2. 1300 Math Formulas
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3. =
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i

4. Preface
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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===
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ii

5. Contents
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1 krj_bo=pbqp=
NKN= pÉí=fÇÉåíáíáÉë==1=
NKO= pÉíë=çÑ=kìãÄÉêë==5=
NKP= _~ëáÅ=fÇÉåíáíáÉë==7=
NKQ= `çãéäÉñ=kìãÄÉêë==8=
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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=
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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

6. 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=
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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

7. 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=
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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=
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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=
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7 ^k^ivqf`=dbljbqov=
TKN= låÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==130=
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8. 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=
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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=
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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

9. VKNO= iáåÉ=fåíÉÖê~ä==275=
VKNP= pìêÑ~ÅÉ=fåíÉÖê~ä==285=
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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=
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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=
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12 mol_^_fifqv=
NOKN= mÉêãìí~íáçåë=~åÇ=`çãÄáå~íáçåë==318=
NOKO= mêçÄ~Äáäáíó=cçêãìä~ë==319=
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vii

10. =
qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK=
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viii

11. Chapter 1
Number Sets
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1.1 Set Identities
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pÉíëW=^I=_I=`=
råáîÉêë~ä=ëÉíW=f=
`çãéäÉãÉåí=W= ^′ =
mêçéÉê=ëìÄëÉíW= ^ ⊂ _ ==
bãéíó=ëÉíW= ∅ =
råáçå=çÑ=ëÉíëW= ^ ∪ _ =
fåíÉêëÉÅíáçå=çÑ=ëÉíëW= ^ ∩ _ =
aáÑÑÉêÉåÅÉ=çÑ=ëÉíëW= ^ y _ =
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1.
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2.
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3.
4.
5.
^ ⊂ f=
^ ⊂ ^=
^ = _ =áÑ= ^ ⊂ _ =~åÇ= _ ⊂ ^ .=
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bãéíó=pÉí=
∅⊂^=
=
råáçå=çÑ=pÉíë==
` = ^ ∪ _ = {ñ ö ñ ∈ ^ çê ñ ∈ _}=
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1

12. CHAPTER 1. NUMBER SETS
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Figure 1.
6.
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7.
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8.
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`çããìí~íáîáíó=
^∪_ = _∪^=
^ëëçÅá~íáîáíó=
^ ∪ (_ ∪ ` ) = (^ ∪ _ ) ∪ ` =
fåíÉêëÉÅíáçå=çÑ=pÉíë=
` = ^ ∪ _ = {ñ ö ñ ∈ ^ ~åÇ ñ ∈ _} =
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=====
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Figure 2.
9.
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10.
=
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`çããìí~íáîáíó=
^∩_ = _∩^=
^ëëçÅá~íáîáíó=
^ ∩ (_ ∩ ` ) = (^ ∩ _ ) ∩ ` =
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2

13. CHAPTER 1. NUMBER SETS
11.
=
12.
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13.
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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 ∉ ^} =
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3

14. CHAPTER 1. NUMBER SETS
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Figure 3.
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19.
_ y ^ = _ y (^ ∩ _ )
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20.
_ y ^ = _ ∩ ^′
21.
^y^=∅
22.
^ y _ = ^ =áÑ= ^ ∩ _ = ∅ .
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Figure 4.
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23.
(^ y _) ∩ ` = (^ ∩ `) y (_ ∩ `)
24.
^′ = f y ^
25.
`~êíÉëá~å=mêçÇìÅí
` = ^ × _ = {(ñ I ó ) ö ñ ∈ ^ ~åÇ ó ∈ _}
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4
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15. CHAPTER 1. NUMBER SETS
1.2 Sets of Numbers
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26.
27.
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28.
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29.
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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

16. 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
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33.
k⊂ w⊂n⊂ o ⊂ `=
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===
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Figure 5.
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6

17. CHAPTER 1. NUMBER SETS
1.3 Basic Identities
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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) =
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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

18. 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
=
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=
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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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 ∈∅ =
=
=
=

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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

48. 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

49. 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

50. 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

51. 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

52. 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

53. 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

54. 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

55. 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

56. 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

57. 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

58. 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

59. 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

60. 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

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

62. 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

63. 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

64. 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

65. 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

66. 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

67. 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

68. 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

69. 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

70. 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

71. 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

72. 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

73. 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

74. 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

75. 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

76. 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

77. 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

78. 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

79. 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

80. 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

81. 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
]

82. 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

83. 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

84. 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

85. 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

86. 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

87. 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

88. 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

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

90. 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 =

91. 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

92. 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

93. 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

94. 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

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

96. 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

97. 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=

98. 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 =

99. 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 =

100. 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

101. 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

102. 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

103. 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

104. 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

105. 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

106. 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

107. 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

108. 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

109. 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

110. 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

111. 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

112. 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=
=
=

113. 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=

114. 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

115. 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

116. 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

117. 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

118. 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

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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

120. 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

121. 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

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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

123. 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

124. 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

125. 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

126. 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

127. 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

128. 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

129. 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

130. 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

131. 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

132. 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

133. 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

134. 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

135. 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

136. 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

137. 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

138. 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

139. 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

140. 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
=

141. 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

142. 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

143. 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

144. 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

145. 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

146. 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
=

147. 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

148. 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

149. 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

150. 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

151. 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

152. CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 102.
=
628. fåíÉêÅÉéí=cçêã=
ñ ó
+ =N=
~ Ä
=
=
=
Figure 103.
=
=
142

153. 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

154. 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

155. 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

156. 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

157. 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

158. 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

159. 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

160. 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

161. 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

162. 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

163. 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

164. 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

165. 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

166. 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

167. 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

168. 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

169. 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

170. 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

171. 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

172. 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

173. CHAPTER 7. ANALYTIC GEOMETRY
========
=
=
Figure 124.
=
=
Figure 125.
163