ( ) β ⋅ α α − β = β − α ëáå ëáå ëáå Åçí Åçí = = 442.       α + π =       α − π = α + α Q ëáå O Q Åçë O ëáå Åçë = = 443.       α + π =       α − π = α − α Q Åçë O Q ëáå O ëáå Åçë = = 444. ( ) β ⋅ α β − α = β + α ëáå Åçë Åçë Åçí í~å = = 445. ( ) β ⋅ α β + α − = β − α ëáå Åçë Åçë Åçí í~å = = 446. O Åçë O Åçë N O α = α + = = 447. O ëáå O Åçë N O α = α − = =
CHAPTER 4. TRIGONOMETRY 97 448.       α − π = α + O Q Åçë O ëáå N O = = 449.       α − π = α − O Q ëáå O ëáå N O = = = = 4.15 Transforming of Trigonometric Expressions to Sum = 450. ( ) ( ) O Åçë Åçë ëáå ëáå β + α − β − α = β ⋅ α = = 451. ( ) ( ) O Åçë Åçë Åçë Åçë β + α + β − α = β ⋅ α = = 452. ( ) ( ) O ëáå ëáå Åçë ëáå β + α + β − α = β ⋅ α = = 453. β + α β + α = β ⋅ α Åçí Åçí í~å í~å í~å í~å = = 454. β + α β + α = β ⋅ α í~å í~å Åçí Åçí Åçí Åçí = = 455. β + α β + α = β ⋅ α í~å Åçí Åçí í~å Åçí í~å = = = =
CHAPTER 4. TRIGONOMETRY 98 4.16 Powers of Trigonometric Functions = 456. O O Åçë N ëáåO α − = α = = 457. Q P ëáå ëáå P ëáåP α − α = α = = 458. U P O Åçë Q Q Åçë ëáåQ + α − α = α = = 459. NS R ëáå P ëáå R ëáå NM ëáåR α + α − α = α = = 460. PO S Åçë Q Åçë S O Åçë NR NM ëáåS α − α + α − = α = = 461. O O Åçë N ÅçëO α + = α = = 462. Q P Åçë Åçë P ÅçëP α + α = α = = 463. U P O Åçë Q Q Åçë ÅçëQ + α + α = α = = 464. NS R Åçë P ëáå R Åçë NM ÅçëR α + α + α = α = = 465. PO S Åçë Q Åçë S O Åçë NR NM ÅçëS α + α + α + = α = =
CHAPTER 4. TRIGONOMETRY 99 4.17 Graphs of Inverse Trigonometric Functions = 466. fåîÉêëÉ=páåÉ=cìåÅíáçå== ñ ~êÅëáå ó = I= N ñ N ≤ ≤ − I= O ñ ~êÅëáå O π ≤ ≤ π − K= = = = Figure 66. = 467. fåîÉêëÉ=`çëáåÉ=cìåÅíáçå== ñ ~êÅÅçë ó = I= N ñ N ≤ ≤ − I= π ≤ ≤ ñ ~êÅÅçë M K= =
CHAPTER 4. TRIGONOMETRY 100 = = Figure 67. = 468. fåîÉêëÉ=q~åÖÉåí=cìåÅíáçå== ñ ~êÅí~å ó = I= ∞ ≤ ≤ ∞ − ñ I= O ñ ~êÅí~å O π < < π − K= = ===== = = Figure 68.
CHAPTER 4. TRIGONOMETRY 101 469. fåîÉêëÉ=`çí~åÖÉåí=cìåÅíáçå== ñ Åçí ~êÅ ó = I= ∞ ≤ ≤ ∞ − ñ I= π < < ñ Åçí ~êÅ M K= ===== = Figure 69. = 470. fåîÉêëÉ=pÉÅ~åí=cìåÅíáçå== ( ] [ ) K I O O I M ñ ëÉÅ ~êÅ I I N N I ñ I ñ = ~êÅëÉÅ ó       π π ∪       π ∈ ∞ ∪ − ∞ − ∈ = = Figure 70.
CHAPTER 4. TRIGONOMETRY 102 471. fåîÉêëÉ=`çëÉÅ~åí=cìåÅíáçå== ( ] [ ) K O I M M I O ñ ÅëÅ ~êÅ I I N N I ñ I ñ ~êÅÅëÅ ó       π ∪       π − ∈ ∞ ∪ − ∞ − ∈ = = = Figure 71. = = 4.18 Principal Values of Inverse Trigonometric Functions 472. ñ = M= O N = O O = O P N= ñ ~êÅëáå = ° M = ° PM = ° QR = ° SM ° VM ñ ~êÅÅçë = ° VM ° SM = ° QR = ° PM ° M = ñ = O N − O O − O P − N − = = ñ ~êÅëáå = ° −PM = ° − QR ° − SM ° − VM = = ñ ~êÅÅçë = ° NOM = ° NPR = ° NRM = ° NUM = =
CHAPTER 4. TRIGONOMETRY 103 473. ñ = M= P P N= P = P P − N − = P − = ñ ~êÅí~å = ° M = ° PM ° QR ° SM ° −PM ° − QR = ° − SM = ñ Åçí ~êÅ = ° VM ° SM ° QR ° PM ° NOM = ° NPR = ° NRM = = = = 4.19 Relations between Inverse Trigonometric Functions = 474. ( ) ñ ~êÅëáå ñ ~êÅëáå − = − = = 475. ñ ~êÅÅçë O ñ ~êÅëáå − π = = = 476. O ñ N ~êÅÅçë ñ ~êÅëáå − = I= N ñ M ≤ ≤ K= = 477. O ñ N ~êÅÅçë ñ ~êÅëáå − − = I= M ñ N ≤ ≤ − K= = 478. O ñ N ñ ~êÅí~å ñ ~êÅëáå − = I= N ñO < K= = 479. ñ ñ N Åçí ~êÅ ñ ~êÅëáå O − = I= N ñ M ≤ < K= = 480. π − − = ñ ñ N Åçí ~êÅ ñ ~êÅëáå O I= M ñ N < ≤ − K= = 481. ( ) ñ ~êÅÅçë ñ ~êÅÅçë − π = − =
CHAPTER 4. TRIGONOMETRY 104 482. ñ ~êÅëáå O ñ ~êÅÅçë − π = = = 483. O ñ N ~êÅëáå ñ ~êÅÅçë − = I= N ñ M ≤ ≤ K= = 484. O ñ N ~êÅëáå ñ ~êÅÅçë − − π = I= M ñ N ≤ ≤ − K= = 485. ñ ñ N ~êÅí~å ñ ~êÅÅçë O − = I= N ñ M ≤ < K= = 486. ñ ñ N ~êÅí~å ñ ~êÅÅçë O − + π = I= M ñ N < ≤ − K= = 487. O ñ N ñ Åçí ~êÅ ñ ~êÅÅçë − = I= N ñ N ≤ ≤ − K= = 488. ( ) ñ ~êÅí~å ñ ~êÅí~å − = − = = 489. ñ Åçí ~êÅ O ñ ~êÅí~å − π = = = 490. O ñ N ñ ~êÅëáå ñ ~êÅí~å + = = = 491. O ñ N N ~êÅÅçë ñ ~êÅí~å + = I= M ñ ≥ K= = 492. O ñ N N ~êÅÅçë ñ ~êÅí~å + − = I= M ñ ≤ K= =
CHAPTER 4. TRIGONOMETRY 105 493. ñ N ~êÅí~å O ñ ~êÅí~å − π = I= M ñ > K= = 494. ñ N ~êÅí~å O ñ ~êÅí~å − π − = I= M ñ < K= = 495. ñ N Åçí ~êÅ ñ ~êÅí~å = I= M ñ > K= = 496. π − = ñ N Åçí ~êÅ ñ ~êÅí~å I= M ñ < K= = 497. ( ) ñ Åçí ~êÅ ñ Åçí ~êÅ − π = − = = 498. ñ ~êÅí~å O ñ Åçí ~êÅ − π = = = 499. O ñ N N ~êÅëáå ñ Åçí ~êÅ + = I= M ñ > K= = 500. O ñ N N ~êÅëáå ñ Åçí ~êÅ + − π = I= M ñ < K= = 501. O ñ N ñ ~êÅÅçë ñ Åçí ~êÅ + = = = 502. ñ N ~êÅí~å ñ Åçí ~êÅ = I= M ñ > K= = 503. ñ N ~êÅí~å ñ Åçí ~êÅ + π = I= M ñ < K= = =
CHAPTER 4. TRIGONOMETRY 106 4.20 Trigonometric Equations = tÜçäÉ=åìãÄÉêW=å= = = 504. ~ ñ ëáå = I= ( ) å ~ ~êÅëáå N ñ å π + − = = = 505. ~ ñ Åçë = I= å O ~ ~êÅÅçë ñ π + ± = = = 506. ~ ñ í~å = I= å ~ ~êÅí~å ñ π + = = = 507. ~ ñ Åçí = I= å ~ Åçí ~êÅ ñ π + = = = = = 4.21 Relations to Hyperbolic Functions = fã~Öáå~êó=ìåáíW=á= = = 508. ( ) ñ ëáåÜ á áñ ëáå = = = 509. ( ) ñ í~åÜ á áñ í~å = = = 510. ( ) ñ ÅçíÜ á áñ Åçí − = = = 511. ( ) ñ ëÉÅÜ áñ ëÉÅ = = = 512. ( ) ñ ÅëÅÜ á áñ ÅëÅ − = = = = =
107 Chapter 5 Matrices and Determinants = = = = j~íêáÅÉëW=Î=_I=`= bäÉãÉåíë=çÑ=~=ã~íêáñW= á ~ I= á Ä I= áà ~ I= áà Ä I= áà Å = aÉíÉêãáå~åí=çÑ=~=ã~íêáñW= ^ ÇÉí = jáåçê=çÑ=~å=ÉäÉãÉåí= áà ~ W= áà j = `çÑ~Åíçê=çÑ=~å=ÉäÉãÉåí= áà ~ W= áà ` = qê~åëéçëÉ=çÑ=~=ã~íêáñW= q ^ I= ^ ú = ^Çàçáåí=çÑ=~=ã~íêáñW= ^ ~Çà = qê~ÅÉ=çÑ=~=ã~íêáñW= ^ íê = fåîÉêëÉ=çÑ=~=ã~íêáñW= N ^− = oÉ~ä=åìãÄÉêW=â= oÉ~ä=î~êá~ÄäÉëW= á ñ = k~íìê~ä=åìãÄÉêëW=ãI=å=== = = 5.1 Determinants = 513. pÉÅçåÇ=lêÇÉê=aÉíÉêãáå~åí= N O O N O O N N Ä ~ Ä ~ Ä ~ Ä ~ ^ ÇÉí − = = = = = = = =
CHAPTER 5. MATRICES AND DETERMINANTS 108 514. qÜáêÇ=lêÇÉê=aÉíÉêãáå~åí= − + + = = PO ON NP PN OP NO PP OO NN PP PO PN OP OO ON NP NO NN ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ^ ÇÉí = PN OO NP PP ON NO PO OP NN ~ ~ ~ ~ ~ ~ ~ ~ ~ − − − = = 515. p~êêìë=oìäÉ=E^êêçï=oìäÉF= = = Figure 72. = 516. k-íÜ=lêÇÉê=aÉíÉêãáå~åí= åå åà O å N å áå áà O á N á å O à O OO ON å N à N NO NN ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ^ ÇÉí K K K K K K K K K K K K K K K K K K K K = = = 517. jáåçê= qÜÉ=ãáåçê= áà j =~ëëçÅá~íÉÇ=ïáíÜ=íÜÉ=ÉäÉãÉåí= áà ~ =çÑ=å-íÜ=çêÇÉê= ã~íêáñ=^=áë=íÜÉ= ( ) N å − -íÜ=çêÇÉê=ÇÉíÉêãáå~åí=ÇÉêáîÉÇ=Ñêçã= íÜÉ=ã~íêáñ=^=Äó=ÇÉäÉíáçå=çÑ=áíë=á-íÜ=êçï=~åÇ=à-íÜ=ÅçäìãåK=== =
CHAPTER 5. MATRICES AND DETERMINANTS 109 518. `çÑ~Åíçê= ( ) áà à á áà j N ` + − = = = 519. i~éä~ÅÉ=bñé~åëáçå=çÑ=å-íÜ=lêÇÉê=aÉíÉêãáå~åí= i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=á-íÜ=êçï= ∑ = = å N à áà áà` ~ ^ ÇÉí I= å I I O I N á K = K= i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=à-íÜ=Åçäìãå= ∑ = = å N á áà áà` ~ ^ ÇÉí I= å I I O I N à K = K== = = = 5.2 Properties of Determinants = 520. qÜÉ==î~äìÉ==çÑ=~=ÇÉíÉêãáå~åí=êÉã~áåë==ìåÅÜ~åÖÉÇ=áÑ=êçïë=~êÉ= ÅÜ~åÖÉÇ=íç=Åçäìãåë=~åÇ=Åçäìãåë=íç=êçïëK= = O O N N O N O N Ä ~ Ä ~ Ä Ä ~ ~ = == = 521. fÑ=íïç==êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áåíÉêÅÜ~åÖÉÇI=íÜÉ=ëáÖå=çÑ= íÜÉ=ÇÉíÉêãáå~åí=áë=ÅÜ~åÖÉÇK= N N O O O O N N Ä ~ Ä ~ Ä ~ Ä ~ − = = = 522. fÑ=íïç=êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áÇÉåíáÅ~äI=íÜÉ=î~äìÉ=çÑ=íÜÉ= ÇÉíÉêãáå~åí=áë=òÉêçK= M ~ ~ ~ ~ O O N N = = =
CHAPTER 5. MATRICES AND DETERMINANTS 110 523. fÑ==íÜÉ===ÉäÉãÉåíë==çÑ==~åó=êçï==Eçê=ÅçäìãåF=~êÉ=ãìäíáéäáÉÇ=Äó===== ~==Åçããçå==Ñ~ÅíçêI==íÜÉ==ÇÉíÉêãáå~åí==áë==ãìäíáéäáÉÇ==Äó==íÜ~í= Ñ~ÅíçêK= O O N N O O N N Ä ~ Ä ~ â Ä ~ âÄ â~ = = = 524. fÑ==íÜÉ==ÉäÉãÉåíë==çÑ==~åó==êçï==Eçê==ÅçäìãåF=~êÉ=áåÅêÉ~ëÉÇ=Eçê= ÇÉÅêÉ~ëÉÇFÄó=Éèì~ä=ãìäíáéäÉë=çÑ=íÜÉ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë= çÑ=~åó=çíÜÉê=êçï==Eçê=ÅçäìãåFI==íÜÉ=î~äìÉ=çÑ=íÜÉ=ÇÉíÉêãáå~åí= áë=ìåÅÜ~åÖÉÇK= O O N N O O O N N N Ä ~ Ä ~ Ä âÄ ~ Ä âÄ ~ = + + = = = = 5.3 Matrices = 525. aÉÑáåáíáçå= ^å= å ã× =ã~íêáñ=^=áë=~=êÉÅí~åÖìä~ê=~êê~ó=çÑ=ÉäÉãÉåíë=Eåìã- ÄÉêë=çê=ÑìåÅíáçåëF=ïáíÜ=ã=êçïë=~åÇ=å=ÅçäìãåëK== [ ]             = = ãå O ã N ã å O OO ON å N NO NN áà ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ^ K M M M K K == = 526. pèì~êÉ=ã~íêáñ=áë=~=ã~íêáñ=çÑ=çêÇÉê= å å× K== = 527. ^=ëèì~êÉ=ã~íêáñ==[ ] áà ~ ==áë==ëóããÉíêáÅ==áÑ== àá áà ~ ~ = I==áKÉK==áí==áë= ëóããÉíêáÅ=~Äçìí=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK== = 528. ^=ëèì~êÉ=ã~íêáñ=[ ] áà ~ =áë=ëâÉï-ëóããÉíêáÅ=áÑ= àá áà ~ ~ − = K== =
CHAPTER 5. MATRICES AND DETERMINANTS 111 529. aá~Öçå~ä=ã~íêáñ==áë==~=ëèì~êÉ==ã~íêáñ=ïáíÜ=~ää==ÉäÉãÉåíë==òÉêç= ÉñÅÉéí=íÜçëÉ=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK== = 530. råáí=ã~íêáñ==áë==~=Çá~Öçå~ä==ã~íêáñ==áå=ïÜáÅÜ=íÜÉ=ÉäÉãÉåíë=çå= íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~ä=~êÉ=~ää=ìåáíóK=qÜÉ=ìåáí=ã~íêáñ=áë=========== ÇÉåçíÉÇ=Äó=fK== = 531. ^=åìää=ã~íêáñ=áë=çåÉ=ïÜçëÉ=ÉäÉãÉåíë=~êÉ=~ää=òÉêçK= = = = 5.4 Operations with Matrices = 532. qïç=ã~íêáÅÉë=^=~åÇ=_=~êÉ=Éèì~ä=áÑI=~åÇ=çåäó=áÑI=íÜÉó=~êÉ=ÄçíÜ= çÑ==íÜÉ==ë~ãÉ==ëÜ~éÉ== å ã× ==~åÇ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=~êÉ= Éèì~äK= = 533. qïç=ã~íêáÅÉë==^=~åÇ=_==Å~å=ÄÉ=~ÇÇÉÇ=Eçê=ëìÄíê~ÅíÉÇF=çÑI=~åÇ= çåäó=áÑI=íÜÉó=Ü~îÉ=íÜÉ=ë~ãÉ=ëÜ~éÉ= å ã× K=fÑ== [ ]             = = ãå O ã N ã å O OO ON å N NO NN áà ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ^ K M M M K K I== [ ]             = = ãå O ã N ã å O OO ON å N NO NN áà Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä _ K M M M K K I== = = = = =
CHAPTER 5. MATRICES AND DETERMINANTS 112 íÜÉå==             + + + + + + + + + = + ãå ãå O ã O ã N ã N ã å O å O OO OO ON ON å N å N NO NO NN NN Ä ~ Ä ~ Ä ~ Ä ~ Ä ~ Ä ~ Ä ~ Ä ~ Ä ~ _ ^ K M M M K K K= = 534. fÑ=â=áë=~=ëÅ~ä~êI=~åÇ= [ ] áà ~ ^ = =áë=~=ã~íêáñI=íÜÉå= [ ]             = = ãå O ã N ã å O OO ON å N NO NN áà â~ â~ â~ â~ â~ â~ â~ â~ â~ â~ â^ K M M M K K K= = 535. jìäíáéäáÅ~íáçå=çÑ=qïç=j~íêáÅÉë= qïç= ã~íêáÅÉë= Å~å= ÄÉ= ãìäíáéäáÉÇ= íçÖÉíÜÉê= çåäó= ïÜÉå= íÜÉ= åìãÄÉê=çÑ=Åçäìãåë=áå=íÜÉ=Ñáêëí=áë=Éèì~ä=íç=íÜÉ=åìãÄÉê=çÑ= êçïë=áå=íÜÉ=ëÉÅçåÇK== = fÑ= [ ]             = = ãå O ã N ã å O OO ON å N NO NN áà ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ^ K M M M K K I== [ ]             = = åâ O å N å â O OO ON â N NO NN áà Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä _ K M M M K K I= = = = = =
CHAPTER 5. MATRICES AND DETERMINANTS 113 íÜÉå==             = = ãâ O ã N ã â O OO ON â N NO NN Å Å Ä Å Å Å Å Å Å ` ^_ K M M M K K I== ïÜÉêÉ== ∑ = λ λ λ = + + + = å N à á åà áå à O O á à N N á áà Ä ~ Ä ~ Ä ~ Ä ~ Å K = E ã I I O I N á K = X â I I O I N à K = FK== = qÜìë=áÑ= [ ]       = = OP OO ON NP NO NN áà ~ ~ ~ ~ ~ ~ ~ ^ I= [ ]           = = P O N á Ä Ä Ä Ä _ I== íÜÉå==       =           ⋅       = P OP O OO N ON P NP O NO N NN P O N OP OO ON NP NO NN Ä ~ Ä ~ Ä ~ Ä ~ Ä ~ Ä ~ Ä Ä Ä ~ ~ ~ ~ ~ ~ ^_ K== = 536. qê~åëéçëÉ=çÑ=~=j~íêáñ= fÑ=íÜÉ=êçïë=~åÇ=Åçäìãåë=çÑ=~=ã~íêáñ=~êÉ=áåíÉêÅÜ~åÖÉÇI=íÜÉå= íÜÉ=åÉï=ã~íêáñ=áë=Å~ääÉÇ=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=çêáÖáå~ä=ã~íêáñK=== fÑ=^=áë=íÜÉ=çêáÖáå~ä=ã~íêáñI=áíë=íê~åëéçëÉ=áë=ÇÉåçíÉÇ= q ^ =çê= ^ ú K== = 537. qÜÉ=ã~íêáñ=^=áë=çêíÜçÖçå~ä=áÑ= f ^^q = K== = 538. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå== ( ) q q q ^ _ ^_ = K= = =
CHAPTER 5. MATRICES AND DETERMINANTS 114 539. ^Çàçáåí=çÑ=j~íêáñ= fÑ=^=áë=~=ëèì~êÉ= å å× ã~íêáñI=áíë=~ÇàçáåíI=ÇÉåçíÉÇ=Äó= ^ ~Çà I= áë=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=ã~íêáñ=çÑ=ÅçÑ~Åíçêë= áà ` =çÑ=^W= [ ]q áà ` ^ ~Çà = K== = 540. qê~ÅÉ=çÑ=~=j~íêáñ= fÑ=^=áë=~=ëèì~êÉ= å å× ã~íêáñI=áíë=íê~ÅÉI=ÇÉåçíÉÇ=Äó= ^ íê I=áë= ÇÉÑáåÉÇ=íç=ÄÉ==íÜÉ=ëìã=çÑ==íÜÉ=íÉêãë=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äW= åå OO NN ~ ~ ~ ^ íê + + + = K K= = 541. fåîÉêëÉ=çÑ=~=j~íêáñ= fÑ=^=áë=~=ëèì~êÉ= å å× ã~íêáñ=ïáíÜ=~=åçåëáåÖìä~ê=ÇÉíÉêãáå~åí= ^ ÇÉí I=íÜÉå=áíë=áåîÉêëÉ= N ^− =áë=ÖáîÉå=Äó= ^ ÇÉí ^ ~Çà ^ N = − K= = 542. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå== ( ) N N N ^ _ ^_ − − − = K= = 543. fÑ==^==áë=~=ëèì~êÉ=== å å× ==ã~íêáñI==íÜÉ==ÉáÖÉåîÉÅíçêë==u===ë~íáëÑó= íÜÉ=Éèì~íáçå= u û λ = I== ïÜáäÉ=íÜÉ=ÉáÖÉåî~äìÉë=λ =ë~íáëÑó=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå= M f ^ = λ − K=== = = = 5.5 Systems of Linear Equations = = s~êá~ÄäÉëW=ñI=óI=òI= N ñ I= K I ñO = oÉ~ä=åìãÄÉêëW= K I ~ I ~ I Ä I ~ I ~ I ~ NO NN N P O N =
CHAPTER 5. MATRICES AND DETERMINANTS 115 aÉíÉêãáå~åíëW=aI= ñ a I= ó a I= ò a == j~íêáÅÉëW=Î=_I=u= = = 544.    = + = + O O O N N N Ç ó Ä ñ ~ Ç ó Ä ñ ~ I== a a ñ ñ = I= a a ó ó = =E`ê~ãÉê∞ë=êìäÉFI== ïÜÉêÉ== N O O N O O N N Ä ~ Ä ~ Ä ~ Ä ~ a − = = I== N O O N O O N N ñ Ä Ç Ä Ç Ä Ç Ä Ç a − = = I== N O O N O O N N ó Ç ~ Ç ~ Ç ~ Ç ~ a − = = K== = 545. fÑ= M a ≠ I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW== a a ñ ñ = I= a a ó ó = K= fÑ= M a = =~åÇ= M añ ≠ Eçê= M aó ≠ FI=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë==åç== ëçäìíáçåK= fÑ= M a a a ó ñ = = = I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = áåÑáåáíÉäó= = ã~åó== ëçäìíáçåëK= = 546.      = + + = + + = + + P P P P O O O O N N N N Ç ò Å ó Ä ñ ~ Ç ò Å ó Ä ñ ~ = Ç ò Å ó Ä ñ ~ I== a a ñ ñ = I= a a ó ó = I= a a ò ò = =E`ê~ãÉê∞ë=êìäÉFI== =
CHAPTER 5. MATRICES AND DETERMINANTS 116 ïÜÉêÉ== P P P O O O N N N Å Ä ~ Å Ä ~ Å Ä ~ a = I= P P P O O O N N N ñ Å Ä Ç Å Ä Ç Å Ä Ç a = I= P P P O O O N N N ó Å Ç ~ Å Ç ~ Å Ç ~ a = I= P P P O O O N N N ò Ç Ä ~ Ç Ä ~ Ç Ä ~ a = K== = 547. fÑ= M a ≠ I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW== a a ñ ñ = I= a a ó ó = I= a a ò ò = K= fÑ= M a = =~åÇ= M añ ≠ Eçê= M aó ≠ =çê= M aò ≠ FI=íÜÉå=íÜÉ=ëóëíÉã= Ü~ë=åç=ëçäìíáçåK= fÑ= M a a a a ò ó ñ = = = = I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= áåÑáåáíÉäó= ã~åó=ëçäìíáçåëK= = 548. j~íêáñ=cçêã=çÑ=~=póëíÉã=çÑ=å=iáåÉ~ê=bèì~íáçåë=áå================= å=råâåçïåë= qÜÉ=ëÉí=çÑ=äáåÉ~ê=Éèì~íáçåë==        = + + + = + + + = + + + å å åå O O å N N å O å å O O OO N ON N å å N O NO N NN Ä ñ ~ ñ ~ ñ ~ Ä ñ ~ ñ ~ ñ ~ Ä ñ ~ ñ ~ ñ ~ K K K K K K K K K K K K K K K = Å~å=ÄÉ=ïêáííÉå=áå=ã~íêáñ=Ñçêã=               =               ⋅               å O N å O N åå O å N å å O OO ON å N NO NN Ä Ä Ä ñ ñ ñ ~ ~ ~ ~ ~ ~ ~ ~ ~ M M K M M M K K I== áKÉK== _ u ^ = ⋅ I==
CHAPTER 5. MATRICES AND DETERMINANTS 117 ïÜÉêÉ==               = åå O å N å å O OO ON å N NO NN ~ ~ ~ ~ ~ ~ ~ ~ ~ ^ K M M M K K I=               = å O N ñ ñ ñ u M I=               = å O N Ä Ä Ä _ M K== = 549. pçäìíáçå=çÑ=~=pÉí=çÑ=iáåÉ~ê=bèì~íáçåë= å å× = _ ^ u N ⋅ = − I== ïÜÉêÉ= N ^− =áë=íÜÉ=áåîÉêëÉ=çÑ=^K= = =
118 Chapter 6 Vectors = = = = sÉÅíçêëW=ì r I= î r I= ï r I= ê r I= → ^_ I=£= sÉÅíçê=äÉåÖíÜW= ì r I= î r I=£= råáí=îÉÅíçêëW= á r I= à r I=â r = kìää=îÉÅíçêW=M r = `ççêÇáå~íÉë=çÑ=îÉÅíçê=ì r W= N N N w I v I u = `ççêÇáå~íÉë=çÑ=îÉÅíçê= î r W= O O O w I v I u = pÅ~ä~êëW=λ Iµ= aáêÉÅíáçå=ÅçëáåÉëW= α Åçë I= β Åçë I= γ Åçë = ^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW=θ = = = 6.1 Vector Coordinates = 550. råáí=sÉÅíçêë= ( ) M I M I N á = r I= ( ) M I N I M à = r I= ( ) N I M I M â = r I= N â à á = = = r r r K= = 551. ( ) ( ) ( )â ò ò à ó ó á ñ ñ ^_ ê M N M N M N r r r r − + − + − = = → = =
CHAPTER 6. VECTORS 119 ======= = = Figure 73. = 552. ( ) ( ) ( )O M N O M N O M N ò ò ó ó ñ ñ ^_ ê − + − + − = = → r = = 553. fÑ= ê ^_ r = → I=íÜÉå= ê _^ r − = → K= = = = Figure 74. = 554. α = Åçë ê u r I= β = Åçë ê v r I= γ = Åçë ê w r K=
CHAPTER 6. VECTORS 120 ===== = = Figure 75. = 555. fÑ= ( ) ( ) N N N N w I v I u ê w I v I u ê r r = I=íÜÉå== N u u = I= N v v = I= N w w = K== == = 6.2 Vector Addition = 556. î ì ï r r r + = = = == = = Figure 76.
CHAPTER 6. VECTORS 121 == = = Figure 77. = 557. å P O N ì ì ì ì ï r K r r r r + + + + = = = == = = Figure 78. = 558. `çããìí~íáîÉ=i~ï= ì î î ì r r r r + = + = = 559. ^ëëçÅá~íáîÉ=i~ï= ( ) ( ) ï î ì ï î ì r r r r r r + + = + + = = 560. ( ) O N O N O N w w I v v I u u î ì + + + = + r r = = = = = = =
CHAPTER 6. VECTORS 122 6.3 Vector Subtraction = 561. î ì ï r r r − = =áÑ= ì ï î r r r = + K= = = = Figure 79. = == = = Figure 80. = 562. ( ) î ì î ì r r r r − + = − = = 563. ( ) M I M I M M ì ì = = − r r r = = 564. M M = r = = 565. ( ) O N O N O N w w I v v I u u î ì − − − = − r r I== = = = 6.4 Scaling Vectors = 566. ì ï r r λ = =
CHAPTER 6. VECTORS 123 = = Figure 81. = 567. ì ï r r ⋅ λ = = = 568. ( ) w I v I u ì λ λ λ = λ r = = 569. λ = λ ì ì r r = = 570. ( ) ì ì ì r r r µ + λ = µ + λ = = 571. ( ) ( ) ( )ì ì ì r r r λµ = λ µ = µ λ = = 572. ( ) î ì î ì r r r r λ + λ = + λ = = = = 6.5 Scalar Product = 573. pÅ~ä~ê=mêçÇìÅí=çÑ=sÉÅíçêë=ì r =~åÇ î r = θ ⋅ ⋅ = ⋅ Åçë î ì î ì r r r r I== ïÜÉêÉ=θ =áë=íÜÉ=~åÖäÉ=ÄÉíïÉÉå=îÉÅíçêë=ì r =~åÇ î r K==== =
CHAPTER 6. VECTORS 124 = = = Figure 82. = 574. pÅ~ä~ê=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã= fÑ= ( ) N N N w I v I u ì = r I= ( ) O O O w I v I u î = r I=íÜÉå== O N O N O N w w v v u u î ì + + = ⋅ r r K= = 575. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë== fÑ= ( ) N N N w I v I u ì = r I= ( ) O O O w I v I u î = r I=íÜÉå== O O O O O O O N O N O N O N O N O N w v u w v u w w v v u u Åçë + + + + + + = θ K= = 576. `çããìí~íáîÉ=mêçéÉêíó= ì î î ì r r r r ⋅ = ⋅ = = 577. ^ëëçÅá~íáîÉ=mêçéÉêíó= ( ) ( ) î ì î ì r r r r ⋅ λµ = µ ⋅ λ = = 578. aáëíêáÄìíáîÉ=mêçéÉêíó= ( ) ï ì î ì ï î ì r r r r r r r ⋅ + ⋅ = + ⋅ = = 579. M î ì = ⋅ r r =áÑ=ì r I î r =~êÉ=çêíÜçÖçå~ä=E O π = θ FK= = 580. M î ì > ⋅ r r =áÑ= O M π < θ < K= =
CHAPTER 6. VECTORS 125 581. M î ì < ⋅ r r =áÑ= π < θ < π O K= = 582. î ì î ì r r r r ⋅ ≤ ⋅ = = 583. î ì î ì r r r r ⋅ = ⋅ =áÑ=ì r I î r =~êÉ=é~ê~ääÉä=E M = θ FK= = 584. fÑ= ( ) N N N w I v I u ì = r I=íÜÉå== O N O N O N O O w v u ì ì ì ì + + = = = ⋅ r r r r K= = 585. N â â à à á á = ⋅ = ⋅ = ⋅ r r r r r r = = 586. M á â â à à á = ⋅ = ⋅ = ⋅ r r r r r r = = = = 6.6 Vector Product = 587. sÉÅíçê=mêçÇìÅí=çÑ=sÉÅíçêë=ì r =~åÇ î r = ï î ì r r r = × I=ïÜÉêÉ== • θ ⋅ ⋅ = ëáå î ì ï r r r I=ïÜÉêÉ= O M π ≤ θ ≤ X= • ì ï r r ⊥ = ~åÇ= î ï r r ⊥ X= • =sÉÅíçêë=ì r I= î r I= ï r =Ñçêã=~=êáÖÜí-Ü~åÇÉÇ=ëÅêÉïK= =
CHAPTER 6. VECTORS 126 ======= = = Figure 83. = 588. O O O N N N w v u w v u â à á î ì ï r r r r r r = × = = = 589.         − = × = O O N N O O N N O O N N v u v u I w u w u I w v w v î ì ï r r r = = 590. θ ⋅ ⋅ = × = ëáå î ì î ì p r r r r =EcáÖKUPF= = 591. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë=EcáÖKUPF= î ì î ì ëáå r r r r ⋅ × = θ = = 592. kçåÅçããìí~íáîÉ=mêçéÉêíó= ( ) ì î î ì r r r r × − = × == = 593. ^ëëçÅá~íáîÉ=mêçéÉêíó= ( ) ( ) î ì î ì r r r r × λµ = µ × λ = = =
CHAPTER 6. VECTORS 127 594. aáëíêáÄìíáîÉ=mêçéÉêíó= ( ) ï ì î ì ï î ì r r r r r r r × + × = + × = = 595. M î ì r r r = × =áÑ=ì r =~åÇ= î r =~êÉ=é~ê~ääÉä=E M = θ FK= = 596. M â â à à á á r r r r r r r = × = × = × = = 597. â à á r r r = × I= á â à r r r = × I= à á â r r r = × = = = = 6.7 Triple Product = 598. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí= [ ] ( ) ( ) ( ) î ì ï ì ï î ï î ì ï î ì r r r r r r r r r r r r × ⋅ = × ⋅ = × ⋅ = = = 599. [ ] [ ] [ ] [ ] [ ] [ ] î ï ì ì î ï ï ì î ì ï î î ì ï ï î ì r r r r r r r r r r r r r r r r r r − = − = − = = = = = 600. ( ) [ ] ï î ì â ï î ì â r r r r r r = × ⋅ = = 601. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã= ( ) P P P O O O N N N w v u w v u w v u ï î ì = × ⋅ r r r I== ïÜÉêÉ== ( ) N N N w I v I u ì = r I= ( ) O O O w I v I u î = r I= ( ) P P P w I v I u ï = r K== = 602. sçäìãÉ=çÑ=m~ê~ääÉäÉéáéÉÇ= ( ) ï î ì s r r r × ⋅ = = =
CHAPTER 6. VECTORS 128 ============ = = Figure 84. = 603. sçäìãÉ=çÑ=móê~ãáÇ= ( ) ï î ì S N s r r r × ⋅ = = = = = Figure 85. = 604. fÑ== ( ) M ï î ì = × ⋅ r r r I=íÜÉå=íÜÉ=îÉÅíçêë==ì r I= î r I=~åÇ= ï r =~êÉ=äáåÉ~êäó= ÇÉéÉåÇÉåí=I=ëç= î ì ï r r r µ + λ = =Ñçê=ëçãÉ=ëÅ~ä~êë=λ =~åÇ=µK== = 605. fÑ== ( ) M ï î ì ≠ × ⋅ r r r I=íÜÉå=íÜÉ=îÉÅíçêë==ì r I= î r I=~åÇ= ï r =~êÉ=äáåÉ~êäó= áåÇÉéÉåÇÉåíK= =
CHAPTER 6. VECTORS 129 606. sÉÅíçê=qêáéäÉ=mêçÇìÅí= ( ) ( ) ( )ï î ì î ï ì ï î ì r r r r r r r r r ⋅ − ⋅ = × × == = = = = = = = =
130 Chapter 7 Analytic Geometry = = = = 7.1 One-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= M ñ I= N ñ I= O ñ I= M ó I= N ó I= O ó = oÉ~ä=åìãÄÉêW=λ == aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= = = 607. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë= O N N O ñ ñ ñ ñ ^_ Ç − = − = = = = = = Figure 86. = 608. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç=λ = λ + λ + = N ñ ñ ñ O N M I= `_ ^` = λ I= N − ≠ λ K= = ======== = = Figure 87.
CHAPTER 7. ANALYTIC GEOMETRY 131 609. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí= O ñ ñ ñ O N M + = I= N = λ K= = = = 7.2 Two-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= M ñ I= N ñ I= O ñ I= M ó I= N ó I= O ó = mçä~ê=ÅççêÇáå~íÉëW= ϕ I ê = oÉ~ä=åìãÄÉêW=λ == mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI== aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= ^êÉ~W=p= = = 610. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë= ( ) ( )O N O O N O ó ó ñ ñ ^_ Ç − + − = = = = = = Figure 88.
CHAPTER 7. ANALYTIC GEOMETRY 132 611. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç=λ = λ + λ + = N ñ ñ ñ O N M I= λ + λ + = N ó ó ó O N M I== `_ ^` = λ I= N − ≠ λ K= = ======= = = Figure 89. = =
CHAPTER 7. ANALYTIC GEOMETRY 133 ======= = = Figure 90. = 612. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí= O ñ ñ ñ O N M + = I= O ó ó ó O N M + = I= N = λ K= = 613. `ÉåíêçáÇ=EfåíÉêëÉÅíáçå=çÑ=jÉÇá~åëF=çÑ=~=qêá~åÖäÉ= P ñ ñ ñ ñ P O N M + + = I= P ó ó ó ó P O N M + + = I== ïÜÉêÉ== ( ) N N ó I ñ ^ I== ( ) O O ó I ñ _ I==~åÇ== ( ) P P ó I ñ ` ==~êÉ=îÉêíáÅÉë=çÑ= íÜÉ=íêá~åÖäÉ= ^_` K= = =
CHAPTER 7. ANALYTIC GEOMETRY 134 ========= = = Figure 91. = 614. fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ= Å Ä ~ Åñ Äñ ~ñ ñ P O N M + + + + = I= Å Ä ~ Åó Äó ~ó ó P O N M + + + + = I== ïÜÉêÉ= _` ~ = I= `^ Ä = I= ^_ Å = K== = ======== = = Figure 92.
CHAPTER 7. ANALYTIC GEOMETRY 135 615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê====================== _áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ= N ó ñ N ó ñ N ó ñ O N ó ó ñ N ó ó ñ N ó ó ñ ñ P P O O N N P O P O P O O O O O N O N O N M + + + = I= N ó ñ N ó ñ N ó ñ O N ó ñ ñ N ó ñ ñ N ó ñ ñ ó P P O O N N O P O P P O O O O O O N O N N M + + + = = = ======== = == Figure 93. = = = = = = =
CHAPTER 7. ANALYTIC GEOMETRY 136 616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ= N ó ñ N ó ñ N ó ñ N ó ñ ñ ó N ó ñ ñ ó N ó ñ ñ ó ñ P P O O N N O P O N P O O N P O O N P O N M + + + = I= N ó ñ N ó ñ N ó ñ N ñ ó ó ñ N ñ ó ó ñ N ñ ó ó ñ ó P P O O N N P O N O P O N P O O N P O O N M + + + = = = ====== = = Figure 94. = 617. ^êÉ~=çÑ=~=qêá~åÖäÉ= ( ) ( ) N P N P N O N O P P O O N N ó ó ñ ñ ó ó ñ ñ O N N ó ñ N ó ñ N ó ñ O N p − − − − ± = ± = = = = =
CHAPTER 7. ANALYTIC GEOMETRY 137 618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä= ( ) ( )( ) ( )( ) [ + + − + + − ± = P O P O O N O N ó ó ñ ñ ó ó ñ ñ O N p = ( )( ) ( )( )] N Q N Q Q P Q P ó ó ñ ñ ó ó ñ ñ + − + + − + = = === = = Figure 95. = kçíÉW=få=Ñçêãìä~ë=SNTI=SNU=ïÉ=ÅÜççëÉ=íÜÉ=ëáÖå=EHF=çê=E¥F=ëç= íÜ~í=íç=ÖÉí=~=éçëáíáîÉ=~åëïÉê=Ñçê=~êÉ~K== = 619. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=áå=mçä~ê=`ççêÇáå~íÉë= ( ) N O O N O O O N Åçë ê ê O ê ê ^_ Ç ϕ − ϕ − + = = = =
CHAPTER 7. ANALYTIC GEOMETRY 138 = = Figure 96. = 620. `çåîÉêíáåÖ=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=íç=mçä~ê=`ççêÇáå~íÉë= ϕ = Åçë ê ñ I= ϕ = ëáå ê ó K= = = = Figure 97. = 621. `çåîÉêíáåÖ=mçä~ê=`ççêÇáå~íÉë=íç=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë= O O ó ñ ê + = I= ñ ó í~å = ϕ K=
CHAPTER 7. ANALYTIC GEOMETRY 139 7.3 Straight Line in Plane = mçáåí=ÅççêÇáå~íÉëW=uI=vI=ñI= M ñ I= N ñ I== M ó I= N ó I= N ~ I= O ~ I=£== oÉ~ä=åìãÄÉêëW=âI=~I=ÄI=éI=íI=Î=_I=Ì= N ^ I= O ^ I=£= ^åÖäÉëW=α I=β = ^åÖäÉ=ÄÉíïÉÉå=íïç=äáåÉëW=ϕ = kçêã~ä=îÉÅíçêW=å r = mçëáíáçå=îÉÅíçêëW= ê r I=~ r I= Ä r = = = 622. dÉåÉê~ä=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ= M ` _ó ^ñ = + + = = 623. kçêã~ä=sÉÅíçê=íç=~=píê~áÖÜí=iáåÉ= qÜÉ=îÉÅíçê= ( ) _ I ^ å r =áë=åçêã~ä=íç=íÜÉ=äáåÉ= M ` _ó ^ñ = + + K= = = = Figure 98. = 624. bñéäáÅáí=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=EpäçéÉ-fåíÉêÅÉéí=cçêãF= Ä âñ ó + = K==
CHAPTER 7. ANALYTIC GEOMETRY 140 qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= α = í~å â K= = = = Figure 99. = 625. dê~ÇáÉåí=çÑ=~=iáåÉ== N O N O ñ ñ ó ó í~å â − − = α = = = = = Figure 100.
CHAPTER 7. ANALYTIC GEOMETRY 141 626. bèì~íáçå=çÑ=~=iáåÉ=dáîÉå=~=mçáåí=~åÇ=íÜÉ=dê~ÇáÉåí= ( ) M M ñ ñ â ó ó − + = I== ïÜÉêÉ=â=áë=íÜÉ=Öê~ÇáÉåíI= ( ) M M ó I ñ m =áë=~=éçáåí=çå=íÜÉ=äáåÉK= = = = Figure 101. = 627. bèì~íáçå=çÑ=~=iáåÉ=qÜ~í=m~ëëÉë=qÜêçìÖÜ=qïç=mçáåíë= N O N N O N ñ ñ ñ ñ ó ó ó ó − − = − − == çê= M N ó ñ N ó ñ N ó ñ O O N N = K= =
CHAPTER 7. ANALYTIC GEOMETRY 142 = = Figure 102. = 628. fåíÉêÅÉéí=cçêã= N Ä ó ~ ñ = + = = = = Figure 103. = =
CHAPTER 7. ANALYTIC GEOMETRY 143 629. kçêã~ä=cçêã= M é ëáå ó Åçë ñ = − β + β = = = = Figure 104. = 630. mçáåí=aáêÉÅíáçå=cçêã= v ó ó u ñ ñ N N − = − I== ïÜÉêÉ= ( ) v I u =áë=íÜÉ=ÇáêÉÅíáçå=çÑ=íÜÉ=äáåÉ=~åÇ= ( ) N N N ó I ñ m =äáÉë= çå=íÜÉ=äáåÉK= =
CHAPTER 7. ANALYTIC GEOMETRY 144 = = Figure 105. = 631. sÉêíáÅ~ä=iáåÉ= ~ ñ = = = 632. eçêáòçåí~ä=iáåÉ= Ä ó = = = 633. sÉÅíçê=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ= Ä í ~ ê r r r + = I== ïÜÉêÉ== l=áë=íÜÉ=çêáÖáå=çÑ=íÜÉ=ÅççêÇáå~íÉëI= u=áë=~åó=î~êá~ÄäÉ=éçáåí=çå=íÜÉ=äáåÉI== ~ r =áë=íÜÉ=éçëáíáçå=îÉÅíçê=çÑ=~=âåçïå=éçáåí=^=çå=íÜÉ=äáåÉ=I= Ä r =áë=~=âåçïå=îÉÅíçê=çÑ=ÇáêÉÅíáçåI=é~ê~ääÉä=íç=íÜÉ=äáåÉI== í=áë=~=é~ê~ãÉíÉêI== → = lu ê r =áë=íÜÉ=éçëáíáçå=îÉÅíçê=çÑ=~åó=éçáåí=u=çå=íÜÉ=äáåÉK== =
CHAPTER 7. ANALYTIC GEOMETRY 145 = = Figure 106. = 634. píê~áÖÜí=iáåÉ=áå=m~ê~ãÉíêáÅ=cçêã=    + = + = O O N N íÄ ~ ó íÄ ~ ñ I== ïÜÉêÉ== ( ) ó I ñ ~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~åó=ìåâåçïå=éçáåí=çå=íÜÉ=äáåÉI== ( ) O N ~ I ~ =~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~=âåçïå=éçáåí=çå=íÜÉ=äáåÉI== ( ) O N Ä I Ä =~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~=îÉÅíçê=é~ê~ääÉä=íç=íÜÉ=äáåÉI== í=áë=~=é~ê~ãÉíÉêK= =
CHAPTER 7. ANALYTIC GEOMETRY 146 = Figure 107. = 635. aáëí~åÅÉ=cêçã=~=mçáåí=qç=~=iáåÉ= qÜÉ=Çáëí~åÅÉ=Ñêçã=íÜÉ=éçáåí= ( ) Ä I ~ m =íç=íÜÉ=äáåÉ= M ` _ó ^ñ = + + =áë== O O _ ^ ` _Ä ^~ Ç + + + = K= = = = Figure 108.
CHAPTER 7. ANALYTIC GEOMETRY 147 636. m~ê~ääÉä=iáåÉë= qïç=äáåÉë= N N Ä ñ â ó + = =~åÇ= O O Ä ñ â ó + = =~êÉ=é~ê~ääÉä=áÑ== O N â â = K= qïç= äáåÉë= M ` ó _ ñ ^ N N N = + + = ~åÇ= M ` ó _ ñ ^ O O O = + + = ~êÉ= é~ê~ääÉä=áÑ= O N O N _ _ ^ ^ = K= = = = Figure 109. = 637. mÉêéÉåÇáÅìä~ê=iáåÉë= qïç=äáåÉë= N N Ä ñ â ó + = =~åÇ= O O Ä ñ â ó + = =~êÉ=éÉêéÉåÇáÅìä~ê=áÑ== N O â N â − = =çêI=Éèìáî~äÉåíäóI= N â â O N − = K= qïç= äáåÉë= M ` ó _ ñ ^ N N N = + + = ~åÇ= M ` ó _ ñ ^ O O O = + + = ~êÉ= éÉêéÉåÇáÅìä~ê=áÑ= M _ _ ^ ^ O N O N = + K= =
CHAPTER 7. ANALYTIC GEOMETRY 148 = = Figure 110. = 638. ^åÖäÉ=_ÉíïÉÉå=qïç=iáåÉë= O N N O â â N â â í~å + − = ϕ I== O O O O O N O N O N O N _ ^ _ ^ _ _ ^ ^ Åçë + ⋅ + + = ϕ K= =
CHAPTER 7. ANALYTIC GEOMETRY 149 = = Figure 111. = 639. fåíÉêëÉÅíáçå=çÑ=qïç=iáåÉë= fÑ=íïç=äáåÉë= M ` ó _ ñ ^ N N N = + + =~åÇ= M ` ó _ ñ ^ O O O = + + =áåíÉê- ëÉÅíI=íÜÉ=áåíÉêëÉÅíáçå=éçáåí=Ü~ë=ÅççêÇáå~íÉë= N O O N N O O N M _ ^ _ ^ _ ` _ ` ñ − + − = I= N O O N N O O N M _ ^ _ ^ ` ^ ` ^ ó − + − = K= = = = 7.4 Circle = o~ÇáìëW=o= `ÉåíÉê=çÑ=ÅáêÅäÉW=( ) Ä I ~ = mçáåí=ÅççêÇáå~íÉëW=ñI=óI= N ñ I= N ó I=£= oÉ~ä=åìãÄÉêëW=Î=_I=Ì=aI=bI=cI=í=
CHAPTER 7. ANALYTIC GEOMETRY 150 640. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ= cçêãF= O O O o ó ñ = + = ====== = = Figure 112. = 641. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=^åó=mçáåí=( ) Ä I ~ ( ) ( ) O O O o Ä ó ~ ñ = − + − Figure 113.
CHAPTER 7. ANALYTIC GEOMETRY 151 642. qÜêÉÉ=mçáåí=cçêã M N ó ñ ó ñ N ó ñ ó ñ N ó ñ ó ñ N ó ñ ó ñ P P O P O P O O O O O O N N O N O N O O = + + + + = = = Figure 114. = 643. m~ê~ãÉíêáÅ=cçêã    = = í ëáå o ó í Åçë o ñ I= π ≤ ≤ O í M K = 644. dÉåÉê~ä=cçêã M c bó añ ^ó ^ñ O O = + + + + =E^=åçåòÉêçI= ^c Q b a O O > + FK== qÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ÅáêÅäÉ=Ü~ë=ÅççêÇáå~íÉë=( ) Ä I ~ I=ïÜÉêÉ== ^ O a ~ − = I= ^ O b Ä − = K= qÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅäÉ=áë
CHAPTER 7. ANALYTIC GEOMETRY 152 ^ O ^c Q b a o O O − + = K = = = 7.5 Ellipse = pÉãáã~àçê=~ñáëW=~= pÉãáãáåçê=~ñáëW=Ä= cçÅáW= ( ) M I Å cN − I= ( ) M I Å cO = aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= = bÅÅÉåíêáÅáíóW=É== oÉ~ä=åìãÄÉêëW=Î=_I=Ì=aI=bI=cI=í= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = 645. bèì~íáçå=çÑ=~å=bääáéëÉ=Epí~åÇ~êÇ=cçêãF N Ä ó ~ ñ O O O O = + = = Figure 115.
CHAPTER 7. ANALYTIC GEOMETRY 153 646. ~ O ê ê O N = + I= ïÜÉêÉ== N ê I== O ê ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó==éçáåí== ( ) ó I ñ m ==çå= íÜÉ=ÉääáéëÉ=íç=íÜÉ=íïç=ÑçÅáK= = = = Figure 116. = 647. O O O Å Ä ~ + = = 648. bÅÅÉåíêáÅáíó N ~ Å É < = = = 649. bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë Å ~ É ~ ñ O ± = ± = = = 650. m~ê~ãÉíêáÅ=cçêã    = = í ëáå Ä ó í Åçë ~ ñ I= π ≤ ≤ O í M K = =
CHAPTER 7. ANALYTIC GEOMETRY 154 651. dÉåÉê~ä=cçêã M c bó añ `ó _ñó ^ñ O O = + + + + + I== ïÜÉêÉ= M ^` Q _O < − K= = 652. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë M c bó añ `ó ^ñ O O = + + + + I== ïÜÉêÉ= M ^` > K = 653. `áêÅìãÑÉêÉåÅÉ ( ) É ~b Q i = I== ïÜÉêÉ==íÜÉ==ÑìåÅíáçå=b==áë==íÜÉ=ÅçãéäÉíÉ==ÉääáéíáÅ=áåíÉÖê~ä==çÑ= íÜÉ=ëÉÅçåÇ=âáåÇK== = 654. ^ééêçñáã~íÉ=cçêãìä~ë=çÑ=íÜÉ=`áêÅìãÑÉêÉåÅÉ ( ) ( ) ~Ä Ä ~ R K N i − + π = I== ( ) O O Ä ~ O i + π = K= = 655. ~Ä p π = = = = = 7.6 Hyperbola = qê~åëîÉêëÉ=~ñáëW=~= `çåàìÖ~íÉ=~ñáëW=Ä= cçÅáW= ( ) M I Å cN − I= ( ) M I Å cO = aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= = bÅÅÉåíêáÅáíóW=É== ^ëóãéíçíÉëW=ëI=í= oÉ~ä=åìãÄÉêëW=Î=_I=Ì=aI=bI=cI=íI=â= = = =
CHAPTER 7. ANALYTIC GEOMETRY 155 656. bèì~íáçå=çÑ=~=eóéÉêÄçä~=Epí~åÇ~êÇ=cçêãF= N Ä ó ~ ñ O O O O = − = = = = Figure 117. = 657. ~ O ê ê O N = − I= ïÜÉêÉ== N ê I== O ê ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó=éçáåí== ( ) ó I ñ m ==çå= íÜÉ=ÜóéÉêÄçä~=íç=íÜÉ=íïç=ÑçÅáK= =
CHAPTER 7. ANALYTIC GEOMETRY 156 = = Figure 118. = 658. bèì~íáçåë=çÑ=^ëóãéíçíÉë= ñ ~ Ä ó ± = = = 659. O O O Ä ~ Å + = = = 660. bÅÅÉåíêáÅáíó N ~ Å É > = = = 661. bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë Å ~ É ~ ñ O ± = ± = = = = =
CHAPTER 7. ANALYTIC GEOMETRY 157 662. m~ê~ãÉíêáÅ=bèì~íáçåë=çÑ=íÜÉ=oáÖÜí=_ê~åÅÜ=çÑ=~=eóéÉêÄçä~=    = = í ëáåÜ Ä ó í ÅçëÜ ~ ñ I= π ≤ ≤ O í M K = 663. dÉåÉê~ä=cçêã M c bó añ `ó _ñó ^ñ O O = + + + + + I== ïÜÉêÉ= M ^` Q _O > − K= = 664. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë M c bó añ `ó ^ñ O O = + + + + I== ïÜÉêÉ= M ^` < K= 665. ^ëóãéíçíáÅ=cçêã= Q É ñó O = I== çê== ñ â ó = I=ïÜÉêÉ= Q É â O = K= få= íÜáë= Å~ëÉ= I= íÜÉ= ~ëóãéíçíÉë= Ü~îÉ= Éèì~íáçåë= M ñ = = ~åÇ= M ó = K== =
CHAPTER 7. ANALYTIC GEOMETRY 158 = = Figure 119. = = = 7.7 Parabola = cçÅ~ä=é~ê~ãÉíÉêW=é= cçÅìëW=c= sÉêíÉñW= ( ) M M ó I ñ j = oÉ~ä=åìãÄÉêëW=Î=_I=Ì=aI=bI=cI=éI=~I=ÄI=Å= = = 666. bèì~íáçå=çÑ=~=m~ê~Äçä~=Epí~åÇ~êÇ=cçêãF éñ O óO = =
CHAPTER 7. ANALYTIC GEOMETRY 159 = = Figure 120. = bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ O é ñ − = I= `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=       M I O é c I= `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= ( ) M I M j K= = 667. dÉåÉê~ä=cçêã M c bó añ `ó _ñó ^ñ O O = + + + + + I== ïÜÉêÉ= M ^` Q _O = − K= = 668. O ~ñ ó = I= ~ O N é = K= bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ
CHAPTER 7. ANALYTIC GEOMETRY 160 O é ó − = I= `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=       O é I M c I= `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= ( ) M I M j K= = = = Figure 121. = 669. dÉåÉê~ä=cçêãI=^ñáë=m~ê~ääÉä=íç=íÜÉ=ó-~ñáë== M c bó añ ^ñO = + + + =EÎ=b=åçåòÉêçFI== Å Äñ ~ñ ó O + + = I= ~ O N é = K== bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ O é ó ó M − = I= `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=
CHAPTER 7. ANALYTIC GEOMETRY 161       + O é ó I ñ c M M I= `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= ~ O Ä ñM − = I= ~ Q Ä ~Å Q Å Äñ ~ñ ó O M O M M − = + + = K= = = = Figure 122. = = = 7.8 Three-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= M ñ I= M ó I= M ò I= N ñ I= N ó I= N ò I=£= oÉ~ä=åìãÄÉêW=λ == aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= ^êÉ~W=p= sçäìãÉW=s= =

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