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# 1300 math formulas

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1300 math formulas

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### 1300 math formulas

1. 1. 1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë=oÉëÉêîÉÇK=
2. 2. = qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK= i
3. 3. Preface = = = = qÜáë= Ü~åÇÄççâ= áë= ~= ÅçãéäÉíÉ= ÇÉëâíçé= êÉÑÉêÉåÅÉ= Ñçê= ëíìÇÉåíë= ~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=== = = ii
4. 4. Contents = = = = 1 krj_bo=pbqp= NKN= pÉí=fÇÉåíáíáÉë==1= NKO= pÉíë=çÑ=kìãÄÉêë==5= NKP= _~ëáÅ=fÇÉåíáíáÉë==7= NKQ= `çãéäÉñ=kìãÄÉêë==8= = 2 ^idb_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= iii
5. 5. 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= iv
6. 6. 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 ^k^ivqf`=dbljbqov= TKN= låÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==130= v
7. 7. 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 afccbobkqf^i=`^i`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 fkqbdo^i=`^i`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= vi
8. 8. VKNO= iáåÉ=fåíÉÖê~ä==275= VKNP= pìêÑ~ÅÉ=fåíÉÖê~ä==285= = = 10 afccbobkqf^i=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= = = = = = vii
9. 9. = qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK= = viii
10. 10. Chapter 1 Number Sets = = = = 1.1 Set Identities = pÉíëW=^I=_I=`= råáîÉêë~ä=ëÉíW=f= `çãéäÉãÉåí=W= ^′ = mêçéÉê=ëìÄëÉíW= ^ ⊂ _ == bãéíó=ëÉíW= ∅ = råáçå=çÑ=ëÉíëW= ^ ∪ _ = fåíÉêëÉÅíáçå=çÑ=ëÉíëW= ^ ∩ _ = aáÑÑÉêÉåÅÉ=çÑ=ëÉíëW= ^ y _ = = = 1. = 2. = 3. 4. 5. ^ ⊂ f= ^ ⊂ ^= ^ = _ =áÑ= ^ ⊂ _ =~åÇ= _ ⊂ ^ .= = bãéíó=pÉí= ∅⊂^= = råáçå=çÑ=pÉíë== ` = ^ ∪ _ = {ñ ö ñ ∈ ^ çê ñ ∈ _}= = 1
11. 11. CHAPTER 1. NUMBER SETS = ===== = Figure 1. 6. = 7. = 8. = `çããìí~íáîáíó= ^∪_ = _∪^= ^ëëçÅá~íáîáíó= ^ ∪ (_ ∪ ` ) = (^ ∪ _ ) ∪ ` = fåíÉêëÉÅíáçå=çÑ=pÉíë= ` = ^ ∪ _ = {ñ ö ñ ∈ ^ ~åÇ ñ ∈ _} = = = ===== = Figure 2. 9. = 10. = = `çããìí~íáîáíó= ^∩_ = _∩^= ^ëëçÅá~íáîáíó= ^ ∩ (_ ∩ ` ) = (^ ∩ _ ) ∩ ` = = 2
12. 12. CHAPTER 1. NUMBER SETS 11. = 12. = 13. = 14. aáëíêáÄìíáîáíó= ^ ∪ (_ ∩ ` ) = (^ ∪ _ ) ∩ (^ ∪ ` ) I= ^ ∩ (_ ∪ ` ) = (^ ∩ _ ) ∪ (^ ∩ ` ) K= fÇÉãéçíÉåÅó= ^ ∩ ^ = ^ I== ^∪^ = ^= açãáå~íáçå= ^ ∩ ∅ = ∅ I= ^∪f= f= fÇÉåíáíó= ^ ∪ ∅ = ^ I== ^∩f= ^ = 15. 16. 17. 18. `çãéäÉãÉåí= ^′ = {ñ ∈ f ö ñ ∉ ^} = `çãéäÉãÉåí=çÑ=fåíÉêëÉÅíáçå=~åÇ=råáçå ^ ∪ ^′ = f I== ^ ∩ ^′ = ∅ = = aÉ=jçêÖ~å∞ë=i~ïë (^ ∪ _ )′ = ^′ ∩ _′ I== (^ ∩ _ )′ = ^′ ∪ _′ = = aáÑÑÉêÉåÅÉ=çÑ=pÉíë ` = _ y ^ = {ñ ö ñ ∈ _ ~åÇ ñ ∉ ^} = = 3
13. 13. CHAPTER 1. NUMBER SETS = ===== = Figure 3. = 19. _ y ^ = _ y (^ ∩ _ ) = 20. _ y ^ = _ ∩ ^′ 21. ^y^=∅ 22. ^ y _ = ^ =áÑ= ^ ∩ _ = ∅ . = = = ===== = Figure 4. = 23. (^ y _) ∩ ` = (^ ∩ `) y (_ ∩ `) 24. ^′ = f y ^ 25. `~êíÉëá~å=mêçÇìÅí ` = ^ × _ = {(ñ I ó ) ö ñ ∈ ^ ~åÇ ó ∈ _} = = 4 =
14. 14. CHAPTER 1. NUMBER SETS 1.2 Sets of Numbers = 26. 27. = 28. = 29. = 30. k~íìê~ä=åìãÄÉêëW=k= tÜçäÉ=åìãÄÉêëW= kM = fåíÉÖÉêëW=w= mçëáíáîÉ=áåíÉÖÉêëW= w + = kÉÖ~íáîÉ=áåíÉÖÉêëW= w − = o~íáçå~ä=åìãÄÉêëW=n= oÉ~ä=åìãÄÉêëW=o== `çãéäÉñ=åìãÄÉêëW=`== = = k~íìê~ä=kìãÄÉêë `çìåíáåÖ=åìãÄÉêëW k = {NI OI PI K} K= tÜçäÉ=kìãÄÉêë `çìåíáåÖ=åìãÄÉêë=~åÇ=òÉêçW= k M = {MI NI OI PI K} K= fåíÉÖÉêë tÜçäÉ=åìãÄÉêë=~åÇ=íÜÉáê=çééçëáíÉë=~åÇ=òÉêçW= w + = k = {NI OI PI K}I= w − = {KI − PI − OI − N} I= w = w − ∪ {M} ∪ w + = {KI − PI − OI − NI MI NI OI PI K} K= o~íáçå~ä=kìãÄÉêë oÉéÉ~íáåÖ=çê=íÉêãáå~íáåÖ=ÇÉÅáã~äëW== ~   n = ñ ö ñ = ~åÇ ~ ∈ w ~åÇ Ä ∈ w ~åÇ Ä ≠ M K= Ä   fêê~íáçå~ä=kìãÄÉêë kçåêÉéÉ~íáåÖ=~åÇ=åçåíÉêãáå~íáåÖ=ÇÉÅáã~äëK = 5
15. 15. CHAPTER 1. NUMBER SETS 31. oÉ~ä=kìãÄÉêë== råáçå=çÑ=ê~íáçå~ä=~åÇ=áêê~íáçå~ä=åìãÄÉêëW=oK= = 32. `çãéäÉñ=kìãÄÉêë ` = {ñ + áó ö ñ ∈ o ~åÇ ó ∈ o}I== ïÜÉêÉ=á=áë=íÜÉ=áã~Öáå~êó=ìåáíK = 33. k⊂ w⊂n⊂ o ⊂ `= = === = = Figure 5. = = = = = = 6
16. 16. CHAPTER 1. NUMBER SETS 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= (~ + Ä) + Å = ~ + (Ä + Å ) = = = 38. aÉÑáåáíáçå=çÑ=pìÄíê~Åíáçå= ~ − Ä = ~ + (− Ä) = = 39. = 40. 41. 42. jìäíáéäáÅ~íáîÉ=fÇÉåíáíó= ~ ⋅N = ~ = jìäíáéäáÅ~íáîÉ=fåîÉêëÉ= N ~ ⋅ = N I= ~ ≠ M ~ = jìäíáéäáÅ~íáçå=qáãÉë=M ~ ⋅M = M = `çããìí~íáîÉ=çÑ=jìäíáéäáÅ~íáçå= ~ ⋅Ä = Ä⋅~ = = 7
17. 17. CHAPTER 1. NUMBER SETS 43. ^ëëçÅá~íáîÉ=çÑ=jìäíáéäáÅ~íáçå= (~ ⋅ Ä)⋅ Å = ~ ⋅ (Ä ⋅ Å ) = aáëíêáÄìíáîÉ=i~ï= ~ (Ä + Å ) = ~Ä + ~Å = 44. = 45. aÉÑáåáíáçå=çÑ=aáîáëáçå= ~ N = ~⋅ = Ä 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= êN I= êO = ^êÖìãÉåí=çÑ=~=ÅçãéäÉñ=åìãÄÉêW= ϕ I= ϕN I= ϕO = = = 46. = 47. = 48. áN = á = á O = −N = á P = −á = áQ = N= áR = á = á S = −N = á T = −á = áU = N = á Q å +N = á = á Q å+ O = −N = á Q å + P = −á = á Qå = N = ò = ~ + Äá = `çãéäÉñ=mä~åÉ= = 8
18. 18. CHAPTER 1. NUMBER SETS = ===== = Figure 6. = 49. = 50. = 51. = (~ + Äá ) − (Å + Çá ) = (~ − Å ) + (Ä − Ç)á = (~ + Äá )(Å + Çá ) = (~Å − ÄÇ ) + (~Ç + ÄÅ )á = ~ + Äá ~Å + ÄÇ ÄÅ − ~Ç = + ⋅á = Å + Çá Å O + Ç O Å O + Ç O 52. = 53. (~ + Äá ) + (Å + Çá ) = (~ + Å ) + (Ä + Ç )á = `çåàìÖ~íÉ=`çãéäÉñ=kìãÄÉêë= ||||||| ~ + Äá = ~ − Äá = = 54. ~ = ê Åçë ϕ I= Ä = ê ëáå ϕ == = 9
19. 19. CHAPTER 1. NUMBER SETS = = Figure 7. 55. = 56. = mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë= ~ + Äá = ê(Åçë ϕ + á ëáå ϕ) = jçÇìäìë=~åÇ=^êÖìãÉåí=çÑ=~=`çãéäÉñ=kìãÄÉê= fÑ= ~ + Äá =áë=~=ÅçãéäÉñ=åìãÄÉêI=íÜÉå= ê = ~ O + ÄO =EãçÇìäìëFI== Ä ϕ = ~êÅí~å =E~êÖìãÉåíFK= ~ = 57. = 58. mêçÇìÅí=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ò N ⋅ ò O = êN (Åçë ϕN + á ëáå ϕN ) ⋅ êO (Åçë ϕO + á ëáå ϕO ) = = êNêO [Åçë(ϕN + ϕO ) + á ëáå(ϕN + ϕO )] = `çåàìÖ~íÉ=kìãÄÉêë=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ||||||||||||||||||||| ê(Åçë ϕ + á ëáå ϕ) = ê[Åçë(− ϕ) + á ëáå(− ϕ)] = = 59. fåîÉêëÉ=çÑ=~=`çãéäÉñ=kìãÄÉê=áå=mçä~ê=oÉéêÉëÉåí~íáçå= N N = [Åçë(− ϕ) + á ëáå(− ϕ)] = ê(Åçë ϕ + á ëáå ϕ) ê 10
20. 20. CHAPTER 1. NUMBER SETS 60. = 61. = 62. = 63. = 64. nìçíáÉåí=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ò N êN (Åçë ϕN + á ëáå ϕN ) êN = [Åçë(ϕN − ϕO ) + á ëáå(ϕN − ϕO )] = = ò O êO (Åçë ϕO + á ëáå ϕO ) êO mçïÉê=çÑ=~=`çãéäÉñ=kìãÄÉê= å ò å = [ê(Åçë ϕ + á ëáå ϕ)] = ê å [Åçë(åϕ) + á ëáå(åϕ)] = cçêãìä~=±aÉ=jçáîêÉ≤= (Åçë ϕ + á ëáå ϕ)å = Åçë(åϕ) + á ëáå(åϕ) = kíÜ=oççí=çÑ=~=`çãéäÉñ=kìãÄÉê= ϕ + Oπâ ϕ + Oπâ   å ò = å ê(Åçë ϕ + á ëáå ϕ) = å ê  Åçë + á ëáå  I== å å   ïÜÉêÉ== â = MI NI OI KI å − N K== bìäÉê∞ë=cçêãìä~= É áñ = Åçë ñ + á ëáå ñ = = = 11
21. 21. Chapter 2 Algebra = = = = 2.1 Factoring Formulas = oÉ~ä=åìãÄÉêëW=~I=ÄI=Å== k~íìê~ä=åìãÄÉêW=å= = = 65. = 66. = 67. = 68. = 69. = 70. = 71. = 72. ~ O − ÄO = (~ + Ä)(~ − Ä) = ~ P − ÄP = (~ − Ä)(~ O + ~Ä + ÄO ) = ~ P + ÄP = (~ + Ä)(~ O − ~Ä + ÄO ) = ~ Q − ÄQ = (~ O − ÄO )(~ O + ÄO ) = (~ − Ä)(~ + Ä)(~ O + ÄO ) = ~ R − ÄR = (~ − Ä)(~ Q + ~ P Ä + ~ O ÄO + ~ÄP + ÄQ ) = ~ R + ÄR = (~ + Ä)(~ Q − ~ P Ä + ~ O ÄO − ~ÄP + ÄQ ) = fÑ=å=áë=çÇÇI=íÜÉå= ~ å + Äå = (~ + Ä)(~ å−N − ~ å −O Ä + ~ å −P ÄO − K − ~Äå −O + Äå −N ) K== fÑ=å=áë=ÉîÉåI=íÜÉå== ~ å − Äå = (~ − Ä)(~ å −N + ~ å −O Ä + ~ å −P ÄO + K + ~Äå−O + Äå −N ) I== 12
22. 22. CHAPTER 2. ALGEBRA ~ å + Äå = (~ + Ä)(~ å−N − ~ å −O Ä + ~ å −P ÄO − K + ~Äå−O − Äå −N ) K= = = = 2.2 Product Formulas 73. = 74. = 75. = 76. = 77. = 78. = 79. = 80. = 81. oÉ~ä=åìãÄÉêëW=~I=ÄI=Å== tÜçäÉ=åìãÄÉêëW=åI=â= = = (~ − Ä)O = ~ O − O~Ä + ÄO = (~ + Ä)O = ~ O + O~Ä + ÄO = (~ − Ä)P = ~ P − P~ O Ä + P~ÄO − ÄP = (~ + Ä)P = ~ P + P~ OÄ + P~ÄO + ÄP = (~ − Ä)Q = ~ Q − Q~ P Ä + S~ O ÄO − Q~ÄP + ÄQ = (~ + Ä)Q = ~ Q + Q~ P Ä + S~ OÄO + Q~ÄP + ÄQ = _áåçãá~ä=cçêãìä~= (~ + Ä)å = å` M~ å + å`N~ å−NÄ + å` O~ å−OÄO + K + å` å−N~Äå−N + å` å Äå I å> ïÜÉêÉ= å ` â = =~êÉ=íÜÉ=Äáåçãá~ä=ÅçÉÑÑáÅáÉåíëK= â> (å − â )> (~ + Ä + Å )O = ~ O + ÄO + Å O + O~Ä + O~Å + OÄÅ = (~ + Ä + Å + K + ì + î )O = ~ O + ÄO + Å O + K + ì O + î O + = + O(~Ä + ~Å + K + ~ì + ~î + ÄÅ + K + Äì + Äî + K + ìî ) = 13
23. 23. CHAPTER 2. ALGEBRA 2.3 Powers = _~ëÉë=EéçëáíáîÉ=êÉ~ä=åìãÄÉêëFW=~I=Ä== mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã= = = ~ ã ~ å = ~ ã+å = 82. = 83. ~ã = ~ ã −å = å ~ = 84. = (~Ä)ã = ~ ã Äã = 85. ~ã ~   = ã = Ä  Ä ã = 86. = 87. = 88. = (~ ) ã å = ~ ãå = ~ M = N I= ~ ≠ M = ~N = N = ~ −ã = 89. N = ~ã = ã å ~ = å ~ã = 90. = = = = = 14
24. 24. CHAPTER 2. ALGEBRA 2.4 Roots = 91. = _~ëÉëW=~I=Ä== mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã= ~ I Ä ≥ M =Ñçê=ÉîÉå=êççíë=E å = Oâ I= â ∈ k F= = = å ~Ä = å ~ å Ä = 92. = å ~ ã Ä = åã ~ ã Äå = 93. å ~ å~ = I= Ä ≠ M = Ä åÄ = 94. = 95. = 96. = ~ åã ~ ã åã ~ ã I= Ä ≠ M K= = = ã Äå Ä åã Äå å (~ ) å ã ( ~) å å é = å ~ ãé = =~= åé 97. = å ~ã = 98. = å ~ =~ = 99. = ã å 100. = ã å ã ~ = ãå ~ = ( ~) å ~ ãé = ã = å ~ã = 15
25. 25. CHAPTER 2. ALGEBRA N å ~ å −N = I= ~ ≠ M K= å ~ ~ 101. = ~± Ä = 102. ~ + ~O − Ä ~ − ~O − Ä ± = O O = N ~m Ä = = ~−Ä ~± Ä 103. = = = 2.5 Logarithms = 104. 105. 106. 107. 108. 109. mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=ñI=óI=~I=ÅI=â= k~íìê~ä=åìãÄÉêW=å== = = aÉÑáåáíáçå=çÑ=içÖ~êáíÜã= ó = äçÖ ~ ñ =áÑ=~åÇ=çåäó=áÑ= ñ = ~ ó I= ~ > M I= ~ ≠ N K= = äçÖ ~ N = M = = äçÖ ~ ~ = N = = − ∞ áÑ ~ > N äçÖ ~ M =  = + ∞ áÑ ~ < N = äçÖ ~ (ñó ) = äçÖ ~ ñ + äçÖ ~ ó = = ñ äçÖ ~ = äçÖ ~ ñ − äçÖ ~ ó = ó 16
26. 26. CHAPTER 2. ALGEBRA 110. äçÖ ~ (ñ å ) = å äçÖ ~ ñ = = N 111. äçÖ ~ å ñ = äçÖ ~ ñ = å = äçÖ Å ñ 112. äçÖ ~ ñ = = äçÖ Å ñ ⋅ äçÖ ~ Å I= Å > M I= Å ≠ N K= äçÖ Å ~ = N 113. äçÖ ~ Å = = äçÖ Å ~ = 114. ñ = ~ äçÖ ~ ñ = = 115. içÖ~êáíÜã=íç=_~ëÉ=NM= äçÖ NM ñ = äçÖ ñ = = 116. k~íìê~ä=içÖ~êáíÜã= äçÖ É ñ = äå ñ I== â  N ïÜÉêÉ= É = äáã N +  = OKTNUOUNUOUK = â →∞  â = N 117. äçÖ ñ = äå ñ = MKQPQOVQ äå ñ = äå NM = N 118. äå ñ = äçÖ ñ = OKPMORUR äçÖ ñ = äçÖ É = = = = = 17
27. 27. CHAPTER 2. ALGEBRA 2.6 Equations = oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=éI=èI=ìI=î= pçäìíáçåëW= ñ N I= ñ O I= ó N I= ó O I= ó P = = = 119. iáåÉ~ê=bèì~íáçå=áå=låÉ=s~êá~ÄäÉ= Ä ~ñ + Ä = M I= ñ = − K== ~ = 120. nì~Çê~íáÅ=bèì~íáçå= − Ä ± ÄO − Q~Å ~ñ + Äñ + Å = M I= ñ NI O = K= O~ = 121. aáëÅêáãáå~åí= a = ÄO − Q~Å = = 122. sáÉíÉ∞ë=cçêãìä~ë= fÑ= ñ O + éñ + è = M I=íÜÉå== ñ N + ñ O = −é K=  ñ Nñ O = è  = Ä 123. ~ñ O + Äñ = M I= ñ N = M I= ñ O = − K= ~ = Å 124. ~ñ O + Å = M I= ñ NI O = ± − K= ~ = 125. `ìÄáÅ=bèì~íáçåK=`~êÇ~åç∞ë=cçêãìä~K== ó P + éó + è = M I== O 18
28. 28. CHAPTER 2. ALGEBRA ó N = ì + î I= ó OI P = − N (ì + î ) ± P (ì + î ) á I== O O ïÜÉêÉ== O ì=P − O O O è è è  é  è  é +   +   I= î = P − −   +   K== O O  O P  O P = = 2.7 Inequalities s~êá~ÄäÉëW=ñI=óI=ò= ~ I ÄI ÅI Ç oÉ~ä=åìãÄÉêëW=  I=ãI=å= ~N I ~ O I ~ P I KI ~ å aÉíÉêãáå~åíëW=aI= añ I= aó I= aò == = = 126. fåÉèì~äáíáÉëI=fåíÉêî~ä=kçí~íáçåë=~åÇ=dê~éÜë== = fåÉèì~äáíó= fåíÉêî~ä=kçí~íáçå= dê~éÜ= [~I Ä]= ~ ≤ ñ ≤ Ä= ~ < ñ ≤ Ä= (~I Ä] = = ~ ≤ ñ < Ä= [~I Ä) = = ~ < ñ < Ä= (~I Ä) = = − ∞ < ñ ≤ Ä I= ñ≤Ä= − ∞ < ñ < Ä I= ñ<Ä= ~ ≤ ñ < ∞ I= ñ≥~= ~ < ñ < ∞ I= ñ >~= (− ∞I Ä] = = = (− ∞I Ä) = = [~I ∞ ) = = (~I ∞ ) = = 19
29. 29. CHAPTER 2. ALGEBRA 127. = 128. = 129. = 130. = 131. = 132. = 133. = fÑ= ~ > Ä I=íÜÉå= Ä < ~ K= fÑ= ~ > Ä I=íÜÉå= ~ − Ä > M =çê= Ä − ~ < M K= fÑ= ~ > Ä I=íÜÉå= ~ + Å > Ä + Å K= fÑ= ~ > Ä I=íÜÉå= ~ − Å > Ä − Å K= fÑ= ~ > Ä =~åÇ= Å > Ç I=íÜÉå= ~ + Å > Ä + Ç K= fÑ= ~ > Ä =~åÇ= Å > Ç I=íÜÉå= ~ − Ç > Ä − Å K= fÑ= ~ > Ä =~åÇ= ã > M I=íÜÉå= ã~ > ãÄ K= 134. fÑ= ~ > Ä =~åÇ= ã > M I=íÜÉå= ~ Ä > K= ã ã = 135. fÑ= ~ > Ä =~åÇ= ã < M I=íÜÉå= ã~ < ãÄ K= = ~ Ä 136. fÑ= ~ > Ä =~åÇ= ã < M I=íÜÉå= < K= ã ã = 137. fÑ= M < ~ < Ä =~åÇ= å > M I=íÜÉå= ~ å < Äå K= = 138. fÑ= M < ~ < Ä =~åÇ= å < M I=íÜÉå= ~ å > Äå K= = 139. fÑ= M < ~ < Ä I=íÜÉå= å ~ < å Ä K= = ~+Ä I== 140. ~Ä ≤ O ïÜÉêÉ= ~ > M =I= Ä > M X=~å=Éèì~äáíó=áë=î~äáÇ=çåäó=áÑ= ~ = Ä K== = N 141. ~ + ≥ O I=ïÜÉêÉ= ~ > M X=~å=Éèì~äáíó=í~âÉë=éä~ÅÉ=çåäó=~í= ~ = N K= ~ 20
30. 30. CHAPTER 2. ALGEBRA 142. å ~N~ O K~ å ≤ ~N + ~ O + K + ~ å I=ïÜÉêÉ= ~N I ~ O I KI ~ å > M K= å = Ä 143. fÑ= ~ñ + Ä > M =~åÇ= ~ > M I=íÜÉå= ñ > − K= ~ = Ä 144. fÑ= ~ñ + Ä > M =~åÇ= ~ < M I=íÜÉå= ñ < − K== ~ = 145. ~ñ O + Äñ + Å > M = = = ~ > M= = = = = a>M= = = = a=M= = = = a<M= = ñ < ñ N I= ñ > ñ O = = ñ N < ñ I= ñ > ñ N = = = −∞< ñ <∞= = 21 ~ <M= = = ñN < ñ < ñ O = = ñ ∈∅ = = = ñ ∈∅ = = = =
31. 31. CHAPTER 2. ALGEBRA ~+Ä ≤ ~ + Ä = 146. = 147. = 148. = 149. = 150. = fÑ= ñ < ~ I=íÜÉå= − ~ < ñ < ~ I=ïÜÉêÉ= ~ > M K= fÑ= ñ > ~ I=íÜÉå= ñ < −~ =~åÇ= ñ > ~ I=ïÜÉêÉ= ~ > M K= fÑ= ñ O < ~ I=íÜÉå= ñ < ~ I=ïÜÉêÉ= ~ > M K= fÑ= ñ O > ~ I=íÜÉå= ñ > ~ I=ïÜÉêÉ= ~ > M K= 151. fÑ= = Ñ (ñ ) ⋅ Ö (ñ ) > M Ñ (ñ ) > M I=íÜÉå=  K= Ö (ñ ) Ö (ñ ) ≠ M Ñ (ñ ) ⋅ Ö (ñ ) < M Ñ (ñ ) < M I=íÜÉå=  K= Ö (ñ ) Ö (ñ ) ≠ M 152. = = = 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  ê ^ = ` N +  =  å = 22
32. 32. CHAPTER 2. ALGEBRA 154. páãéäáÑáÉÇ=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~= fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=çåÅÉ=éÉê=óÉ~êI=íÜÉå=íÜÉ=éêÉîáçìë= Ñçêãìä~=ëáãéäáÑáÉë=íçW= í ^ = `(N + ê ) K= = 155. `çåíáåìçìë=`çãéçìåÇ=fåíÉêÉëí= fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=Åçåíáåì~ääó=E å → ∞ FI=íÜÉå== ^ = `É êí K= = = 23
33. 33. Chapter 3 Geometry = = = = 3.1 Right Triangle = iÉÖë=çÑ=~=êáÖÜí=íêá~åÖäÉW=~I=Ä= eóéçíÉåìëÉW=Å= ^äíáíìÇÉW=Ü= jÉÇá~åëW= ã ~ I= ã Ä I= ã Å = ^åÖäÉëW= α I β = o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= ^êÉ~W=p= = = = = Figure 8. = 156. α + β = VM° = = 24
34. 34. CHAPTER 3. GEOMETRY 157. ëáå α = ~ = Åçë β = Å = 158. Åçë α = Ä = ëáå β = Å = 159. í~å α = ~ = Åçí β = Ä = Ä 160. Åçí α = = í~å β = ~ = Å 161. ëÉÅ α = = Åçë ÉÅ β = Ä = 162. Åçë ÉÅ α = Å = ëÉÅ β = ~ = 163. móíÜ~ÖçêÉ~å=qÜÉçêÉã= ~ O + ÄO = Å O = = 164. ~ = ÑÅ I= Ä = ÖÅ I== ïÜÉêÉ= Ñ= ~åÇ= Å= ~êÉ= éêçàÉÅíáçåë= çÑ= íÜÉ= äÉÖë= ~= ~åÇ= ÄI= êÉëéÉÅíáîÉäóI=çåíç=íÜÉ=ÜóéçíÉåìëÉ=ÅK= = O = O = ===== Figure 9. = 25
35. 35. CHAPTER 3. GEOMETRY 165. Ü O = ÑÖ I=== ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK== = O O ~ Ä 166. ã O = ÄO − I= ã O = ~ O − I=== ~ Ä Q Q ïÜÉêÉ= ã ~ =~åÇ= ã Ä =~êÉ=íÜÉ=ãÉÇá~åë=íç=íÜÉ=äÉÖë=~=~åÇ=ÄK== = = = Figure 10. = Å 167. ã Å = I== O ïÜÉêÉ= ã Å =áë=íÜÉ=ãÉÇá~å=íç=íÜÉ=ÜóéçíÉåìëÉ=ÅK= = Å 168. o = = ã Å = O = ~ +Ä−Å ~Ä = = 169. ê = O ~ +Ä+Å = 170. ~Ä = ÅÜ = = = 26
36. 36. CHAPTER 3. GEOMETRY 171. p = ~Ä ÅÜ = = O O = = = 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. β = VM° − α = O = 173. Ü O = ÄO − O ~ = Q 27
37. 37. CHAPTER 3. GEOMETRY 174. i = ~ + OÄ = = 175. p = O ~Ü Ä = ëáå α = O 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. Ü = ~ P = O = 28
38. 38. CHAPTER 3. GEOMETRY O ~ P = 177. o = Ü = P P = N ~ P o = = 178. ê = Ü = P S O = 179. i = P~ = = 180. p = O ~Ü ~ P = = O Q = = = 3.4 Scalene Triangle E^=íêá~åÖäÉ=ïáíÜ=åç=íïç=ëáÇÉë=Éèì~äF= = = páÇÉë=çÑ=~=íêá~åÖäÉW=~I=ÄI=Å= ~ +Ä+Å == pÉãáéÉêáãÉíÉêW= é = O ^åÖäÉë=çÑ=~=íêá~åÖäÉW= αI βI γ = ^äíáíìÇÉë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= Ü ~ I Ü Ä I Ü Å = jÉÇá~åë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= ã ~ I ã Ä I ã Å = _áëÉÅíçêë=çÑ=íÜÉ=~åÖäÉë= αI βI γ W= í ~ I í Ä I í Å = o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= ^êÉ~W=p= = = 29
39. 39. CHAPTER 3. GEOMETRY = ===== = Figure 13. = 181. α + β + γ = NUM° = 182. ~ + Ä > Å I== Ä + Å > ~ I== ~ + Å > Ä K= = 183. ~ − Ä < Å I== Ä − Å < ~ I== ~ − Å < Ä K= = = 184. jáÇäáåÉ= ~ è = I= è öö ~ K= O = = = ===== Figure 14. = 30
40. 40. CHAPTER 3. GEOMETRY 185. i~ï=çÑ=`çëáåÉë= ~ O = ÄO + Å O − OÄÅ Åçë α I= ÄO = ~ O + Å O − O~Å Åçë β I= Å O = ~ O + ÄO − O~Ä Åçë γ K= = 186. i~ï=çÑ=páåÉë= ~ Ä Å = = = Oo I== ëáå α ëáå β ëáå γ ïÜÉêÉ=o=áë=íÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉK== = ~ Ä Å ÄÅ ~Å ~Ä ~ÄÅ = = = = = = 187. o = = O ëáå α O ëáå β O ëáå γ OÜ ~ OÜ Ä OÜ Å Qp = (é − ~ )(é − Ä)(é − Å ) I== 188. ê O = é N N N N = + + K= ê Ü~ ÜÄ ÜÅ = (é − Ä)(é − Å ) I= α 189. ëáå = O ÄÅ Åçë α é(é − ~ ) I= = O ÄÅ í~å α = O (é − Ä)(é − Å ) K= é(é − ~ ) = O 190. Ü ~ = é(é − ~ )(é − Ä)(é − Å ) I= ~ O é(é − ~ )(é − Ä)(é − Å ) I= ÜÄ = Ä O ÜÅ = é(é − ~ )(é − Ä)(é − Å ) K= Å 31
41. 41. CHAPTER 3. GEOMETRY 191. Ü ~ = Ä ëáå γ = Å ëáå β I= Ü Ä = ~ ëáå γ = Å ëáå α I= Ü Å = ~ ëáå β = Ä ëáå α K= = Ä +Å ~ − I== O Q O O ~ + Å ÄO ãO = − I== Ä O Q O O ~ + Ä ÅO O ãÅ = − K= O Q 192. ã O = ~ O O O = = = ===== Figure 15. = O O O 193. ^j = ã ~ I= _j = ã Ä I= `j = ã Å =EcáÖKNRFK= P P P = QÄÅé(é − ~ ) 194. í O = I== ~ (Ä + Å )O Q~Åé(é − Ä) íO = I== Ä (~ + Å )O Q~Äé(é − Å ) íO = K= Å (~ + Ä)O = 32
42. 42. CHAPTER 3. GEOMETRY ~Ü ~ ÄÜ Ä ÅÜ Å = = I== O O O ~Ä ëáå γ ~Å ëáå β ÄÅ ëáå α I== p= = = O O O p = é(é − ~ )(é − Ä)(é − Å ) =EeÉêçå∞ë=cçêãìä~FI= p = éê I== ~ÄÅ p= I= Qo p = Oo O ëáå α ëáå β ëáå γ I= α β γ p = éO í~å í~å í~å K= O O O 195. p = = = = 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. 33
43. 43. CHAPTER 3. GEOMETRY 196. Ç = ~ O == = 197. o = Ç ~ O = = O O = ~ 198. ê = = O 199. i = Q~ = = = 200. p = ~ = = = = O 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. Ç = ~ O + ÄO == 34
44. 44. CHAPTER 3. GEOMETRY 202. o = Ç = O = 203. i = O(~ + Ä) = = 204. p = ~Ä = = = = 3.7 Parallelogram = páÇÉë=çÑ=~=é~ê~ääÉäçÖê~ãW=~I=Ä= aá~Öçå~äëW= ÇN I Ç O = `çåëÉÅìíáîÉ=~åÖäÉëW= αI β = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ = ^äíáíìÇÉW=Ü== mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = ===== = Figure 18. = 205. α + β = NUM° = 206. Ç + Ç = O(~ + Ä ) = O N O O O = O = 35
45. 45. CHAPTER 3. GEOMETRY 207. Ü = Ä ëáå α = Ä ëáå β = 208. i = O(~ + Ä) = 209. p = ~Ü = ~Ä ëáå α I== N p = ÇNÇ O ëáå ϕ K= O = = = = = 3.8 Rhombus = páÇÉ=çÑ=~=êÜçãÄìëW=~= aá~Öçå~äëW= ÇN I Ç O = `çåëÉÅìíáîÉ=~åÖäÉëW= αI β = ^äíáíìÇÉW=e= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = ===== Figure 19. = 36
46. 46. CHAPTER 3. GEOMETRY 210. α + β = NUM° = = 211. Ç + Ç = Q~ = O N O O O = 212. Ü = ~ ëáå α = ÇNÇ O = O~ = Ü ÇÇ ~ ëáå α 213. ê = = N O = = O Q~ O = 214. i = Q~ = = 215. p = ~Ü = ~ ëáå α I== N p = ÇNÇ O K= O = = = O 3.9 Trapezoid = _~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä= jáÇäáåÉW=è= ^äíáíìÇÉW=Ü= ^êÉ~W=p= = = 37
47. 47. CHAPTER 3. GEOMETRY = = Figure 20. = 216. è = 217. p = ~+Ä = O ~+Ä ⋅ Ü = èÜ = O = = = = 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= = = 38
48. 48. CHAPTER 3. GEOMETRY = = Figure 21. = 218. è = ~+Ä = O = 219. Ç = ~Ä + Å = = N O 220. Ü = Å O − (Ä − ~ ) = Q O = Å ~Ä + Å O = (OÅ − ~ + Ä)(OÅ + ~ − Ä) = ~+Ä 222. p = ⋅ Ü = èÜ = O = = = = = = 221. o = 39
49. 49. CHAPTER 3. GEOMETRY 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. ~ + Ä = OÅ = = ~+Ä 224. è = =Å= O = 225. Ç = Ü + Å = O O O = 40
50. 50. CHAPTER 3. GEOMETRY 226. ê = Ü ~Ä = = O O = Ä ÅÇ ÅÇ Å Å Å ~+Ä ~ N+ ÜO + Å O = = = = +S+ = OÜ Qê O ~Ä OÜ U Ä ~ = 228. i = O(~ + Ä) = QÅ = = (~ + Ä) ~Ä = èÜ = ÅÜ = iê == ~+Ä ⋅Ü = 229. p = O O O = = = 227. o = O 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= ÇN I Ç O = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ = o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= ^êÉ~W=p= = 41
51. 51. CHAPTER 3. GEOMETRY = = Figure 23. = 230. ~ + Ä = Å + Ç = ~+Ä Å+Ç = = 231. è = O O 232. i = O(~ + Ä) = O(Å + Ç ) = = = = ~+Ä Å+Ç ⋅Ü = ⋅ Ü = èÜ I== O O N p = ÇNÇ O ëáå ϕ K= O 233. p = = = = 3.13 Kite = páÇÉë=çÑ=~=âáíÉW=~I=Ä= aá~Öçå~äëW= ÇN I Ç O = ^åÖäÉëW= αI βI γ = mÉêáãÉíÉêW=i= ^êÉ~W=p= = = 42
52. 52. CHAPTER 3. GEOMETRY = = Figure 24. = 234. α + β + Oγ = PSM° = 235. i = O(~ + Ä) = = = 236. p = ÇNÇ O = O = = = 3.14 Cyclic Quadrilateral páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç= aá~Öçå~äëW= ÇN I Ç O = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ = fåíÉêå~ä=~åÖäÉëW= αI βI γ I δ = o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= pÉãáéÉêáãÉíÉêW=é== ^êÉ~W=p= 43
53. 53. CHAPTER 3. GEOMETRY = = Figure 25. = 237. α + γ = β + δ = NUM° = = 238. míçäÉãó∞ë=qÜÉçêÉã= ~Å + ÄÇ = ÇNÇ O = 239. i = ~ + Ä + Å + Ç = = = N (~Å + ÄÇ )(~Ç + ÄÅ )(~Ä + ÅÇ ) I== 240. o = Q (é − ~ )(é − Ä)(é − Å )(é − Ç ) i ïÜÉêÉ= é = K= O = N 241. p = ÇNÇ O ëáå ϕ I== O p = (é − ~ )(é − Ä)(é − Å )(é − Ç ) I== i ïÜÉêÉ= é = K= O = = = 44
54. 54. CHAPTER 3. GEOMETRY 3.15 Tangential Quadrilateral = páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç= aá~Öçå~äëW= ÇN I Ç O = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ = o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= pÉãáéÉêáãÉíÉêW=é== ^êÉ~W=p= = = = = Figure 26. = 242. ~ + Å = Ä + Ç = = 243. i = ~ + Ä + Å + Ç = O(~ + Å ) = O(Ä + Ç ) = = O ÇN Ç O − (~ − Ä) (~ + Ä − é ) O I== Oé i ïÜÉêÉ= é = K== O = O O 244. ê = 45
55. 55. CHAPTER 3. GEOMETRY N 245. p = éê = ÇNÇ O ëáå ϕ = O = = = 3.16 General Quadrilateral = páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç= aá~Öçå~äëW= ÇN I Ç O = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ = fåíÉêå~ä=~åÖäÉëW= αI βI γ I δ = mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = ======= Figure 27. = 246. α + β + γ + δ = PSM° = 247. i = ~ + Ä + Å + Ç = = = 46
56. 56. CHAPTER 3. GEOMETRY N 248. p = ÇNÇ O ëáå ϕ = O = = = 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. ê = ã = ~ P = O 47
57. 57. CHAPTER 3. GEOMETRY 251. o = ~ = = 252. i = S~ = = O ~ P P I== O i ïÜÉêÉ= é = K= O = = = 253. p = éê = 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= = = 48
58. 58. CHAPTER 3. GEOMETRY = = Figure 29. = 254. α = 255. α = å−O ⋅ NUM° = O = å−O ⋅ NUM° = O = 256. o = ~ π O ëáå å = = 257. ê = ã = ~ O í~å π å = oO − ~O = Q = 258. i = å~ = = 259. p = åo Oπ ëáå I== O å O p = éê = é o O − ~O I== Q 49
59. 59. CHAPTER 3. GEOMETRY ïÜÉêÉ= é = i K== O = = = 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= = = α 260. ~ = Oo ëáå = O = = = Figure 30. = 50
60. 60. CHAPTER 3. GEOMETRY 261. ~N~ O = ÄNÄO = = = = Figure 31. = 262. ÉÉN = ÑÑN = = = = ===== Figure 32. = 263. Ö O = ÑÑN = = 51
61. 61. CHAPTER 3. GEOMETRY = ===== = Figure 33. = 264. β = α = O = = = Figure 34. = 265. i = Oπo = πÇ = = 266. p = πo O = io πÇ = == Q O O = 52
62. 62. CHAPTER 3. GEOMETRY 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. ë = oñ = 268. ë = = πoα = NUM° = 269. i = ë + Oo = = 270. p = oë o ñ πo α = = == O O PSM° O O = = 53
63. 63. CHAPTER 3. GEOMETRY 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 OÜo − Ü O = = N 272. Ü = o − Qo O − ~ O I= Ü < o = O = 273. i = ë + ~ = = 54
64. 64. CHAPTER 3. GEOMETRY O O N [ëo − ~(o − Ü )] = o  απ − ëáå α  = o (ñ − ëáå ñ ) I==   O O  NUM°  O O p ≈ Ü~ K= P 274. p = = = = 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. Ç = ~ P = = ~ 276. ê = = O = 55
65. 65. CHAPTER 3. GEOMETRY 277. o = ~ P = O = 278. p = S~ = O = 279. s = ~ == = = = P 3.23 Rectangular Parallelepiped = bÇÖÉëW=~I=ÄI=Å== aá~Öçå~äW=Ç= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = ===== Figure 38. = 280. Ç = ~ O + ÄO + Å O = 281. p = O(~Ä + ~Å + ÄÅ ) = 282. s = ~ÄÅ == = = 56
66. 66. CHAPTER 3. GEOMETRY 3.24 Prism = i~íÉê~ä=ÉÇÖÉW=ä= eÉáÖÜíW=Ü= i~íÉê~ä=~êÉ~W= p i = ^êÉ~=çÑ=Ä~ëÉW= p_ = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = ===== Figure 39. = 283. p = p i + Op_ K== = 284. i~íÉê~ä=^êÉ~=çÑ=~=oáÖÜí=mêáëã= p i = (~ N + ~ O + ~ P + K + ~ å )ä = = 285. i~íÉê~ä=^êÉ~=çÑ=~å=lÄäáèìÉ=mêáëã= p i = éä I== ïÜÉêÉ=é=áë=íÜÉ=éÉêáãÉíÉê=çÑ=íÜÉ=Åêçëë=ëÉÅíáçåK= = 57
67. 67. CHAPTER 3. GEOMETRY 286. s = p_ Ü = = 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. Ü = O ~= P = 58
68. 68. CHAPTER 3. GEOMETRY 289. p_ = P~ O = Q = 290. p = P~ = = N ~P 291. s = p_ Ü = K== P S O = = = O 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= pi = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 59
69. 69. CHAPTER 3. GEOMETRY = = Figure 41. = 292. ã = ÄO − ~O = Q = 293. Ü = π O −~ å = π O ëáå å QÄO ëáå O = N N 294. p i = å~ã = å~ QÄO − ~ O = éã = O Q = 295. p_ = éê = = 296. p = p_ + p i = = N N 297. s = p_ Ü = éêÜ == P P = = = 60
70. 70. CHAPTER 3. GEOMETRY 3.27 Frustum of a Regular Pyramid = ~N I ~ O I ~ P IKI ~ å = _~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW=  ÄN I ÄO I ÄP IKI Äå eÉáÖÜíW=Ü= pä~åí=ÜÉáÖÜíW=ã== ^êÉ~=çÑ=Ä~ëÉëW= pN I= pO = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i = mÉêáãÉíÉê=çÑ=Ä~ëÉëW= mN I= mO = pÅ~äÉ=Ñ~ÅíçêW=â= qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 42. = 298. ÄN ÄO ÄP Ä Ä = = =K= å = = â = ~N ~ O ~ P ~å ~ = 61
71. 71. CHAPTER 3. GEOMETRY 299. pO = âO = pN = ã(mN + mO ) = 300. p i = O = 301. p = p i + pN + pO = = Ü 302. s = pN + pNpO + pO = P = O Üp  Ä  Ä   Üp 303. s = N N + +    = N N + â + â O = P  ~ ~  P   = = = ( ) [ ] 3.28 Rectangular Right Wedge = páÇÉë=çÑ=Ä~ëÉW=~I=Ä= qçé=ÉÇÖÉW=Å= eÉáÖÜíW=Ü= i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i = ^êÉ~=çÑ=Ä~ëÉW= p_ = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 62
72. 72. CHAPTER 3. GEOMETRY = = Figure 43. = N (~ + Å ) QÜO + ÄO + Ä ÜO + (~ − Å )O = O = 305. p_ = ~Ä = = 306. p = p_ + p i = = ÄÜ (O~ + Å ) = 307. s = S = = = 304. p i = 3.29 Platonic Solids = bÇÖÉW=~= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 63
73. 73. CHAPTER 3. GEOMETRY 308. cáîÉ=mä~íçåáÅ=pçäáÇë= qÜÉ= éä~íçåáÅ= ëçäáÇë= ~êÉ= ÅçåîÉñ= éçäóÜÉÇê~= ïáíÜ= Éèìáî~äÉåí= Ñ~ÅÉë=ÅçãéçëÉÇ=çÑ=ÅçåÖêìÉåí=ÅçåîÉñ=êÉÖìä~ê=éçäóÖçåëK== = kìãÄÉê= kìãÄÉê= pÉÅíáçå= pçäáÇ= kìãÄÉê= çÑ=sÉêíáÅÉë çÑ=bÇÖÉë= çÑ=c~ÅÉë= 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. ê = ~ S = S = 310. o = ~ O = O = 64
74. 74. CHAPTER 3. GEOMETRY 311. p = O~ O P = = ~P O 312. s = = P = = Icosahedron = = = Figure 45. = 313. ê = ( = 314. o = ) ~ P P+ R = NO ( ) ~ O R+ R = Q = 315. p = R~ O P = = R~ P P + R 316. s = = NO = = ( ) 65
75. 75. CHAPTER 3. GEOMETRY Dodecahedron = = = Figure 46. 317. ê = ( ~ NM OR + NN R = O = 318. o = ) = ( ) ~ P N+ R = Q = ( ) 319. p = P~ O R R + O R = = ~ P NR + T R 320. s = = Q = = = ( ) 3.30 Right Circular Cylinder = o~Çáìë=çÑ=Ä~ëÉW=o= aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç= 66
76. 76. CHAPTER 3. GEOMETRY eÉáÖÜíW=e= i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i = ^êÉ~=çÑ=Ä~ëÉW= p_ = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = ===== = Figure 47. = 321. p i = Oπoe = = Ç  322. p = p i + Op_ = Oπo(e + o ) = πÇ e +  = O  = 323. s = p_ e = πo O e = = = = 67
77. 77. CHAPTER 3. GEOMETRY 3.31 Right Circular Cylinder with an Oblique Plane Face = o~Çáìë=çÑ=Ä~ëÉW=o= qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= ÜN = qÜÉ=ëÜçêíÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= Ü O = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i = ^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= p_ = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 48. = 324. p i = πo(ÜN + Ü O ) = = O  Ü − ÜO  325. p_ = πo + πo o +  N  =  O  = O O 68
78. 78. CHAPTER 3. GEOMETRY O   ÜN − Ü O   O 326. p = p i + p_ = πo ÜN + Ü O + o + o +   =  O      = πo O (ÜN + ÜO ) = 327. s = O = = = 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= pi = ^êÉ~=çÑ=Ä~ëÉW= p_ = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 49. 69
79. 79. CHAPTER 3. GEOMETRY 328. e = ã O − o O = = πãÇ 329. p i = πoã = = O = 330. p_ = πo O = = N  Ç 331. p = p i + p_ = πo (ã + o ) = πÇ ã +  = O  O = N N 332. s = p_ e = πo O e = P P = = = 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= pN I= pO = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 70
80. 80. CHAPTER 3. GEOMETRY = = Figure 50. = 333. e = ã O − (o − ê ) = = o 334. =â= ê = p oO 335. O = O = â O = pN ê = 336. p i = πã(o + ê ) = = 337. p = pN + pO + p i = π o O + ê O + ã(o + ê ) = = Ü 338. s = pN + pNpO + pO = P = O ÜpN  o  o   ÜpN 339. s = N+ â + âO = N + +    = P  ê ê  P   = = = O [ ( ] ) [ 71 ]
81. 81. CHAPTER 3. GEOMETRY 3.34 Sphere = o~ÇáìëW=o= aá~ãÉíÉêW=Ç= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = Figure 51. = 340. p = Qπo O = = Q N N 341. s = πo P e = πÇ P = po = P S P = = = 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= 72
82. 82. CHAPTER 3. GEOMETRY = = Figure 52. = 342. o = ê O + ÜO = OÜ = 343. p_ = πê O = = 344. p` = π(Ü O + ê O )= = 345. p = p_ + p` = π(Ü O + Oê O ) = π(OoÜ + ê O ) = = π π 346. s = Ü O (Po − Ü ) = Ü(Pê O + Ü O ) = S S = = = 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= = 73
83. 83. CHAPTER 3. GEOMETRY ====== = === = Figure 53. = 347. p = πo(OÜ + ê ) = = O 348. s = πo O Ü = P = 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= êN I= êO = eÉáÖÜíW=Ü= ^êÉ~=çÑ=ëéÜÉêáÅ~ä=ëìêÑ~ÅÉW= pp = ^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= pN I= pO = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = 74
84. 84. CHAPTER 3. GEOMETRY = ===== = Figure 54. = 349. pp = OπoÜ = = 350. p = pp + pN + pO = π(OoÜ + êNO + êOO ) = = N 351. s = πÜ(PêNO + PêOO + Ü O )= S = = = 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= p i = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 75
85. 85. CHAPTER 3. GEOMETRY = = Figure 55. = 352. p i = πo O α = Oo O ñ = VM = 353. p = πo O + πo O α = πo O + Oo O ñ = VM = 354. s = πoP O α = oP ñ = OTM P = = = 3.39 Ellipsoid = pÉãá-~ñÉëW=~I=ÄI=Å= sçäìãÉW=s= 76
86. 86. CHAPTER 3. GEOMETRY = ======= = Figure 56. = Q 355. s = π~ÄÅ = P = = = Prolate Spheroid = pÉãá-~ñÉëW=~I=ÄI=Ä=E ~ > Ä F= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = ~ ~êÅëáå É   356. p = OπÄ Ä +  I== É   ïÜÉêÉ= É = ~ O − ÄO K= ~ = Q 357. s = πÄO~ = P = 77
87. 87. CHAPTER 3. GEOMETRY Oblate Spheroid = pÉãá-~ñÉëW=~I=ÄI=Ä=E ~ < Ä F= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = =   ÄÉ   ~ ~êÅëáåÜ      ~   I== 358. p = OπÄ Ä +   ÄÉ L ~     ïÜÉêÉ= É = ÄO − ~ O K= Ä = Q 359. s = πÄO~ = P = = = 3.40 Circular Torus = j~àçê=ê~ÇáìëW=o= jáåçê=ê~ÇáìëW=ê= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = 78
88. 88. CHAPTER 3. GEOMETRY == Picture 57. = 360. p = QπOoê = = 361. s = OπOoê O = = = 79 =
89. 89. 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. N ê~Ç = = 363. N° = = 364. N D = = 365. N ? = = 366. = = = = = NUM° ≈ RT°NT DQR? = π π ê~Ç ≈ MKMNTQRP ê~Ç = NUM π ê~Ç ≈ MKMMMOVN ê~Ç = NUM ⋅ SM π ê~Ç ≈ MKMMMMMR ê~Ç = NUM ⋅ PSMM ^åÖäÉ= EÇÉÖêÉÉëF= ^åÖäÉ= Eê~Çá~åëF= M= PM= QR= SM= VM= NUM= OTM= PSM= M= π = S π = Q 80 π = P π = O π= Pπ = O Oπ =
90. 90. CHAPTER 4. TRIGONOMETRY 4.2 Definitions and Graphs of Trigonometric Functions = = = = Figure 58. = 367. ëáå α = ó = ê = 368. Åçë α = ñ = ê = 369. í~å α = ó = ñ = 370. Åçí α = ñ = ó = 81
91. 91. CHAPTER 4. TRIGONOMETRY 371. ëÉÅ α = ê = ñ = 372. ÅçëÉÅ α = ê = ó = 373. páåÉ=cìåÅíáçå= ó = ëáå ñ I= − N ≤ ëáå ñ ≤ N K= = = Figure 59. = 374. `çëáåÉ=cìåÅíáçå== ó = Åçë ñ I= − N ≤ Åçë ñ ≤ N K= 82
92. 92. CHAPTER 4. TRIGONOMETRY = = Figure 60. = 375. q~åÖÉåí=cìåÅíáçå= π ó = í~å ñ I= ñ ≠ (Oâ + N) I= − ∞ ≤ í~å ñ ≤ ∞K = O = = = Figure 61. = 83
93. 93. CHAPTER 4. TRIGONOMETRY 376. `çí~åÖÉåí=cìåÅíáçå== ó = Åçí ñ I= ñ ≠ âπ I== − ∞ ≤ Åçí ñ ≤ ∞ K= = = = Figure 62. = 377. pÉÅ~åí=cìåÅíáçå= π ó = ëÉÅ ñ I= ñ ≠ (Oâ + N) K= O == 84
94. 94. CHAPTER 4. TRIGONOMETRY = = Figure 63. = 378. `çëÉÅ~åí=cìåÅíáçå== ó = Åçë ÉÅ ñ I= ñ ≠ âπ K= = Figure 64. 85
95. 95. CHAPTER 4. TRIGONOMETRY 4.3. Signs of Trigonometric Functions 379. = = = = 380. = nì~Çê~åí= = f= ff= fff= fs= páå α= H= H= = = `çë α= H= = = H= q~å α= H= = H= = `çí α= H= = H= = pÉÅ α= H= = = H= `çëÉÅ= α= H= H= = = = = Figure 65. = = = = = = = = = = 86
96. 96. CHAPTER 4. TRIGONOMETRY 4.4 Trigonometric Functions of Common Angles 381. = α° = α ê~Ç = M= M= π = PM= S π = QR= Q π = SM= P π = VM= O Oπ = NOM= P NUM= π= Pπ = OTM= O PSM= Oπ = = = = = = = = = = = = = = O = O P = O Åçë α = N= P = O O = O N = O N= M= P = O M= N − = O − N= − N= M= ëáå α = M= N = O í~å α = Åçí α M= ∞= N = P= P ëÉÅ α = N= O = P ÅçëÉÅ α = ∞= O= N= N= P= N = P O= O = P M= ∞= N= ∞= O= O= M= N P ∞= − N= O = P ∞= M= ∞= M= ∞= − N= N= M= ∞= N= ∞= − P= 87 − −O=
97. 97. CHAPTER 4. TRIGONOMETRY 382. = α° = α ê~Ç = π = NR= NO ëáå α = Åçë α = í~å α = Åçí α = S− O = Q S+ O = Q O− P = O+ P = R−O R = R R+O R = NU= π = NM R −N = Q NM + O R Q PS= π = R NM − O R Q R +N = Q RQ= Pπ = NM R +N = Q NM − O R Q TO= Oπ = R NM + O R Q R −N = Q TR= Rπ = NO S+ O = Q S− O = Q = = = 4.5 Most Important Formulas = 383. ëáå O α + Åçë O α = N = = 384. ëÉÅ O α − í~å O α = N = = 385. ÅëÅ O α − Åçí O α = N = = ëáå α = 386. í~å α = Åçë α 88 NM − O R R +N R +N NM − O R R +N NM − O R = NM − O R R +N = R+O R = R−O R R = O+ P = O− P =
98. 98. CHAPTER 4. TRIGONOMETRY 387. Åçí α = Åçë α = ëáå α = 388. í~å α ⋅ Åçí α = N = = N 389. ëÉÅ α = = Åçë α = N 390. ÅçëÉÅ α = = ëáå α = = = 4.6 Reduction Formulas = 391. = = = = = = = β= −α= VM° − α = VM° + α = NUM° − α NUM° + α OTM° − α OTM° + α PSM° − α = PSM° + α ëáå β = − ëáå α = + Åçë α = + Åçë α = + ëáå α = − ëáå α = − Åçë α = − Åçë α = − ëáå α = + ëáå α = 89 Åçë β = + Åçë α = + ëáå α = − ëáå α = − Åçë α = − Åçë α = − ëáå α = + ëáå α = + Åçë α = + Åçë α = í~å β = − í~å α = + Åçí α = − Åçí α = − í~å α = + í~å α = + Åçí α = − Åçí α = − í~å α = + í~å α = Åçí β = − Åçí α = + í~å α = − í~å α = − Åçí α = + Åçí α = + í~å α = − í~å α = − Åçí α = + Åçí α =
99. 99. CHAPTER 4. TRIGONOMETRY 4.7 Periodicity of Trigonometric Functions = 392. ëáå(α ± Oπå ) = ëáå α I=éÉêáçÇ= Oπ =çê= PSM° K= = 393. Åçë(α ± Oπå ) = Åçë α I=éÉêáçÇ= Oπ =çê= PSM° K= = 394. í~å(α ± πå ) = í~å α I=éÉêáçÇ= π =çê= NUM° K= = 395. Åçí(α ± πå ) = Åçí α I=éÉêáçÇ= π =çê= NUM° K= = = = 4.8 Relations between Trigonometric Functions = 396. ëáå α = ± N − Åçë O α = ± α O = = α N + í~å O O N (N − Åçë Oα ) = O Åçë O  α − π  − N =   O  O Q O í~å = = 397. Åçë α = ± N − ëáå O α = ± α O= = α N + í~å O O N (N + Åçë Oα ) = O Åçë O α − N = O O N − í~å O = = 398. í~å α = ëáå α ëáå Oα N − Åçë Oα = ± ëÉÅ O α − N = = = Åçë α N + Åçë Oα ëáå Oα 90
100. 100. CHAPTER 4. TRIGONOMETRY α N − Åçë Oα O = =± = N + Åçë Oα O α N + í~å O O í~å = = Åçë α N + Åçë Oα ëáå Oα = ± ÅëÅ O α − N = = = ëáå α ëáå Oα N − Åçë Oα α N − í~å O N + Åçë Oα O= = = =± α N − Åçë Oα O í~å O 399. Åçí α = = α N O= 400. ëÉÅ α = = ± N + í~å O α = α Åçë α N − í~å O O = α N + í~å O N O= 401. ÅëÅ α = = ± N + Åçí O α = α ëáå α O í~å O = = = N + í~å O 4.9 Addition and Subtraction Formulas = 402. ëáå(α + β) = ëáå α Åçë β + ëáå β Åçë α = = 403. ëáå(α − ó ) = ëáå α Åçë β − ëáå β Åçë α = = 404. Åçë(α + β ) = Åçë α Åçë β − ëáå α ëáå β = = 405. Åçë(α − β ) = Åçë α Åçë β + ëáå α ëáå β = 91
101. 101. CHAPTER 4. TRIGONOMETRY 406. í~å(α + β ) = = 407. í~å(α − β ) = = 408. Åçí(α + β) = = 409. Åçí(α − β) = í~å α + í~å β = N − í~å α í~å β í~å α − í~å β = N + í~å α í~å β N − í~å α í~å β = í~å α + í~å β N + í~å α í~å β = í~å α − í~å β = = = 4.10 Double Angle Formulas = 410. ëáå Oα = O ëáå α ⋅ Åçë α = = 411. Åçë Oα = Åçë O α − ëáå O α = N − O ëáå O α = O Åçë O α − N = = O í~å α O 412. í~å Oα = = = O N − í~å α Åçí α − í~å α = Åçí O α − N Åçí α − í~å α = = 413. Åçí Oα = O Åçí α O = = = = = = 92
102. 102. CHAPTER 4. TRIGONOMETRY 4.11 Multiple Angle Formulas = 414. ëáå Pα = P ëáå α − Q ëáå P α = P Åçë O α ⋅ ëáå α − ëáåP α = = 415. ëáå Qα = Q ëáå α ⋅ Åçë α − U ëáå P α ⋅ Åçë α = = 416. ëáå Rα = R ëáå α − OM ëáå P α + NS ëáå R α = = 417. Åçë Pα = Q ÅçëP α − P Åçë α = Åçë P α − P Åçë α ⋅ ëáå O α = = 418. Åçë Qα = U Åçë Q α − U Åçë O α + N = = 419. Åçë Rα = NS Åçë R α − OM Åçë P α + R Åçë α = = P í~å α − í~å P α 420. í~å Pα = = N − P í~å O α = Q í~å α − Q í~å P α = 421. í~å Qα = N − S í~å O α + í~å Q α = í~å R α − NM í~å P α + R í~å α = 422. í~å Rα = N − NM í~å O α + R í~å Q α = Åçí P α − P Åçí α 423. Åçí Pα = = P Åçí O α − N = N − S í~å O α + í~å Q α == 424. Åçí Qα = Q í~å α − Q í~å P α = 93
103. 103. CHAPTER 4. TRIGONOMETRY 425. Åçí Rα = N − NM í~å O α + R í~å Q α = í~å R α − NM í~å P α + R í~å α = = = 4.12 Half Angle Formulas = 426. ëáå α N − Åçë α = =± O O = 427. Åçë α N + Åçë α = =± O O = 428. í~å α N − Åçë α ëáå α N − Åçë α =± = = = ÅëÅ α − Åçí α = O N + Åçë α N + Åçë α ëáå α = 429. Åçí α N + Åçë α ëáå α N + Åçë α =± = = = ÅëÅ α + Åçí α = O N − Åçë α N − Åçë α ëáå α = = = 4.13 Half Angle Tangent Identities = α O = 430. ëáå α = α N + í~å O O = O í~å 94
104. 104. CHAPTER 4. TRIGONOMETRY α O= 431. Åçë α = O α N + í~å O = α O í~å O = 432. í~å α = α N − í~å O O = α N − í~å O O= 433. Åçí α = α O í~å O = = = N − í~å O 4.14 Transforming of Trigonometric Expressions to Product = 434. ëáå α + ëáå β = O ëáå = 435. ëáå α − ëáå β = O Åçë α+β α −β = Åçë O O α +β α −β = ëáå O O = 436. Åçë α + Åçë β = O Åçë α+β α −β = Åçë O O = 437. Åçë α − Åçë β = −O ëáå α +β α −β = ëáå O O = 95
105. 105. CHAPTER 4. TRIGONOMETRY 438. í~å α + í~å β = = 439. í~å α − í~å β = = 440. Åçí α + Åçí β = = 441. Åçí α − Åçí β = ëáå(α + β ) = Åçë α ⋅ Åçë β ëáå(α − β ) = Åçë α ⋅ Åçë β ëáå(β + α ) = ëáå α ⋅ ëáå β ëáå(β − α ) = ëáå α ⋅ ëáå β = π  π  442. Åçë α + ëáå α = O Åçë − α  = O ëáå + α  = Q  Q  = π  π  443. Åçë α − ëáå α = O ëáå − α  = O Åçë + α  = Q  Q  = Åçë(α − β) = 444. í~å α + Åçí β = Åçë α ⋅ ëáå β = Åçë(α + β ) = 445. í~å α − Åçí β = − Åçë α ⋅ ëáå β = α 446. N + Åçë α = O Åçë O = O = α 447. N − Åçë α = O ëáå O = O = 96
106. 106. CHAPTER 4. TRIGONOMETRY π α 448. N + ëáå α = O Åçë O  −  = Q O = π α 449. N − ëáå α = O ëáå O  −  = Q O = = = 4.15 Transforming of Trigonometric Expressions to Sum = 450. ëáå α ⋅ ëáå β = Åçë(α − β) − Åçë(α + β ) = O = 451. Åçë α ⋅ Åçë β = = 452. ëáå α ⋅ Åçë β = = 453. í~å α ⋅ í~å β = = 454. Åçí α ⋅ Åçí β = = 455. í~å α ⋅ Åçí β = Åçë(α − β ) + Åçë(α + β ) = O ëáå(α − β ) + ëáå(α + β ) = O í~å α + í~å β = Åçí α + Åçí β Åçí α + Åçí β = í~å α + í~å β í~å α + Åçí β = Åçí α + í~å β = = = 97
107. 107. CHAPTER 4. TRIGONOMETRY 4.16 Powers of Trigonometric Functions = 456. ëáå O α = = 457. ëáå P α = = 458. ëáå Q α = = 459. ëáå R α = = 460. ëáå S α = = 461. Åçë O α = = 462. Åçë P α = = 463. Åçë Q α = = 464. Åçë R α = = 465. Åçë S α = N − Åçë Oα = O P ëáå α − ëáå Pα = Q Åçë Qα − Q Åçë Oα + P = U NM ëáå α − R ëáå Pα + ëáå Rα = NS NM − NR Åçë Oα + S Åçë Qα − Åçë Sα = PO N + Åçë Oα = O P Åçë α + Åçë Pα = Q Åçë Qα + Q Åçë Oα + P = U NM Åçë α + R ëáå Pα + Åçë Rα = NS NM + NR Åçë Oα + S Åçë Qα + Åçë Sα = PO = 98
108. 108. CHAPTER 4. TRIGONOMETRY 4.17 Graphs of Inverse Trigonometric Functions = 466. fåîÉêëÉ=páåÉ=cìåÅíáçå== ó = ~êÅëáå ñ I= − N ≤ ñ ≤ N I= − π π ≤ ~êÅëáå ñ ≤ K= O O = = = Figure 66. = 467. fåîÉêëÉ=`çëáåÉ=cìåÅíáçå== ó = ~êÅÅçë ñ I= − N ≤ ñ ≤ N I= M ≤ ~êÅÅçë ñ ≤ π K= = 99
109. 109. CHAPTER 4. TRIGONOMETRY = = Figure 67. = 468. fåîÉêëÉ=q~åÖÉåí=cìåÅíáçå== ó = ~êÅí~å ñ I= − ∞ ≤ ñ ≤ ∞ I= − π π < ~êÅí~å ñ < K= O O = = = ===== Figure 68. 100
110. 110. CHAPTER 4. TRIGONOMETRY 469. fåîÉêëÉ=`çí~åÖÉåí=cìåÅíáçå== ó = ~êÅ Åçí ñ I= − ∞ ≤ ñ ≤ ∞ I= M < ~êÅ Åçí ñ < π K= ===== = Figure 69. = 470. fåîÉêëÉ=pÉÅ~åí=cìåÅíáçå==  π  π  ó = ~êÅëÉÅ=ñ I ñ ∈ (− ∞I − N] ∪ [NI ∞ )I ~êÅ ëÉÅ ñ ∈ MI  ∪  I πK  O  O  = Figure 70. 101
111. 111. CHAPTER 4. TRIGONOMETRY 471. fåîÉêëÉ=`çëÉÅ~åí=cìåÅíáçå==  π   π ó = ~êÅÅëÅ ñ I ñ ∈ (− ∞I − N] ∪ [NI ∞ )I ~êÅ ÅëÅ ñ ∈ − I M  ∪  MI K  O   O = = Figure 71. = = 4.18 Principal Values of Inverse Trigonometric Functions 472. ñ= M= N = O PM° = SM° = O − O ~êÅëáå ñ = M° = ~êÅÅçë ñ = VM° N − ñ= O − PM° ~êÅëáå ñ = − QR° = NOM° ~êÅÅçë ñ = NPR° = = O = O QR° = QR° = P − O P O SM° PM° VM° M° = − N= = − VM° = NUM° NRM° = = − SM° 102 N= = =
112. 112. CHAPTER 4. TRIGONOMETRY 473. ñ= M= P P N= ~êÅí~å ñ = M° = PM° QR° SM° ~êÅ Åçí ñ = VM° SM° QR° PM° P= − P P 4.19 Relations between Inverse Trigonometric Functions = 474. ~êÅëáå(− ñ ) = − ~êÅëáå ñ = = π 475. ~êÅëáå ñ = − ~êÅÅçë ñ = O = 476. ~êÅëáå ñ = ~êÅÅçë N − ñ O I= M ≤ ñ ≤ N K= = 477. ~êÅëáå ñ = − ~êÅÅçë N − ñ O I= − N ≤ ñ ≤ M K= = ñ O I= ñ < N K= 478. ~êÅëáå ñ = ~êÅí~å O N− ñ = N− ñO I= M < ñ ≤ N K= ñ = 480. ~êÅëáå ñ = ~êÅ Åçí N− ñO − π I= − N ≤ ñ < M K= ñ = 481. ~êÅÅçë(− ñ ) = π − ~êÅÅçë ñ = 103 − P= − QR° − SM° = = NPR° NOM° = NRM° = = − PM° = = = 479. ~êÅëáå ñ = ~êÅ Åçí − N=
113. 113. CHAPTER 4. TRIGONOMETRY 482. ~êÅÅçë ñ = π − ~êÅëáå ñ = O = 483. ~êÅÅçë ñ = ~êÅëáå N − ñ O I= M ≤ ñ ≤ N K= = 484. ~êÅÅçë ñ = π − ~êÅëáå N − ñ O I= − N ≤ ñ ≤ M K= = 485. ~êÅÅçë ñ = ~êÅí~å N− ñO I= M < ñ ≤ N K= ñ = N− ñO I= − N ≤ ñ < M K= ñ 486. ~êÅÅçë ñ = π + ~êÅí~å = 487. ~êÅÅçë ñ = ~êÅ Åçí ñ N− ñO I= − N ≤ ñ ≤ N K= = 488. ~êÅí~å(− ñ ) = − ~êÅí~å ñ = = π 489. ~êÅí~å ñ = − ~êÅ Åçí ñ = O = ñ = 490. ~êÅí~å ñ = ~êÅëáå N+ ñO = N I= ñ ≥ M K= 491. ~êÅí~å ñ = ~êÅÅçë N+ ñO = N I= ñ ≤ M K= 492. ~êÅí~å ñ = − ~êÅÅçë N+ ñO = 104
114. 114. CHAPTER 4. TRIGONOMETRY 493. ~êÅí~å ñ = π N − ~êÅí~å I= ñ > M K= O ñ = π N 494. ~êÅí~å ñ = − − ~êÅí~å I= ñ < M K= O ñ = N 495. ~êÅí~å ñ = ~êÅ Åçí I= ñ > M K= ñ = N 496. ~êÅí~å ñ = ~êÅ Åçí − π I= ñ < M K= ñ = 497. ~êÅ Åçí(− ñ ) = π − ~êÅ Åçí ñ = = π 498. ~êÅ Åçí ñ = − ~êÅí~å ñ = O = N I= ñ > M K= 499. ~êÅ Åçí ñ = ~êÅëáå N+ ñO = N I= ñ < M K= 500. ~êÅ Åçí ñ = π − ~êÅëáå N+ ñO = ñ = 501. ~êÅ Åçí ñ = ~êÅÅçë N+ ñO = N 502. ~êÅ Åçí ñ = ~êÅí~å I= ñ > M K= ñ = N 503. ~êÅ Åçí ñ = π + ~êÅí~å I= ñ < M K= ñ = = 105
115. 115. CHAPTER 4. TRIGONOMETRY 4.20 Trigonometric Equations 504. 505. 506. 507. = tÜçäÉ=åìãÄÉêW=å= = = å ëáå ñ = ~ I= ñ = (− N) ~êÅëáå ~ + πå = = Åçë ñ = ~ I= ñ = ± ~êÅÅçë ~ + Oπå = = í~å ñ = ~ I= ñ = ~êÅí~å ~ + πå = = Åçí ñ = ~ I= ñ = ~êÅ Åçí ~ + πå = = = = 4.21 Relations to Hyperbolic Functions 508. 509. 510. 511. 512. = fã~Öáå~êó=ìåáíW=á= = = ëáå(áñ ) = á ëáåÜ ñ = = í~å(áñ ) = á í~åÜ ñ = = Åçí(áñ ) = −á ÅçíÜ ñ = = ëÉÅ(áñ ) = ëÉÅÜ ñ = = ÅëÅ(áñ ) = −á ÅëÅÜ ñ = = = = 106
116. 116. Chapter 5 Matrices and Determinants = = = = j~íêáÅÉëW=^I=_I=`= bäÉãÉåíë=çÑ=~=ã~íêáñW= ~ á I= Äá I= ~ áà I= Äáà I= Å áà = aÉíÉêãáå~åí=çÑ=~=ã~íêáñW= ÇÉí ^ = jáåçê=çÑ=~å=ÉäÉãÉåí= ~ áà W= j áà = `çÑ~Åíçê=çÑ=~å=ÉäÉãÉåí= ~ áà W= ` áà = ú qê~åëéçëÉ=çÑ=~=ã~íêáñW= ^ q I= ^ = ^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= ñ á = k~íìê~ä=åìãÄÉêëW=ãI=å=== = = 5.1 Determinants = 513. pÉÅçåÇ=lêÇÉê=aÉíÉêãáå~åí= ~ ÄN ÇÉí ^ = N = ~ N Ä O − ~ O ÄN = ~ O ÄO = = = = = 107
117. 117. CHAPTER 5. MATRICES AND DETERMINANTS 514. qÜáêÇ=lêÇÉê=aÉíÉêãáå~åí= ~NN ~NO ~NP ÇÉí ^ = ~ ON ~ OO ~ OP = ~NN~ OO~ PP + ~NO~ OP~ PN + ~ NP~ ON~ PO − = ~ PN ~ PO ~ PP − ~NN~ OP~ PO − ~NO~ ON~ PP − ~ NP~ OO~ PN = = 515. p~êêìë=oìäÉ=E^êêçï=oìäÉF= = = Figure 72. = 516. k-íÜ=lêÇÉê=aÉíÉêãáå~åí= ~NN ~NO K ~Nà ~ ON ~ OO K ~ O à K K K K ÇÉí ^ = ~ áN ~ á O K ~ áà K K K K ~ åN ~ å O K ~ åà K ~Nå K ~ Oå K K K ~ áå = K K K ~ åå = 517. jáåçê= qÜÉ=ãáåçê= j áà =~ëëçÅá~íÉÇ=ïáíÜ=íÜÉ=ÉäÉãÉåí= ~ áà =çÑ=å-íÜ=çêÇÉê= ã~íêáñ= ^= áë= íÜÉ= (å − N) -íÜ= çêÇÉê= ÇÉíÉêãáå~åí= ÇÉêáîÉÇ= Ñêçã= íÜÉ=ã~íêáñ=^=Äó=ÇÉäÉíáçå=çÑ=áíë=á-íÜ=êçï=~åÇ=à-íÜ=ÅçäìãåK=== = 108
118. 118. CHAPTER 5. MATRICES AND DETERMINANTS 518. `çÑ~Åíçê= á +à ` áà = (− N) j áà = = 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 ÇÉí ^ = ∑ ~ áà` áà I= á = NI OI KI å K= à=N i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=à-íÜ=Åçäìãå= å ÇÉí ^ = ∑ ~ áà` áà I= à = NI OI KI å K== á =N = = = 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 ~N ÄN = == = N ÄN ÄO ~ O ÄO = 521. fÑ=íïç==êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áåíÉêÅÜ~åÖÉÇI=íÜÉ=ëáÖå=çÑ= íÜÉ=ÇÉíÉêãáå~åí=áë=ÅÜ~åÖÉÇK= ~N ÄN ~ ÄO =− O = ~ O ÄO ~N ÄN = 522. fÑ=íïç=êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áÇÉåíáÅ~äI=íÜÉ=î~äìÉ=çÑ=íÜÉ= ÇÉíÉêãáå~åí=áë=òÉêçK= ~N ~N = M= ~O ~O = 109
119. 119. CHAPTER 5. MATRICES AND DETERMINANTS 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= â~ N âÄN ~ ÄN =â N = ~ O ÄO ~ O ÄO = 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= ~N + âÄN ÄN ~N ÄN = = ~ O + âÄO ÄO ~ O ÄO = = = 5.3 Matrices = 525. aÉÑáåáíáçå= ^å= ã × å =ã~íêáñ=^=áë=~=êÉÅí~åÖìä~ê=~êê~ó=çÑ=ÉäÉãÉåíë=EåìãÄÉêë=çê=ÑìåÅíáçåëF=ïáíÜ=ã=êçïë=~åÇ=å=ÅçäìãåëK==  ~ NN ~ NO K ~ Nå  ~ ~ OO K ~ Oå   ==  ON ^ = ~ áà =  M M M    ~ ãN ~ ã O K ~ ãå  = 526. pèì~êÉ=ã~íêáñ=áë=~=ã~íêáñ=çÑ=çêÇÉê= å× å K== = 527. ^=ëèì~êÉ=ã~íêáñ== ~ áà ==áë==ëóããÉíêáÅ==áÑ== ~ áà = ~ àá I==áKÉK==áí==áë= [ ] [ ] ëóããÉíêáÅ=~Äçìí=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK== = 528. ^=ëèì~êÉ=ã~íêáñ= ~ áà =áë=ëâÉï-ëóããÉíêáÅ=áÑ= ~ áà = −~ àá K== = [ ] 110
120. 120. CHAPTER 5. MATRICES AND DETERMINANTS 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== ã × å ==~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= ã × å K=fÑ==  ~NN ~NO K ~Nå  ~ ~ OO K ~ Oå   I== ^ = ~ áà =  ON  M M M    ~ ãN ~ ã O K ~ ãå   ÄNN ÄNO K ÄNå  Ä ÄOO K ÄOå   I== _ = Äáà =  ON  M M M    ÄãN Äã O K Äãå  = = = = = [ ] [ ] 111
121. 121. CHAPTER 5. MATRICES AND DETERMINANTS íÜÉå== ~NO + ÄNO K ~Nå + ÄNå   ~NN + ÄNN ~ +Ä ~ OO + ÄOO K ~ Oå + ÄOå   K=  ON ON ^+_=   M M M   ~ ãN + ÄãN ~ ã O + Äã O K ~ ãå + Äãå  = 534. fÑ=â=áë=~=ëÅ~ä~êI=~åÇ= ^ = ~ áà =áë=~=ã~íêáñI=íÜÉå= [ ]  â~NN â~NO K â~Nå   â~ â~ OO K â~ Oå   K=  ON â^ = â~ áà =  M M M    â~ ãN â~ ã O K â~ ãå  = 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=  ~NN ~NO K ~Nå  ~ ~ OO K ~ Oå   I== ^ = ~ áà =  ON  M M M    ~ ãN ~ ã O K ~ ãå   ÄNN ÄNO K ÄNâ  Ä ÄOO K ÄO â   I= _ = Äáà =  ON  M M M    ÄåN Äå O K Äåâ  = = = = = [ ] [ ] [ ] 112
122. 122. CHAPTER 5. MATRICES AND DETERMINANTS íÜÉå==  ÅNN ÅNO K ÅNâ  Å Å OO K Å O â   I==  ON ^_ = ` =  M M M    Ä ãN Å ã O K Å ãâ  ïÜÉêÉ== å Å áà = ~ áNÄNà + ~ á O ÄO à + K + ~ áå Äåà = ∑ ~ á λ Äλ à = E á = NI OI KI ã X à = NI OI KI â FK== = qÜìë=áÑ= [ ] ~ NN ^ = ~ áà =  ~ ON ~ NO ~ OO λ =N  ÄN  ~ NP     I= _ = [Ä á ] = Ä O  I== ~ OP   ÄP    íÜÉå== ~ NN ~ NO ^_ =  ~ ON ~ OO Ä  ~ NP   N  ~ NNÄN ⋅ Ä = ~ OP   O  ~ ONÄN  Ä    P ~ NO Ä O ~ OO Ä O ~ NP ÄP  K== ~ OP ÄP   = 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= ^^ q = f K== = 538. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå== (^_ )q = _ q ^ q K= = = 113
123. 123. CHAPTER 5. MATRICES AND DETERMINANTS 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= [ ] ~Çà ^ = ` áà 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= íê ^ = ~NN + ~ OO + K + ~ åå 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= ~Çà ^ ^ −N = K= ÇÉí ^ = 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 = λu I== ïÜáäÉ=íÜÉ=ÉáÖÉåî~äìÉë= λ =ë~íáëÑó=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå= ^ − λf = M K=== = = = q 5.5 Systems of Linear Equations = = s~êá~ÄäÉëW=ñI=óI=òI= ñ N I= ñ O I K = oÉ~ä=åìãÄÉêëW= ~ N I ~ O I ~ P I ÄN I ~ NN I ~ NO I K = 114
124. 124. CHAPTER 5. MATRICES AND DETERMINANTS aÉíÉêãáå~åíëW=aI= añ I= aó I= aò == j~íêáÅÉëW=^I=_I=u= = = ~ ñ + ÄNó = ÇN I== 544.  N ~ O ñ + ÄO ó = Ç O aó a =E`ê~ãÉê∞ë=êìäÉFI== ñ = ñ I= ó = a a ïÜÉêÉ== ~ ÄN a= N = ~NÄO − ~ O ÄN I== ~ O ÄO Ç ÄN añ = N = ÇNÄO − Ç O ÄN I== Ç O ÄO ~ ÇN aó = N = ~NÇ O − ~ OÇN K== ~ O ÇO = 545. fÑ= a ≠ M I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW== aó a K= ñ = ñ I= ó = a a fÑ= a = M = ~åÇ= añ ≠ M Eçê= aó ≠ M FI= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = åç== ëçäìíáçåK= fÑ= a = añ = aó = M I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = áåÑáåáíÉäó= = ã~åó== ëçäìíáçåëK= = ~Nñ + ÄNó + ÅNò = ÇN=  546. ~ O ñ + ÄO ó + Å Oò = Ç O I== ~ ñ + Ä ó + Å ò = Ç P P P  P ñ= aó añ a I= ó = I= ò = ò =E`ê~ãÉê∞ë=êìäÉFI== a a a = 115
125. 125. CHAPTER 5. MATRICES AND DETERMINANTS ïÜÉêÉ== ~N ÄN a = ~ O ÄO ~ P ÄP ÅN ÇN ÄN ÅN Å O I= añ = Ç O ÄO Å O I= ÅP ÄP ÅP ÇP ~N ÇN ÅN ~N ÄN ÇN aó = ~ O ~P ÇO ÇP Å O I= aò = ~ O ÅP ~P ÄO ÄP Ç O K== ÇP = 547. fÑ= a ≠ M I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW== aó a a I= ò = ò K= ñ = ñ I= ó = a a a fÑ= a = M =~åÇ= añ ≠ M Eçê= aó ≠ M =çê= aò ≠ M FI=íÜÉå=íÜÉ=ëóëíÉã= Ü~ë=åç=ëçäìíáçåK= fÑ= a = añ = aó = aò = M 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== ~NNñ N + ~ NO ñ O + K + ~ Nå ñ å = ÄN ~ ñ + ~ ñ + K + ~ ñ = Ä  ON N OO O Oå å O =  KKKKKKKKKKKK  ~ åNñ N + ~ å O ñ O + K + ~ åå ñ å = Äå  Å~å=ÄÉ=ïêáííÉå=áå=ã~íêáñ=Ñçêã=  ~ NN ~ NO K ~ Nå   ñ N   ÄN         ~ ON ~ OO K ~ Oå   ñ O   Ä O  I== = ⋅  M M M   M   M             ~  åN ~ å O K ~ åå   ñ å   Ä å  áKÉK== ^ ⋅ u = _ I== 116
126. 126. CHAPTER 5. MATRICES AND DETERMINANTS ïÜÉêÉ==  ~ NN  ~ ^ =  ON M  ~  åN ~ NO K ~ Nå   ñN   ÄN       ~ OO K ~ Oå   ñO  Ä  I= u =   I= _ =  O  K== M M  M M       ñ  Ä  ~ å O K ~ åå   å  å = 549. pçäìíáçå=çÑ=~=pÉí=çÑ=iáåÉ~ê=bèì~íáçåë= å × å = u = ^ −N ⋅ _ I== ïÜÉêÉ= ^ −N =áë=íÜÉ=áåîÉêëÉ=çÑ=^K= = = 117
127. 127. Chapter 6 Vectors = = = = r r r r → sÉÅíçêëW= ì I= î I= ï I= ê I= ^_ I=£= r r sÉÅíçê=äÉåÖíÜW= ì I= î I=£= r r r råáí=îÉÅíçêëW= á I= à I= â = r kìää=îÉÅíçêW= M = r `ççêÇáå~íÉë=çÑ=îÉÅíçê= ì W= uN I vN I wN = r `ççêÇáå~íÉë=çÑ=îÉÅíçê= î W= u O I vO I wO = pÅ~ä~êëW= λ I µ = aáêÉÅíáçå=ÅçëáåÉëW= Åçë α I= Åçë β I= Åçë γ = ^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW= θ = = = 6.1 Vector Coordinates = 550. råáí=sÉÅíçêë= r á = (NI MI M) I= r à = (MI NI M) I= r â = (MI MI N) I= r r r á = à = â = N K= = r r r r → 551. ê = ^_ = (ñ N − ñ M ) á + (ó N − ó M ) à + (ò N − ò M ) â = = 118
128. 128. CHAPTER 6. VECTORS ======= = = Figure 73. = → r ê = ^_ = 552. (ñ N − ñ M )O + (óN − ó M )O + (òN − ò M )O = = → → r r 553. fÑ= ^_ = ê I=íÜÉå= _^ = − ê K= = = = Figure 74. r 554. u = ê Åçë α I= r v = ê Åçë β I= r w = ê Åçë γ K= = 119
129. 129. CHAPTER 6. VECTORS = ===== = Figure 75. = r r 555. fÑ= ê (uI v I w ) = êN (uN I vN I wN ) I=íÜÉå== u = uN I= v = vN I= w = wN K== == = 6.2 Vector Addition = r r r 556. ï = ì + î = = = == = Figure 76. 120
130. 130. CHAPTER 6. VECTORS = == = Figure 77. = r r r r r 557. ï = ìN + ì O + ìP + K + ì å = = = = == Figure 78. = 558. `çããìí~íáîÉ=i~ï= r r r r ì+ î =î+ì= = 559. ^ëëçÅá~íáîÉ=i~ï= r r r r r r (ì + î ) + ï = ì + (î + ï ) = = r r 560. ì + î = (uN + u O I vN + vO I wN + wO ) = = = = = = = 121
131. 131. CHAPTER 6. VECTORS 6.3 Vector Subtraction = r r r r r r 561. ï = ì − î =áÑ= î + ï = ì K= = = = Figure 79. = = == = Figure 80. = r r r r 562. ì − î = ì + (− î ) = = r r r 563. ì − ì = M = (MI MI M ) = = r 564. M = M = = r r 565. ì − î = (uN − u O I vN − vO I wN − w O ) I== = = = 6.4 Scaling Vectors = r r 566. ï = λì = 122
132. 132. CHAPTER 6. VECTORS = = Figure 81. = 567. r r ï = λ⋅ì= = r 568. λì = (λuI λv I λw ) = = r r 569. λì = ìλ = = r r r 570. (λ + µ ) ì = λì + µì = = r r r 571. λ(µì ) = µ(λì ) = (λµ )ì = = r r r r 572. λ(ì + î ) = λì + λî = = = = 6.5 Scalar Product = r r 573. pÅ~ä~ê=mêçÇìÅí=çÑ=sÉÅíçêë= ì =~åÇ î = r r r r ì ⋅ î = ì ⋅ î ⋅ Åçë θ I== r r ïÜÉêÉ= θ =áë=íÜÉ=~åÖäÉ=ÄÉíïÉÉå=îÉÅíçêë= ì =~åÇ î K==== = 123
133. 133. CHAPTER 6. VECTORS = = = Figure 82. = 574. pÅ~ä~ê=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã= r r fÑ= ì = (uN I vN I wN ) I= î = (u O I vO I w O ) I=íÜÉå== r r ì ⋅ î = uNu O + vNvO + wNwO K= = 575. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë== r r fÑ= ì = (uN I vN I wN ) I= î = (u O I vO I w O ) I=íÜÉå== uNu O + vNvO + wNw O K= Åçë θ = O O O O O O uN + vN + wN u O + vO + w O = 576. `çããìí~íáîÉ=mêçéÉêíó= r r r r ì⋅î = î ⋅ì= = 577. ^ëëçÅá~íáîÉ=mêçéÉêíó= r r r r (λì ) ⋅ (µî ) = λµì ⋅ î = = 578. aáëíêáÄìíáîÉ=mêçéÉêíó= r r r r r r r ì ⋅ (î + ï ) = ì ⋅ î + ì ⋅ ï = = π r r r r 579. ì ⋅ î = M =áÑ= ì I î =~êÉ=çêíÜçÖçå~ä=E θ = FK= O = π r r 580. ì ⋅ î > M =áÑ= M < θ < K= O = 124
134. 134. CHAPTER 6. VECTORS π r r 581. ì ⋅ î < M =áÑ= < θ < π K= O = r r r r 582. ì ⋅ î ≤ ì ⋅ î = = r r r r r r 583. ì ⋅ î = ì ⋅ î =áÑ= ì I î =~êÉ=é~ê~ääÉä=E θ = M FK= = r 584. fÑ= ì = (uN I vN I wN ) I=íÜÉå== r r r rO O O O ì ⋅ ì = ì O = ì = uN + vN + wN K= = r r r r r r 585. á ⋅ á = à ⋅ à = â ⋅ â = N = = r r r r r r 586. á ⋅ à = à ⋅ â = â ⋅ á = M = = = = 6.6 Vector Product = r r 587. sÉÅíçê=mêçÇìÅí=çÑ=sÉÅíçêë= ì =~åÇ î = r r r ì × î = ï I=ïÜÉêÉ== π r r r • ï = ì ⋅ î ⋅ ëáå θ I=ïÜÉêÉ= M ≤ θ ≤ X= O r r r r • ï ⊥ì= ~åÇ= ï ⊥ î X= r r r • =sÉÅíçêë= ì I= î I= ï =Ñçêã=~=êáÖÜí-Ü~åÇÉÇ=ëÅêÉïK= = 125
135. 135. CHAPTER 6. VECTORS = ======= = Figure 83. = r á r r r 588. ï = ì × î = u N uO r à vN vO r â wN = wO = uN wN uN vN  r r r  v wN = 589. ï = ì × î =  N I− I v w u O w O u O vO  O O   = r r r r 590. p = ì × î = ì ⋅ î ⋅ ëáå θ =EcáÖKUPF= = 591. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë=EcáÖKUPF= r r ì× î ëáå θ = r r = ì⋅î = 592. kçåÅçããìí~íáîÉ=mêçéÉêíó= r r r r ì × î = −(î × ì ) == = 593. ^ëëçÅá~íáîÉ=mêçéÉêíó= r r r r (λì )× (µî ) = λµì × î = = = 126
136. 136. CHAPTER 6. VECTORS 594. aáëíêáÄìíáîÉ=mêçéÉêíó= r r r r r r r ì × (î + ï ) = ì × î + ì × ï = = r r r r r 595. ì × î = M =áÑ= ì =~åÇ= î =~êÉ=é~ê~ääÉä=E θ = M FK= = r r r r r r r 596. á × á = à × à = â × â = M = = r r r r r r r r r 597. á × à = â I= à × â = á I= â × á = à = = = = 6.7 Triple Product 598. 599. 600. 601. = pÅ~ä~ê=qêáéäÉ=mêçÇìÅí= rr r r r r r r r r r r [ìîï ] = ì ⋅ (î × ï ) = î ⋅ (ï × ì ) = ï ⋅ (ì × î ) = = rr r r rr rr r rr r r rr rrr [ìîï ] = [ïìî ] = [îïì] = −[îìï ] = −[ïîì] = −[ìïî ] = = r r r rr r âì ⋅ (î × ï ) = â[ìîï ] = = pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã= uN vN wN r r r ì ⋅ (î × ï ) = u O vO w O I== uP vP wP ïÜÉêÉ== r r r ì = (uN I vN I wN ) I= î = (u O I vO I w O ) I= ï = (uP I vP I wP ) K== = 602. sçäìãÉ=çÑ=m~ê~ääÉäÉéáéÉÇ= r r r s = ì ⋅ (î × ï ) = = 127
137. 137. CHAPTER 6. VECTORS = ============ = Figure 84. = 603. sçäìãÉ=çÑ=móê~ãáÇ= Nr r r s = ì ⋅ (î × ï ) = S = = = Figure 85. = r r r r r r 604. fÑ== ì ⋅ (î × ï ) = M I=íÜÉå=íÜÉ=îÉÅíçêë== ì I= î I=~åÇ= ï =~êÉ=äáåÉ~êäó= r r r ÇÉéÉåÇÉåí=I=ëç= ï = λì + µî =Ñçê=ëçãÉ=ëÅ~ä~êë= λ =~åÇ= µ K== = r r r r r r 605. fÑ== ì ⋅ (î × ï ) ≠ M I=íÜÉå=íÜÉ=îÉÅíçêë== ì I= î I=~åÇ= ï =~êÉ=äáåÉ~êäó= áåÇÉéÉåÇÉåíK= = 128
138. 138. CHAPTER 6. VECTORS 606. sÉÅíçê=qêáéäÉ=mêçÇìÅí= r r r r r r r r r ì × (î × ï ) = (ì ⋅ ï )î − (ì ⋅ î )ï == = = = = = = = = 129
139. 139. Chapter 7 Analytic Geometry = = = = 7.1 One-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= ñ M I= ñ N I= ñ O I= ó M I= ó N I= ó O = oÉ~ä=åìãÄÉêW= λ == aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= = = 607. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë= Ç = ^_ = ñ O − ñ N = ñ N − ñ O = = = = Figure 86. = 608. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ = ñ + λñ O ^` I= λ = ñM = N I= λ ≠ −N K= N+ λ `_ = = ======== Figure 87. 130 =
140. 140. CHAPTER 7. ANALYTIC GEOMETRY 609. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí= ñ + ñO ñM = N I= λ = N K= O = = = 7.2 Two-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= ñ M I= ñ N I= ñ O I= ó M I= ó N I= ó O = mçä~ê=ÅççêÇáå~íÉëW= êI ϕ = 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= Ç = ^_ = = (ñ O − ñ N )O + (ó O − óN )O = = = Figure 88. 131
141. 141. CHAPTER 7. ANALYTIC GEOMETRY 611. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ = ñ + λñ O ó + λó O ñM = N I= ó M = N I== N+ λ N+ λ ^` λ= I= λ ≠ −N K= `_ = ======= = = Figure 89. = = 132
142. 142. CHAPTER 7. ANALYTIC GEOMETRY ======= = = Figure 90. = 612. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí= ñ + ñO ó + óO I= ó M = N I= λ = N K= ñM = N O O = 613. `ÉåíêçáÇ=EfåíÉêëÉÅíáçå=çÑ=jÉÇá~åëF=çÑ=~=qêá~åÖäÉ= ñ + ñ O + ñP ó + óO + óP I= ó M = N ñM = N I== P P ïÜÉêÉ== ^(ñ N I ó N ) I== _(ñ O I ó O ) I==~åÇ== `(ñ P I ó P ) ==~êÉ=îÉêíáÅÉë=çÑ= íÜÉ=íêá~åÖäÉ= ^_` K= = = 133
143. 143. CHAPTER 7. ANALYTIC GEOMETRY ========= = = Figure 91. = 614. fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ= ~ñ + Äñ O + Åñ P ~ó + Äó O + Åó P I= ó M = N ñM = N I== ~ +Ä+Å ~ +Ä+Å ïÜÉêÉ= ~ = _` I= Ä = `^ I= Å = ^_ K== = ======== = = Figure 92. 134
144. 144. CHAPTER 7. ANALYTIC GEOMETRY 615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê====================== _áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ= O O O O ñN + óN óN N ñN ñN + óN N ñO + óO óO N ñO ñO + óO N O O O O O O O O ñP + óP óP N ñP ñP + óP N ñM = I= ó M = = ñN óN N ñN óN N O ñO ñP óO N óP N O ñO ñP óO N óP N = = ======== == Figure 93. = = = = = = = 135
145. 145. CHAPTER 7. ANALYTIC GEOMETRY 616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ= O O óN ñ O ñ P + óN N ñN + ó OóP ñN N ó O ñPñN + ó O N ñ O + ó P óN ñ O N O O O O ó P ñ Nñ O + ó P N ñ P + ó Nó O ñ P N I= ó M = = ñM = ñN óN N ñN óN N ñO óO N ñO óO N ñP óP N ñP óP N = = ====== = Figure 94. = 617. ^êÉ~=çÑ=~=qêá~åÖäÉ= ñ N óN N N N ñ O − ñN p = (± ) ñ O ó O N = (± ) O O ñ P − ñN ñP óP N = = = 136 ó O − óN ó P − óN =
146. 146. CHAPTER 7. ANALYTIC GEOMETRY 618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä= N p = (± ) [(ñ N − ñ O )(ó N + ó O ) + (ñ O − ñ P )(ó O + ó P ) + = O + (ñ P − ñ Q )(ó P + ó Q ) + (ñ Q − ñ N )(ó Q + ó N )] = = === = = 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= Ç = ^_ = êNO + êOO − OêNêO Åçë(ϕ O − ϕN ) = = 137
147. 147. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 96. = 620. `çåîÉêíáåÖ=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=íç=mçä~ê=`ççêÇáå~íÉë= ñ = ê Åçë ϕ I= ó = ê ëáå ϕ K= = = = Figure 97. = 621. `çåîÉêíáåÖ=mçä~ê=`ççêÇáå~íÉë=íç=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë= ó ê = ñ O + ó O I= í~å ϕ = K= ñ 138
148. 148. CHAPTER 7. ANALYTIC GEOMETRY 7.3 Straight Line in Plane = mçáåí=ÅççêÇáå~íÉëW=uI=vI=ñI= ñ M I= ñ N I== ó M I= ó N I= ~N I= ~ O I=£== oÉ~ä=åìãÄÉêëW=âI=~I=ÄI=éI=íI=^I=_I=`I= ^N I= ^ O I=£= ^åÖäÉëW= α I= β = ^åÖäÉ=ÄÉíïÉÉå=íïç=äáåÉëW= ϕ = r kçêã~ä=îÉÅíçêW= å = r r r mçëáíáçå=îÉÅíçêëW= ê I= ~ I= Ä = = = 622. dÉåÉê~ä=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ= ^ñ + _ó + ` = M = = 623. kçêã~ä=sÉÅíçê=íç=~=píê~áÖÜí=iáåÉ= r qÜÉ=îÉÅíçê= å(^I _ ) =áë=åçêã~ä=íç=íÜÉ=äáåÉ= ^ñ + _ó + ` = M K= = = = Figure 98. = 624. bñéäáÅáí=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=EpäçéÉ-fåíÉêÅÉéí=cçêãF= ó = âñ + Ä K== 139
149. 149. CHAPTER 7. ANALYTIC GEOMETRY qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= â = í~å α K= = = = Figure 99. = 625. dê~ÇáÉåí=çÑ=~=iáåÉ== ó − óN â = í~å α = O = ñ O − ñN = = = Figure 100. 140
150. 150. CHAPTER 7. ANALYTIC GEOMETRY 626. bèì~íáçå=çÑ=~=iáåÉ=dáîÉå=~=mçáåí=~åÇ=íÜÉ=dê~ÇáÉåí= ó = ó M + â (ñ − ñ M ) I== ïÜÉêÉ=â=áë=íÜÉ=Öê~ÇáÉåíI= m(ñ M I ó M ) =áë=~=éçáåí=çå=íÜÉ=äáåÉK= = = = Figure 101. = 627. bèì~íáçå=çÑ=~=iáåÉ=qÜ~í=m~ëëÉë=qÜêçìÖÜ=qïç=mçáåíë= ó − óN ñ − ñN = == ó O − óN ñ O − ñN çê= ñ ó N ñ N ó N N = M K= ñO óO N = 141
151. 151. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 102. = 628. fåíÉêÅÉéí=cçêã= ñ ó + =N= ~ Ä = = = Figure 103. = = 142
152. 152. CHAPTER 7. ANALYTIC GEOMETRY 629. kçêã~ä=cçêã= ñ Åçë β + ó ëáå β − é = M = = = = Figure 104. = 630. mçáåí=aáêÉÅíáçå=cçêã= ñ − ñ N ó − óN = I== u v ïÜÉêÉ= (uI v ) = áë= íÜÉ= ÇáêÉÅíáçå= çÑ= íÜÉ= äáåÉ= ~åÇ= mN (ñ N I ó N ) = äáÉë= çå=íÜÉ=äáåÉK= = 143
153. 153. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 105. = 631. sÉêíáÅ~ä=iáåÉ= ñ =~= = 632. eçêáòçåí~ä=iáåÉ= ó=Ä= = 633. sÉÅíçê=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ= r r r ê = ~ + íÄ 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= r Ä =áë=~=âåçïå=îÉÅíçê=çÑ=ÇáêÉÅíáçåI=é~ê~ääÉä=íç=íÜÉ=äáåÉI== í=áë=~=é~ê~ãÉíÉêI== r → ê = lu =áë=íÜÉ=éçëáíáçå=îÉÅíçê=çÑ=~åó=éçáåí=u=çå=íÜÉ=äáåÉK== = 144
154. 154. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 106. = 634. píê~áÖÜí=iáåÉ=áå=m~ê~ãÉíêáÅ=cçêã= ñ = ~N + íÄN I==  ó = ~ O + íÄO ïÜÉêÉ== (ñ I ó ) ~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~åó=ìåâåçïå=éçáåí=çå=íÜÉ=äáåÉI== (~N I ~ O ) =~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~=âåçïå=éçáåí=çå=íÜÉ=äáåÉI== (ÄN I ÄO ) =~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~=îÉÅíçê=é~ê~ääÉä=íç=íÜÉ=äáåÉI== í=áë=~=é~ê~ãÉíÉêK= = 145
155. 155. CHAPTER 7. ANALYTIC GEOMETRY = Figure 107. = 635. aáëí~åÅÉ=cêçã=~=mçáåí=qç=~=iáåÉ= qÜÉ=Çáëí~åÅÉ=Ñêçã=íÜÉ=éçáåí= m(~ I Ä) =íç=íÜÉ=äáåÉ= ^ñ + _ó + ` = M =áë== ^~ + _Ä + ` K= Ç= ^ O + _O = = = Figure 108. 146
156. 156. CHAPTER 7. ANALYTIC GEOMETRY 636. m~ê~ääÉä=iáåÉë= qïç=äáåÉë= ó = â Nñ + ÄN =~åÇ= ó = â O ñ + ÄO =~êÉ=é~ê~ääÉä=áÑ== â N = â O K= qïç= äáåÉë= ^Nñ + _Nó + `N = M = ~åÇ= ^ O ñ + _O ó + ` O = M = ~êÉ= é~ê~ääÉä=áÑ= ^N _N = K= ^ O _O = = = Figure 109. = 637. mÉêéÉåÇáÅìä~ê=iáåÉë= qïç=äáåÉë= ó = â Nñ + ÄN =~åÇ= ó = â O ñ + ÄO =~êÉ=éÉêéÉåÇáÅìä~ê=áÑ== N â O = − =çêI=Éèìáî~äÉåíäóI= â Nâ O = −N K= âN qïç= äáåÉë= ^Nñ + _Nó + `N = M = ~åÇ= ^ O ñ + _ O ó + ` O = M = ~êÉ= éÉêéÉåÇáÅìä~ê=áÑ= ^N^ O + _N_ O = M K= = 147
157. 157. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 110. = 638. ^åÖäÉ=_ÉíïÉÉå=qïç=iáåÉë= â − âN í~å ϕ = O I== N + â Nâ O ^N^ O + _N_ O Åçë ϕ = K= O O ^N + _N ⋅ ^ O + _ O O O = 148
158. 158. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 111. = 639. fåíÉêëÉÅíáçå=çÑ=qïç=iáåÉë= fÑ=íïç=äáåÉë= ^Nñ + _Nó + `N = M =~åÇ= ^ O ñ + _ O ó + ` O = M =áåíÉêëÉÅíI=íÜÉ=áåíÉêëÉÅíáçå=éçáåí=Ü~ë=ÅççêÇáå~íÉë= − `N_ O + ` O_N − ^N` O + ^ O`N ñM = I= ó M = K= ^N_ O − ^ O_N ^N_ O − ^ O_N = = = 7.4 Circle = o~ÇáìëW=o= `ÉåíÉê=çÑ=ÅáêÅäÉW= (~ I Ä) = mçáåí=ÅççêÇáå~íÉëW=ñI=óI= ñ N I= ó N I=£= oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í= 149
159. 159. CHAPTER 7. ANALYTIC GEOMETRY 640. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ= cçêãF= ñ O + ó O = oO = ====== = = Figure 112. = 641. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=^åó=mçáåí= (~I Ä) (ñ − ~ )O + (ó − Ä)O = o O Figure 113. 150
160. 160. CHAPTER 7. ANALYTIC GEOMETRY 642. qÜêÉÉ=mçáåí=cçêã ñO + óO ñ ó N O O ñN + óN ñN óN N =M ñO + óO ñO óO N O O O O ñP + óP ñP óP N = = = Figure 114. = 643. m~ê~ãÉíêáÅ=cçêã ñ = o Åçë í I= M ≤ í ≤ Oπ K  ó = o ëáå í = 644. dÉåÉê~ä=cçêã ^ñ O + ^ó O + añ + bó + c = M =E^=åçåòÉêçI= aO + b O > Q ^c FK== qÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ÅáêÅäÉ=Ü~ë=ÅççêÇáå~íÉë= (~ I Ä) I=ïÜÉêÉ== a b ~=− I= Ä = − K= O^ O^ qÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅäÉ=áë 151
161. 161. CHAPTER 7. ANALYTIC GEOMETRY o= aO + b O − Q ^c K O^ = = = 7.5 Ellipse = pÉãáã~àçê=~ñáëW=~= pÉãáãáåçê=~ñáëW=Ä= cçÅáW= cN (− ÅI M) I= cO (ÅI M) = aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= = bÅÅÉåíêáÅáíóW=É== oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = 645. bèì~íáçå=çÑ=~å=bääáéëÉ=Epí~åÇ~êÇ=cçêãF ñO óO + =N ~ O ÄO = = Figure 115. 152
162. 162. CHAPTER 7. ANALYTIC GEOMETRY 646. êN + êO = O~ I= ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó==éçáåí== m(ñ I ó ) ==çå= íÜÉ=ÉääáéëÉ=íç=íÜÉ=íïç=ÑçÅáK= = = = Figure 116. = 647. ~ O = ÄO + Å O = 648. bÅÅÉåíêáÅáíó Å É = <N= ~ = 649. bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë ~ ~O ñ=± =± = É Å = 650. m~ê~ãÉíêáÅ=cçêã ñ = ~ Åçë í I= M ≤ í ≤ Oπ K  ó = Ä ëáå í = = 153
163. 163. CHAPTER 7. ANALYTIC GEOMETRY 651. dÉåÉê~ä=cçêã ^ñ O + _ñó + `ó O + añ + bó + c = M I== ïÜÉêÉ= _ O − Q ^` < M K= = 652. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë ^ñ O + `ó O + añ + bó + c = M I== ïÜÉêÉ= ^` > M K = 653. `áêÅìãÑÉêÉåÅÉ i = Q~b(É ) I== ïÜÉêÉ==íÜÉ==ÑìåÅíáçå=b==áë==íÜÉ=ÅçãéäÉíÉ==ÉääáéíáÅ=áåíÉÖê~ä==çÑ= íÜÉ=ëÉÅçåÇ=âáåÇK== = 654. ^ééêçñáã~íÉ=cçêãìä~ë=çÑ=íÜÉ=`áêÅìãÑÉêÉåÅÉ i = π NKR(~ + Ä) − ~Ä I== ( i = π O(~ O + ÄO ) K= = 655. p = π~Ä = = = = ) 7.6 Hyperbola = qê~åëîÉêëÉ=~ñáëW=~= `çåàìÖ~íÉ=~ñáëW=Ä= cçÅáW= cN (− ÅI M) I= cO (ÅI M) = aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= = bÅÅÉåíêáÅáíóW=É== ^ëóãéíçíÉëW=ëI=í= oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=íI=â= = = = 154
164. 164. CHAPTER 7. ANALYTIC GEOMETRY 656. bèì~íáçå=çÑ=~=eóéÉêÄçä~=Epí~åÇ~êÇ=cçêãF= ñO óO − = N= ~ O ÄO = = = Figure 117. = 657. êN − êO = O~ I= ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó=éçáåí== m(ñ I ó ) ==çå= íÜÉ=ÜóéÉêÄçä~=íç=íÜÉ=íïç=ÑçÅáK= = 155
165. 165. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 118. 658. 659. 660. 661. = bèì~íáçåë=çÑ=^ëóãéíçíÉë= Ä ó=± ñ= ~ = Å O = ~ O + ÄO = = bÅÅÉåíêáÅáíó Å É = > N= ~ = bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë ~ ~O ñ=± =± = É Å = = = 156
166. 166. CHAPTER 7. ANALYTIC GEOMETRY 662. m~ê~ãÉíêáÅ=bèì~íáçåë=çÑ=íÜÉ=oáÖÜí=_ê~åÅÜ=çÑ=~=eóéÉêÄçä~= ñ = ~ ÅçëÜ í I= M ≤ í ≤ Oπ K  ó = Ä ëáåÜ í = 663. dÉåÉê~ä=cçêã ^ñ O + _ñó + `ó O + añ + bó + c = M I== ïÜÉêÉ= _ O − Q ^` > M K= = 664. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë ^ñ O + `ó O + añ + bó + c = M I== ïÜÉêÉ= ^` < M K= 665. ^ëóãéíçíáÅ=cçêã= ÉO ñó = I== Q çê== ÉO â ó = I=ïÜÉêÉ= â = K= ñ Q få= íÜáë= Å~ëÉ= I= íÜÉ= ~ëóãéíçíÉë= Ü~îÉ= Éèì~íáçåë= ñ = M = ~åÇ= ó = M K== = 157
167. 167. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 119. = = = 7.7 Parabola = cçÅ~ä=é~ê~ãÉíÉêW=é= cçÅìëW=c= sÉêíÉñW= j(ñ M I ó M ) = oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=éI=~I=ÄI=Å= = = 666. bèì~íáçå=çÑ=~=m~ê~Äçä~=Epí~åÇ~êÇ=cçêãF ó O = Oéñ = 158
168. 168. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 120. = bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ é ñ = − I= O `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë= é  c I M  I= O  `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= j(MI M) K= = 667. dÉåÉê~ä=cçêã ^ñ O + _ñó + `ó O + añ + bó + c = M I== ïÜÉêÉ= _ O − Q ^` = M K= = N 668. ó = ~ñ O I= é = K= O~ bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ 159
169. 169. CHAPTER 7. ANALYTIC GEOMETRY é ó = − I= O `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=  é c MI  I=  O `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= j(MI M) K= = = = Figure 121. = 669. dÉåÉê~ä=cçêãI=^ñáë=m~ê~ääÉä=íç=íÜÉ=ó-~ñáë== ^ñ O + añ + bó + c = M =E^I=b=åçåòÉêçFI== N ó = ~ñ O + Äñ + Å I= é = K== O~ bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ é ó = ó M − I= O `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë= 160
170. 170. CHAPTER 7. ANALYTIC GEOMETRY é  c ñ M I ó M +  I= O  `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= Ä Q~Å − ÄO K= ñ M = − I= ó M = ~ñ O + Äñ M + Å = M O~ Q~ = = = Figure 122. = = = 7.8 Three-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= ñ M I= ó M I= ò M I= ñ N I= ó N I= ò N I=£= oÉ~ä=åìãÄÉêW= λ == aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= ^êÉ~W=p= sçäìãÉW=s= = 161
171. 171. CHAPTER 7. ANALYTIC GEOMETRY 670. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë= Ç = ^_ = = = (ñ O − ñ N )O + (ó O − óN )O + (ò O − òN )O = = === Figure 123. = 671. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ = ñ + λñ O ó + λó O ò + λò O ñM = N I= ó M = N I= ò M = N I== N+ λ N+ λ N+ λ ïÜÉêÉ= ^` λ= I= λ ≠ −N K= `_ = 162
172. 172. CHAPTER 7. ANALYTIC GEOMETRY ======== = = Figure 124. = = Figure 125. 163