1300 Math Formulas

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Published on March 8, 2014

Author: ajooani

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1300 Math Formulas - Alex Svirin

1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë=oÉëÉêîÉÇK=

= qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK= i

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

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

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

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

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

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

= qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK= = viii

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

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

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

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 =

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ∈∅ = = = =

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

CHAPTER 3. GEOMETRY 261. ~N~ O = ÄNÄO = = = = Figure 31. = 262. ÉÉN = ÑÑN = = = = ===== Figure 32. = 263. Ö O = ÑÑN = = 51

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ]

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

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

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

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

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

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

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

CHAPTER 3. GEOMETRY == Picture 57. = 360. p = QπOoê = = 361. s = OπOoê O = = = 79 =

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 =

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

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

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

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

CHAPTER 4. TRIGONOMETRY = = Figure 63. = 378. `çëÉÅ~åí=cìåÅíáçå== ó = Åçë ÉÅ ñ I= ñ ≠ âπ K= = Figure 64. 85

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

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=

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 =

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 =

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

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

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

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

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

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

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

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

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

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

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

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

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= = =

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=

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

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

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

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

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

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=

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