1300 MATHS FORMULA.pdf

2. 1300 Math Formulas
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3. i
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4. ii
Preface
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5. iii
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=

6. iv
PKNP= háíÉ==42=
PKNQ= `óÅäáÅ=nì~Çêáä~íÉê~ä==43=
PKNR= q~åÖÉåíá~ä=nì~Çêáä~íÉê~ä==45=
PKNS= dÉåÉê~ä=nì~Çêáä~íÉê~ä==46=
PKNT= oÉÖìä~ê=eÉñ~Öçå==47=
PKNU= oÉÖìä~ê=mçäóÖçå==48=
PKNV= `áêÅäÉ==50=
PKOM= pÉÅíçê=çÑ=~=`áêÅäÉ==53=
PKON= pÉÖãÉåí=çÑ=~=`áêÅäÉ==54=
PKOO= `ìÄÉ==55=
PKOP= oÉÅí~åÖìä~ê=m~ê~ääÉäÉéáéÉÇ==56=
PKOQ= mêáëã==57=
PKOR= oÉÖìä~ê=qÉíê~ÜÉÇêçå==58=
PKOS= oÉÖìä~ê=móê~ãáÇ==59=
PKOT= cêìëíìã=çÑ=~=oÉÖìä~ê=móê~ãáÇ==61=
PKOU= oÉÅí~åÖìä~ê=oáÖÜí=tÉÇÖÉ==62=
PKOV= mä~íçåáÅ=pçäáÇë==63=
PKPM= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê==66=
PKPN= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê=ïáíÜ=~å=lÄäáèìÉ=mä~åÉ=c~ÅÉ==68=
PKPO= oáÖÜí=`áêÅìä~ê=`çåÉ==69=
PKPP= cêìëíìã=çÑ=~=oáÖÜí=`áêÅìä~ê=`çåÉ==70=
PKPQ= péÜÉêÉ==72=
PKPR= péÜÉêáÅ~ä=`~é==72=
PKPS= péÜÉêáÅ~ä=pÉÅíçê==73=
PKPT= péÜÉêáÅ~ä=pÉÖãÉåí==74=
PKPU= péÜÉêáÅ~ä=tÉÇÖÉ==75=
PKPV= bääáéëçáÇ==76=
PKQM= `áêÅìä~ê=qçêìë==78=
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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=

7. v
QKS= oÉÇìÅíáçå=cçêãìä~ë==89=
QKT= mÉêáçÇáÅáíó=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90=
QKU= oÉä~íáçåë=ÄÉíïÉÉå=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90=
QKV= ^ÇÇáíáçå=~åÇ=pìÄíê~Åíáçå=cçêãìä~ë==91=
QKNM= açìÄäÉ=^åÖäÉ=cçêãìä~ë==92=
QKNN= jìäíáéäÉ=^åÖäÉ=cçêãìä~ë==93=
QKNO= e~äÑ=^åÖäÉ=cçêãìä~ë==94=
QKNP= e~äÑ=^åÖäÉ=q~åÖÉåí=fÇÉåíáíáÉë==94=
QKNQ= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=mêçÇìÅí==95=
QKNR= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=pìã==97===
QKNS= mçïÉêë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==98=
QKNT= dê~éÜë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==99=
QKNU= mêáåÅáé~ä=s~äìÉë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==102=
QKNV= oÉä~íáçåë=ÄÉíïÉÉå=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==103=
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8. vi
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VKNN= qêáéäÉ=fåíÉÖê~ä==269=

9. vii
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10. viii
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11. 1
Chapter 1
Number Sets
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1.1 Set Identities
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12. CHAPTER 1. NUMBER SETS
2
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Figure 1.
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Figure 2.
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13. CHAPTER 1. NUMBER SETS
3
11. aáëíêáÄìíáîáíó=
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14. CHAPTER 1. NUMBER SETS
4
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Figure 3.
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15. CHAPTER 1. NUMBER SETS
5
1.2 Sets of Numbers
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tÜçäÉ=åìãÄÉêëW= M
k =
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w =
kÉÖ~íáîÉ=áåíÉÖÉêëW= −
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K
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kM = K=
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{ }
K
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{ }
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K
K I
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29. o~íáçå~ä=kìãÄÉêë
oÉéÉ~íáåÖ=çê=íÉêãáå~íáåÖ=ÇÉÅáã~äëW==
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30. fêê~íáçå~ä=kìãÄÉêë
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16. CHAPTER 1. NUMBER SETS
6
31. oÉ~ä=kìãÄÉêë==
råáçå=çÑ=ê~íáçå~ä=~åÇ=áêê~íáçå~ä=åìãÄÉêëW=oK=
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17. CHAPTER 1. NUMBER SETS
7
1.3 Basic Identities
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=
=
34. ^ÇÇáíáîÉ=fÇÉåíáíó=
~
M
~ =
+ =
=
35. ^ÇÇáíáîÉ=fåîÉêëÉ=
( ) M
~
~ =
−
+ =
=
36. `çããìí~íáîÉ=çÑ=^ÇÇáíáçå=
~
Ä
Ä
~ +
=
+ =
=
37. ^ëëçÅá~íáîÉ=çÑ=^ÇÇáíáçå=
( ) ( )
Å
Ä
~
Å
Ä
~ +
+
=
+
+ =
=
38. aÉÑáåáíáçå=çÑ=pìÄíê~Åíáçå=
( )
Ä
~
Ä
~ −
+
=
− =
=
39. jìäíáéäáÅ~íáîÉ=fÇÉåíáíó=
~
N
~ =
⋅ =
=
40. jìäíáéäáÅ~íáîÉ=fåîÉêëÉ=
N
~
N
~ =
⋅ I= M
~ ≠
=
41. jìäíáéäáÅ~íáçå=qáãÉë=M
M
M
~ =
⋅
=
42. `çããìí~íáîÉ=çÑ=jìäíáéäáÅ~íáçå=
~
Ä
Ä
~ ⋅
=
⋅
=
=

18. CHAPTER 1. NUMBER SETS
8
43. ^ëëçÅá~íáîÉ=çÑ=jìäíáéäáÅ~íáçå=
( ) ( )
Å
Ä
~
Å
Ä
~ ⋅
⋅
=
⋅
⋅
=
44. aáëíêáÄìíáîÉ=i~ï=
( ) ~Å
~Ä
Å
Ä
~ +
=
+ =
=
45. aÉÑáåáíáçå=çÑ=aáîáëáçå=
Ä
N
~
Ä
~
⋅
= =
=
=
=
1.4 Complex Numbers
=
k~íìê~ä=åìãÄÉêW=å=
fã~Öáå~êó=ìåáíW=á=
`çãéäÉñ=åìãÄÉêW=ò=
oÉ~ä=é~êíW=~I=Å=
fã~Öáå~êó=é~êíW=ÄáI=Çá=
jçÇìäìë=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=êI= N
ê I= O
ê =
^êÖìãÉåí=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=ϕ I= N
ϕ I= O
ϕ =
=
=
á
áN
= = á
áR
= = á
á N
å
Q
=
+
=
N
áO
−
= = N
áS
−
= = N
á O
å
Q
−
=
+
=
á
áP
−
= = á
áT
−
= = á
á P
å
Q
−
=
+
=
46.
N
áQ
= = N
áU
= = N
á å
Q
= =
=
47. Äá
~
ò +
= =
=
48. `çãéäÉñ=mä~åÉ=
=

19. CHAPTER 1. NUMBER SETS
9
===== =
=
Figure 6.
=
49. ( ) ( ) ( ) ( )á
Ç
Ä
Å
~
Çá
Å
Äá
~ +
+
+
=
+
+
+ =
=
50. ( ) ( ) ( ) ( )á
Ç
Ä
Å
~
Çá
Å
Äá
~ −
+
−
=
+
−
+ =
=
51. ( )( ) ( ) ( )á
ÄÅ
~Ç
ÄÇ
~Å
Çá
Å
Äá
~ +
+
−
=
+
+ =
=
52. á
Ç
Å
~Ç
ÄÅ
Ç
Å
ÄÇ
~Å
Çá
Å
Äá
~
O
O
O
O
⋅
+
−
+
+
+
=
+
+
=
=
53. `çåàìÖ~íÉ=`çãéäÉñ=kìãÄÉêë=
Äá
~
Äá
~
|||||||
−
=
+ =
=
54. ϕ
= Åçë
ê
~ I= ϕ
= ëáå
ê
Ä ==
=

20. CHAPTER 1. NUMBER SETS
10
=
=
Figure 7.
=
55. mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë=
( )
ϕ
+
ϕ
=
+ ëáå
á
Åçë
ê
Äá
~ =
=
56. jçÇìäìë=~åÇ=^êÖìãÉåí=çÑ=~=`çãéäÉñ=kìãÄÉê=
fÑ= Äá
~ + =áë=~=ÅçãéäÉñ=åìãÄÉêI=íÜÉå=
O
O
Ä
~
ê +
= =EãçÇìäìëFI==
~
Ä
~êÅí~å
=
ϕ =E~êÖìãÉåíFK=
=
57. mêçÇìÅí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
( ) ( )
O
O
O
N
N
N
O
N ëáå
á
Åçë
ê
ëáå
á
Åçë
ê
ò
ò ϕ
+
ϕ
⋅
ϕ
+
ϕ
=
⋅ =
( ) ( )
[ ]
O
N
O
N
O
N ëáå
á
Åçë
ê
ê ϕ
+
ϕ
+
ϕ
+
ϕ
= =
=
58. `çåàìÖ~íÉ=kìãÄÉêë=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
( ) ( ) ( )
[ ]
ϕ
−
+
ϕ
−
=
ϕ
+
ϕ ëáå
á
Åçë
ê
ëáå
á
Åçë
ê
|
||||||||||
||||||||||
=
=
59. fåîÉêëÉ=çÑ=~=`çãéäÉñ=kìãÄÉê=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
( )
( ) ( )
[ ]
ϕ
−
+
ϕ
−
=
ϕ
+
ϕ
ëáå
á
Åçë
ê
N
ëáå
á
Åçë
ê
N
=

21. CHAPTER 1. NUMBER SETS
11
60. nìçíáÉåí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
( )
( )
( ) ( )
[ ]
O
N
O
N
O
N
O
O
O
N
N
N
O
N
ëáå
á
Åçë
ê
ê
ëáå
á
Åçë
ê
ëáå
á
Åçë
ê
ò
ò
ϕ
−
ϕ
+
ϕ
−
ϕ
=
ϕ
+
ϕ
ϕ
+
ϕ
= =
=
61. mçïÉê=çÑ=~=`çãéäÉñ=kìãÄÉê=
( )
[ ] ( ) ( )
[ ]
ϕ
+
ϕ
=
ϕ
+
ϕ
= å
ëáå
á
å
Åçë
ê
ëáå
á
Åçë
ê
ò å
å
å
=
=
62. cçêãìä~=±aÉ=jçáîêÉ≤=
( ) ( ) ( )
ϕ
+
ϕ
=
ϕ
+
ϕ å
ëáå
á
å
Åçë
ëáå
á
Åçë
å
=
=
63. kíÜ=oççí=çÑ=~=`çãéäÉñ=kìãÄÉê=
( ) 




 π
+
ϕ
+
π
+
ϕ
=
ϕ
+
ϕ
=
å
â
O
ëáå
á
å
â
O
Åçë
ê
ëáå
á
Åçë
ê
ò å
å
å
I==
ïÜÉêÉ==
N
å
I
I
O
I
N
I
M
â −
= K K==
=
64. bìäÉê∞ë=cçêãìä~=
ñ
ëáå
á
ñ
Åçë
Éáñ
+
= =
=
=

22. 12
Chapter 2
Algebra
=
=
=
=
2.1 Factoring Formulas
=
oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==
k~íìê~ä=åìãÄÉêW=å=
=
=
65. ( )( )
Ä
~
Ä
~
Ä
~ O
O
−
+
=
− =
=
66. ( )( )
O
O
P
P
Ä
~Ä
~
Ä
~
Ä
~ +
+
−
=
− =
=
67. ( )( )
O
O
P
P
Ä
~Ä
~
Ä
~
Ä
~ +
−
+
=
+ =
=
68. ( )( ) ( )( )( )
O
O
O
O
O
O
Q
Q
Ä
~
Ä
~
Ä
~
Ä
~
Ä
~
Ä
~ +
+
−
=
+
−
=
− =
=
69. ( )( )
Q
P
O
O
P
Q
R
R
Ä
~Ä
Ä
~
Ä
~
~
Ä
~
Ä
~ +
+
+
+
−
=
− =
=
70. ( )( )
Q
P
O
O
P
Q
R
R
Ä
~Ä
Ä
~
Ä
~
~
Ä
~
Ä
~ +
−
+
−
+
=
+ =
=
71. fÑ=å=áë=çÇÇI=íÜÉå=
( )( )
N
å
O
å
O
P
å
O
å
N
å
å
å
Ä
~Ä
Ä
~
Ä
~
~
Ä
~
Ä
~ −
−
−
−
−
+
−
−
+
−
+
=
+ K K==
=
72. fÑ=å=áë=ÉîÉåI=íÜÉå==
( )( )
N
å
O
å
O
P
å
O
å
N
å
å
å
Ä
~Ä
Ä
~
Ä
~
~
Ä
~
Ä
~ −
−
−
−
−
+
+
+
+
+
−
=
− K I==

23. CHAPTER 2. ALGEBRA
13
( )( )
N
å
O
å
O
P
å
O
å
N
å
å
å
Ä
~Ä
Ä
~
Ä
~
~
Ä
~
Ä
~ −
−
−
−
−
−
+
−
+
−
+
=
+ K K=
=
=
=
2.2 Product Formulas
oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==
tÜçäÉ=åìãÄÉêëW=åI=â=
=
=
73. ( ) O
O
O
Ä
~Ä
O
~
Ä
~ +
−
=
− =
=
74. ( ) O
O
O
Ä
~Ä
O
~
Ä
~ +
+
=
+ =
=
75. ( ) P
O
O
P
P
Ä
~Ä
P
Ä
~
P
~
Ä
~ −
+
−
=
− =
=
76. ( ) P
O
O
P
P
Ä
~Ä
P
Ä
~
P
~
Ä
~ +
+
+
=
+ =
=
77. ( ) Q
P
O
O
P
Q
Q
Ä
~Ä
Q
Ä
~
S
Ä
~
Q
~
Ä
~ +
−
+
−
=
− =
=
78. ( ) Q
P
O
O
P
Q
Q
Ä
~Ä
Q
Ä
~
S
Ä
~
Q
~
Ä
~ +
+
+
+
=
+ =
=
79. _áåçãá~ä=cçêãìä~=
( ) I
Ä
`
~Ä
`
Ä
~
`
Ä
~
`
~
`
Ä
~ å
å
å
N
å
N
å
å
O
O
å
O
å
N
å
N
å
å
M
å
å
+
+
+
+
+
=
+ −
−
−
−
K
ïÜÉêÉ=
( )>
â
å
>
â
>
å
`â
å
−
= =~êÉ=íÜÉ=Äáåçãá~ä=ÅçÉÑÑáÅáÉåíëK=
=
80. ( ) ÄÅ
O
~Å
O
~Ä
O
Å
Ä
~
Å
Ä
~ O
O
O
O
+
+
+
+
+
=
+
+ =
=
81. ( ) +
+
+
+
+
+
=
+
+
+
+
+ O
O
O
O
O
O
î
ì
Å
Ä
~
î
ì
Å
Ä
~ K
K =
( )
ìî
Äî
Äì
ÄÅ
~î
~ì
~Å
~Ä
O +
+
+
+
+
+
+
+
+
+
+ K
K
K =

24. CHAPTER 2. ALGEBRA
14
2.3 Powers
=
_~ëÉë=EéçëáíáîÉ=êÉ~ä=åìãÄÉêëFW=~I=Ä==
mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=
=
=
82. å
ã
å
ã
~
~
~ +
= =
=
83. å
ã
å
ã
~
~
~ −
= =
=
84. ( ) ã
ã
ã
Ä
~
~Ä = =
=
85. ã
ã
ã
Ä
~
Ä
~
=






=
=
86. ( ) ãå
å
ã
~
~ = =
=
87. N
~M
= I= M
~ ≠ =
=
88. N
~N
= =
=
89. ã
ã
~
N
~ =
−
=
=
90. å ã
å
ã
~
~ = =
=
=
=
=
=

25. CHAPTER 2. ALGEBRA
15
2.4 Roots
=
_~ëÉëW=~I=Ä==
mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=
M
Ä
I
~ ≥ =Ñçê=ÉîÉå=êççíë=E â
O
å = I= k
â∈ F=
=
=
91. å
å
å
Ä
~
~Ä = =
=
92. åã å
ã
ã
å
Ä
~
Ä
~ = =
=
93. å
å
å
Ä
~
Ä
~
= I= M
Ä ≠ =
=
94. åã
å
ã
åã å
åã ã
ã
å
Ä
~
Ä
~
Ä
~
=
= I= M
Ä ≠ K=
=
95. ( ) å ãé
é
å ã
~
~ = =
=
96. ( ) ~
~
å
å
= =
=
97.
åé ãé
å ã
~
~ = =
=
98. å
ã
å ã
~
~ = =
=
99. ãå
ã å
~
~ = =
=
100. ( ) å ã
ã
å
~
~ = =
=

26. CHAPTER 2. ALGEBRA
16
101.
~
~
~
N å N
å
å
−
= I= M
~ ≠ K=
=
102.
O
Ä
~
~
O
Ä
~
~
Ä
~
O
O
−
−
±
−
+
=
± =
=
103.
Ä
~
Ä
~
Ä
~
N
−
=
±
m
=
=
=
=
2.5 Logarithms
=
mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=ñI=óI=~I=ÅI=â=
k~íìê~ä=åìãÄÉêW=å==
=
=
104. aÉÑáåáíáçå=çÑ=içÖ~êáíÜã=
ñ
äçÖ
ó ~
= =áÑ=~åÇ=çåäó=áÑ= ó
~
ñ = I= M
~ > I= N
~ ≠ K=
=
105. M
N
äçÖ~ = =
=
106. N
~
äçÖ~ = =
=
107.



<
∞
+
>
∞
−
=
N
~
áÑ
N
~
áÑ
M
äçÖ~ =
=
108. ( ) ó
äçÖ
ñ
äçÖ
ñó
äçÖ ~
~
~ +
= =
=
109. ó
äçÖ
ñ
äçÖ
ó
ñ
äçÖ ~
~
~ −
= =

27. CHAPTER 2. ALGEBRA
17
110. ( ) ñ
äçÖ
å
ñ
äçÖ ~
å
~ = =
=
111. ñ
äçÖ
å
N
ñ
äçÖ ~
å
~ = =
=
112. Å
äçÖ
ñ
äçÖ
~
äçÖ
ñ
äçÖ
ñ
äçÖ ~
Å
Å
Å
~ ⋅
=
= I= M
Å > I= N
Å ≠ K=
=
113.
~
äçÖ
N
Å
äçÖ
Å
~ = =
=
114. ñ
äçÖ~
~
ñ = =
=
115. içÖ~êáíÜã=íç=_~ëÉ=NM=
ñ
äçÖ
ñ
äçÖNM = =
=
116. k~íìê~ä=içÖ~êáíÜã=
ñ
äå
ñ
äçÖÉ = I==
ïÜÉêÉ= K
TNUOUNUOU
K
O
â
N
N
äáã
É
â
â
=






+
=
∞
→
=
=
117. ñ
äå
QPQOVQ
K
M
ñ
äå
NM
äå
N
ñ
äçÖ =
= =
=
118. ñ
äçÖ
PMORUR
K
O
ñ
äçÖ
É
äçÖ
N
ñ
äå =
= =
=
=
=
=
=

28. CHAPTER 2. ALGEBRA
18
2.6 Equations
=
oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=éI=èI=ìI=î=
pçäìíáçåëW= N
ñ I= O
ñ I= N
ó I= O
ó I= P
ó =
=
=
119. iáåÉ~ê=bèì~íáçå=áå=låÉ=s~êá~ÄäÉ=
M
Ä
~ñ =
+ I=
~
Ä
ñ −
= K==
=
120. nì~Çê~íáÅ=bèì~íáçå=
M
Å
Äñ
~ñO
=
+
+ I=
~
O
~Å
Q
Ä
Ä
ñ
O
O
I
N
−
±
−
= K=
=
121. aáëÅêáãáå~åí=
~Å
Q
Ä
a O
−
= =
=
122. sáÉíÉ∞ë=cçêãìä~ë=
fÑ= M
è
éñ
ñO
=
+
+ I=íÜÉå==



=
−
=
+
è
ñ
ñ
é
ñ
ñ
O
N
O
N
K=
=
123. M
Äñ
~ñO
=
+ I= M
ñN = I=
~
Ä
ñO −
= K=
=
124. M
Å
~ñO
=
+ I=
~
Å
ñ O
I
N −
±
= K=
=
125. `ìÄáÅ=bèì~íáçåK=`~êÇ~åç∞ë=cçêãìä~K==
M
è
éó
óP
=
+
+ I==

29. CHAPTER 2. ALGEBRA
19
î
ì
óN +
= I= ( ) ( )á
î
ì
O
P
î
ì
O
N
ó P
I
O +
±
+
−
= I==
ïÜÉêÉ==
P
O
O
P
é
O
è
O
è
ì 





+






+
−
= I= P
O
O
P
é
O
è
O
è
î 





+






−
−
= K==
=
=
2.7 Inequalities
s~êá~ÄäÉëW=ñI=óI=ò=
oÉ~ä=åìãÄÉêëW=



å
P
O
N ~
I
I
~
I
~
I
~
Ç
I
Å
I
Ä
I
~
K
I=ãI=å=
aÉíÉêãáå~åíëW=aI= ñ
a I= ó
a I= ò
a ==
=
=
126. fåÉèì~äáíáÉëI=fåíÉêî~ä=kçí~íáçåë=~åÇ=dê~éÜë==
=
fåÉèì~äáíó= fåíÉêî~ä=kçí~íáçå= dê~éÜ=
Ä
ñ
~ ≤
≤ = [ ]
Ä
I
~ =
=
Ä
ñ
~ ≤
< = ( ]
Ä
I
~ =
=
Ä
ñ
~ <
≤ = [ )
Ä
I
~ =
=
Ä
ñ
~ <
< = ( )
Ä
I
~ =
=
Ä
ñ ≤
<
∞
− I=
Ä
ñ ≤ =
( ]
Ä
I
∞
− =
=
Ä
ñ <
<
∞
− I=
Ä
ñ < =
( )
Ä
I
∞
− =
=
∞
<
≤ ñ
~ I=
~
ñ ≥ =
[ )
∞
I
~ =
=
∞
<
< ñ
~ I=
~
ñ > =
( )
∞
I
~ =
=

30. CHAPTER 2. ALGEBRA
20
127. fÑ= Ä
~ > I=íÜÉå= ~
Ä < K=
=
128. fÑ= Ä
~ > I=íÜÉå= M
Ä
~ >
− =çê= M
~
Ä <
− K=
=
129. fÑ= Ä
~ > I=íÜÉå= Å
Ä
Å
~ +
>
+ K=
=
130. fÑ= Ä
~ > I=íÜÉå= Å
Ä
Å
~ −
>
− K=
=
131. fÑ= Ä
~ > =~åÇ= Ç
Å > I=íÜÉå= Ç
Ä
Å
~ +
>
+ K=
=
132. fÑ= Ä
~ > =~åÇ= Ç
Å > I=íÜÉå= Å
Ä
Ç
~ −
>
− K=
=
133. fÑ= Ä
~ > =~åÇ= M
ã > I=íÜÉå= ãÄ
ã~ > K=
=
134. fÑ= Ä
~ > =~åÇ= M
ã > I=íÜÉå=
ã
Ä
ã
~
> K=
=
135. fÑ= Ä
~ > =~åÇ= M
ã < I=íÜÉå= ãÄ
ã~ < K=
=
136. fÑ= Ä
~ > =~åÇ= M
ã < I=íÜÉå=
ã
Ä
ã
~
< K=
=
137. fÑ= Ä
~
M <
< =~åÇ= M
å > I=íÜÉå= å
å
Ä
~ < K=
=
138. fÑ= Ä
~
M <
< =~åÇ= M
å < I=íÜÉå= å
å
Ä
~ > K=
=
139. fÑ= Ä
~
M <
< I=íÜÉå= å
å
Ä
~ < K=
=
140.
O
Ä
~
~Ä
+
≤ I==
ïÜÉêÉ= M
~ > =I= M
Ä > X=~å=Éèì~äáíó=áë=î~äáÇ=çåäó=áÑ= Ä
~ = K==
=
141. O
~
N
~ ≥
+ I=ïÜÉêÉ= M
~ > X=~å=Éèì~äáíó=í~âÉë=éä~ÅÉ=çåäó=~í= N
~ = K=

31. CHAPTER 2. ALGEBRA
21
142.
å
~
~
~
~
~
~ å
O
N
å
å
O
N
+
+
+
≤
K
K I=ïÜÉêÉ= M
~
I
I
~
I
~ å
O
N >
K K=
=
143. fÑ= M
Ä
~ñ >
+ =~åÇ= M
~ > I=íÜÉå=
~
Ä
ñ −
> K=
=
144. fÑ= M
Ä
~ñ >
+ =~åÇ= M
~ < I=íÜÉå=
~
Ä
ñ −
< K==
=
145. M
Å
Äñ
~ñO
>
+
+ =
=
= M
~ > = M
~ < =
=
=
=
M
a > =
=
=
N
ñ
ñ < I= O
ñ
ñ > =
=
=
=
O
N ñ
ñ
ñ <
< =
=
=
=
M
a = =
=
ñ
ñN < I= N
ñ
ñ > =
=
=
∅
∈
ñ =
=
=
=
M
a< =
=
=
∞
<
<
∞
− ñ =
=
=
=
∅
∈
ñ =
=

32. CHAPTER 2. ALGEBRA
22
146. Ä
~
Ä
~ +
≤
+ =
=
147. fÑ= ~
ñ < I=íÜÉå= ~
ñ
~ <
<
− I=ïÜÉêÉ= M
~ > K=
=
148. fÑ= ~
ñ > I=íÜÉå= ~
ñ −
< =~åÇ= ~
ñ > I=ïÜÉêÉ= M
~ > K=
=
149. fÑ= ~
ñO
< I=íÜÉå= ~
ñ < I=ïÜÉêÉ= M
~ > K=
=
150. fÑ= ~
ñO
> I=íÜÉå= ~
ñ > I=ïÜÉêÉ= M
~ > K=
=
151. fÑ=
( )
( )
M
ñ
Ö
ñ
Ñ
> I=íÜÉå=
( ) ( )
( )



≠
>
⋅
M
ñ
Ö
M
ñ
Ö
ñ
Ñ
K=
=
152.
( )
( )
M
ñ
Ö
ñ
Ñ
< I=íÜÉå=
( ) ( )
( )



≠
<
⋅
M
ñ
Ö
M
ñ
Ö
ñ
Ñ
K=
=
=
=
2.8 Compound Interest Formulas
=
cìíìêÉ=î~äìÉW=^=
fåáíá~ä=ÇÉéçëáíW=`=
^ååì~ä=ê~íÉ=çÑ=áåíÉêÉëíW=ê=
kìãÄÉê=çÑ=óÉ~êë=áåîÉëíÉÇW=í=
kìãÄÉê=çÑ=íáãÉë=ÅçãéçìåÇÉÇ=éÉê=óÉ~êW=å=
=
=
153. dÉåÉê~ä=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=
åí
å
ê
N
`
^ 





+
= =
=

33. CHAPTER 2. ALGEBRA
23
154. páãéäáÑáÉÇ=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=
fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=çåÅÉ=éÉê=óÉ~êI=íÜÉå=íÜÉ=éêÉîáçìë=
Ñçêãìä~=ëáãéäáÑáÉë=íçW=
( )í
ê
N
`
^ +
= K=
=
155. `çåíáåìçìë=`çãéçìåÇ=fåíÉêÉëí=
fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=Åçåíáåì~ääó=E ∞
→
å FI=íÜÉå==
êí
`É
^ = K=
=
=

34. 24
Chapter 3
Geometry
=
=
=
=
3.1 Right Triangle
=
iÉÖë=çÑ=~=êáÖÜí=íêá~åÖäÉW=~I=Ä=
eóéçíÉåìëÉW=Å=
^äíáíìÇÉW=Ü=
jÉÇá~åëW= ~
ã I= Ä
ã I= Å
ã =
^åÖäÉëW=α Iβ =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
^êÉ~W=p=
=
=
=
=
Figure 8.
=
156. °
=
β
+
α VM =
=

35. CHAPTER 3. GEOMETRY
25
157. β
=
=
α Åçë
Å
~
ëáå =
=
158. β
=
=
α ëáå
Å
Ä
Åçë =
=
159. β
=
=
α Åçí
Ä
~
í~å =
=
160. β
=
=
α í~å
~
Ä
Åçí =
=
161. β
=
=
α ÉÅ
Åçë
Ä
Å
ëÉÅ =
=
162. β
=
=
α ëÉÅ
~
Å
ÉÅ
Åçë =
=
163. móíÜ~ÖçêÉ~å=qÜÉçêÉã=
O
O
O
Å
Ä
~ =
+ =
=
164. ÑÅ
~O
= I= ÖÅ
ÄO
= I==
ïÜÉêÉ= Ñ= ~åÇ= Å= ~êÉ= éêçàÉÅíáçåë= çÑ= íÜÉ= äÉÖë= ~= ~åÇ= ÄI= êÉëéÉÅ-
íáîÉäóI=çåíç=íÜÉ=ÜóéçíÉåìëÉ=ÅK=
=
===== =
=
Figure 9.
=

36. CHAPTER 3. GEOMETRY
26
165. ÑÖ
ÜO
= I===
ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK==
=
166.
Q
~
Ä
ã
O
O
O
~ −
= I=
Q
Ä
~
ã
O
O
O
Ä −
= I===
ïÜÉêÉ= ~
ã =~åÇ= Ä
ã =~êÉ=íÜÉ=ãÉÇá~åë=íç=íÜÉ=äÉÖë=~=~åÇ=ÄK==
=
=
=
Figure 10.
=
167.
O
Å
ãÅ = I==
ïÜÉêÉ= Å
ã =áë=íÜÉ=ãÉÇá~å=íç=íÜÉ=ÜóéçíÉåìëÉ=ÅK=
=
168. Å
ã
O
Å
o =
= =
=
169.
Å
Ä
~
~Ä
O
Å
Ä
~
ê
+
+
=
−
+
= =
=
170. ÅÜ
~Ä = =
=
=

37. CHAPTER 3. GEOMETRY
27
171.
O
ÅÜ
O
~Ä
p =
= =
=
=
=
3.2 Isosceles Triangle
=
_~ëÉW=~=
iÉÖëW=Ä=
_~ëÉ=~åÖäÉW=β =
sÉêíÉñ=~åÖäÉW=α =
^äíáíìÇÉ=íç=íÜÉ=Ä~ëÉW=Ü=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 11.
=
172.
O
VM
α
−
°
=
β =
=
173.
Q
~
Ä
Ü
O
O
O
−
= =

38. CHAPTER 3. GEOMETRY
28
174. Ä
O
~
i +
= =
=
175. α
=
= ëáå
O
Ä
O
~Ü
p
O
=
=
=
=
3.3 Equilateral Triangle
=
páÇÉ=çÑ=~=Éèìáä~íÉê~ä=íêá~åÖäÉW=~=
^äíáíìÇÉW=Ü=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 12.
=
176.
O
P
~
Ü = =
=

39. CHAPTER 3. GEOMETRY
29
177.
P
P
~
Ü
P
O
o =
= =
=
178.
O
o
S
P
~
Ü
P
N
ê =
=
= =
=
179. ~
P
i = =
=
180.
Q
P
~
O
~Ü
p
O
=
= =
=
=
=
3.4 Scalene Triangle
E^=íêá~åÖäÉ=ïáíÜ=åç=íïç=ëáÇÉë=Éèì~äF=
=
=
páÇÉë=çÑ=~=íêá~åÖäÉW=~I=ÄI=Å=
pÉãáéÉêáãÉíÉêW=
O
Å
Ä
~
é
+
+
= ==
^åÖäÉë=çÑ=~=íêá~åÖäÉW= γ
β
α I
I =
^äíáíìÇÉë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= Å
Ä
~ Ü
I
Ü
I
Ü =
jÉÇá~åë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= Å
Ä
~ ã
I
ã
I
ã =
_áëÉÅíçêë=çÑ=íÜÉ=~åÖäÉë= γ
β
α I
I W= Å
Ä
~ í
I
í
I
í =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
^êÉ~W=p=
=
=

40. CHAPTER 3. GEOMETRY
30
===== =
=
Figure 13.
=
181. °
=
γ
+
β
+
α NUM =
=
182. Å
Ä
~ >
+ I==
~
Å
Ä >
+ I==
Ä
Å
~ >
+ K=
=
183. Å
Ä
~ <
− I==
~
Å
Ä <
− I==
Ä
Å
~ <
− K=
=
184. jáÇäáåÉ=
O
~
è = I= ~
öö
è K=
=
===== =
=
Figure 14.
=

41. CHAPTER 3. GEOMETRY
31
185. i~ï=çÑ=`çëáåÉë=
α
−
+
= Åçë
ÄÅ
O
Å
Ä
~ O
O
O
I=
β
−
+
= Åçë
~Å
O
Å
~
Ä O
O
O
I=
γ
−
+
= Åçë
~Ä
O
Ä
~
Å O
O
O
K=
=
186. i~ï=çÑ=páåÉë=
o
O
ëáå
Å
ëáå
Ä
ëáå
~
=
γ
=
β
=
α
I==
ïÜÉêÉ=o=áë=íÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉK==
=
187.
p
Q
~ÄÅ
Ü
O
~Ä
Ü
O
~Å
Ü
O
ÄÅ
ëáå
O
Å
ëáå
O
Ä
ëáå
O
~
o
Å
Ä
~
=
=
=
=
γ
=
β
=
α
= =
=
188.
( )( )( )
é
Å
é
Ä
é
~
é
êO −
−
−
= I==
Å
Ä
~ Ü
N
Ü
N
Ü
N
ê
N
+
+
= K=
=
189.
( )( )
ÄÅ
Å
é
Ä
é
O
ëáå
−
−
=
α
I=
( )
ÄÅ
~
é
é
O
Åçë
−
=
α
I=
( )( )
( )
~
é
é
Å
é
Ä
é
O
í~å
−
−
−
=
α
K=
=
190. ( )( )( )
Å
é
Ä
é
~
é
é
~
O
Ü~ −
−
−
= I=
( )( )( )
Å
é
Ä
é
~
é
é
Ä
O
ÜÄ −
−
−
= I=
( )( )( )
Å
é
Ä
é
~
é
é
Å
O
ÜÅ −
−
−
= K=

42. CHAPTER 3. GEOMETRY
32
191. β
=
γ
= ëáå
Å
ëáå
Ä
Ü~ I=
α
=
γ
= ëáå
Å
ëáå
~
ÜÄ I=
α
=
β
= ëáå
Ä
ëáå
~
ÜÅ K=
=
192.
Q
~
O
Å
Ä
ã
O
O
O
O
~ −
+
= I==
Q
Ä
O
Å
~
ã
O
O
O
O
Ä −
+
= I==
Q
Å
O
Ä
~
ã
O
O
O
O
Å −
+
= K=
=
===== =
=
Figure 15.
=
193. ~
ã
P
O
^j = I= Ä
ã
P
O
_j = I= Å
ã
P
O
`j = =EcáÖKNRFK=
=
194.
( )
( )O
O
~
Å
Ä
~
é
ÄÅé
Q
í
+
−
= I==
( )
( )O
O
Ä
Å
~
Ä
é
~Åé
Q
í
+
−
= I==
( )
( )O
O
Å
Ä
~
Å
é
~Äé
Q
í
+
−
= K=
=

43. CHAPTER 3. GEOMETRY
33
195.
O
ÅÜ
O
ÄÜ
O
~Ü
p Å
Ä
~
=
=
= I==
O
ëáå
ÄÅ
O
ëáå
~Å
O
ëáå
~Ä
p
α
=
β
=
γ
= I==
( )( )( )
Å
é
Ä
é
~
é
é
p −
−
−
= =EeÉêçå∞ë=cçêãìä~FI=
éê
p = I==
o
Q
~ÄÅ
p = I=
γ
β
α
= ëáå
ëáå
ëáå
o
O
p O
I=
O
í~å
O
í~å
O
í~å
é
p O γ
β
α
= K=
=
=
=
3.5 Square
páÇÉ=çÑ=~=ëèì~êÉW=~=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
Figure 16.

44. CHAPTER 3. GEOMETRY
34
196. O
~
Ç = ==
=
197.
O
O
~
O
Ç
o =
= =
=
198.
O
~
ê = =
=
199. ~
Q
i = =
=
200. O
~
p = =
=
=
=
3.6 Rectangle
=
páÇÉë=çÑ=~=êÉÅí~åÖäÉW=~I=Ä=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 17.
=
201. O
O
Ä
~
Ç +
= ==

45. CHAPTER 3. GEOMETRY
35
202.
O
Ç
o = =
=
203. ( )
Ä
~
O
i +
= =
=
204. ~Ä
p = =
=
=
=
3.7 Parallelogram
=
páÇÉë=çÑ=~=é~ê~ääÉäçÖê~ãW=~I=Ä=
aá~Öçå~äëW= O
N Ç
I
Ç =
`çåëÉÅìíáîÉ=~åÖäÉëW= β
αI =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
^äíáíìÇÉW=Ü==
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
===== =
=
Figure 18.
=
205. °
=
β
+
α NUM =
=
206. ( )
O
O
O
O
O
N Ä
~
O
Ç
Ç +
=
+ =
=

46. CHAPTER 3. GEOMETRY
36
207. β
=
α
= ëáå
Ä
ëáå
Ä
Ü =
=
208. ( )
Ä
~
O
i +
= =
=
209. α
=
= ëáå
~Ä
~Ü
p I==
ϕ
= ëáå
Ç
Ç
O
N
p O
N K=
=
=
=
3.8 Rhombus
=
páÇÉ=çÑ=~=êÜçãÄìëW=~=
aá~Öçå~äëW= O
N Ç
I
Ç =
`çåëÉÅìíáîÉ=~åÖäÉëW= β
αI =
^äíáíìÇÉW=e=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
===== =
=
Figure 19.
=

47. CHAPTER 3. GEOMETRY
37
210. °
=
β
+
α NUM =
=
211. O
O
O
O
N ~
Q
Ç
Ç =
+ =
=
212.
~
O
Ç
Ç
ëáå
~
Ü O
N
=
α
= =
=
213.
O
ëáå
~
~
Q
Ç
Ç
O
Ü
ê O
N α
=
=
= =
=
214. ~
Q
i = =
=
215. α
=
= ëáå
~
~Ü
p O
I==
O
NÇ
Ç
O
N
p = K=
=
=
=
3.9 Trapezoid
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
^êÉ~W=p=
=
=

48. CHAPTER 3. GEOMETRY
38
=
=
Figure 20.
=
216.
O
Ä
~
è
+
= =
=
217. èÜ
Ü
O
Ä
~
p =
⋅
+
= =
=
=
=
3.10 Isosceles Trapezoid
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
iÉÖW=Å=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
^êÉ~W=p=
=
=

49. CHAPTER 3. GEOMETRY
39
=
=
Figure 21.
=
218.
O
Ä
~
è
+
= =
=
219. O
Å
~Ä
Ç +
= =
=
220. ( )O
O
~
Ä
Q
N
Å
Ü −
−
= =
=
221.
( )( )
Ä
~
Å
O
Ä
~
Å
O
Å
~Ä
Å
o
O
−
+
+
−
+
= =
=
222. èÜ
Ü
O
Ä
~
p =
⋅
+
= =
=
=
=
=
=
=

50. CHAPTER 3. GEOMETRY
40
3.11 Isosceles Trapezoid with
Inscribed Circle
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
iÉÖW=Å=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 22.
=
223. Å
O
Ä
~ =
+ =
=
224. Å
O
Ä
~
è =
+
= =
=
225. O
O
O
Å
Ü
Ç +
= =
=

51. CHAPTER 3. GEOMETRY
41
226.
O
~Ä
O
Ü
ê =
= =
=
227.
~
Ä
S
Ä
~
U
Ä
~
Å
Ü
Ü
O
Å
~Ä
Å
N
O
Å
ê
Q
ÅÇ
Ü
O
ÅÇ
o O
O
O
+
+
+
=
+
=
+
=
=
= =
=
228. ( ) Å
Q
Ä
~
O
i =
+
= =
=
229.
( )
O
iê
ÅÜ
èÜ
O
~Ä
Ä
~
Ü
O
Ä
~
p =
=
=
+
=
⋅
+
= ==
=
=
=
3.12 Trapezoid with Inscribed Circle
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
i~íÉê~ä=ëáÇÉëW=ÅI=Ç=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äëW= O
N Ç
I
Ç =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=

52. CHAPTER 3. GEOMETRY
42
=
=
Figure 23.
=
230. Ç
Å
Ä
~ +
=
+ =
=
231.
O
Ç
Å
O
Ä
~
è
+
=
+
= =
=
232. ( ) ( )
Ç
Å
O
Ä
~
O
i +
=
+
= =
=
233. èÜ
Ü
O
Ç
Å
Ü
O
Ä
~
p =
⋅
+
=
⋅
+
= I==
ϕ
= ëáå
Ç
Ç
O
N
p O
N K=
=
=
=
3.13 Kite
=
páÇÉë=çÑ=~=âáíÉW=~I=Ä=
aá~Öçå~äëW= O
N Ç
I
Ç =
^åÖäÉëW= γ
β
α I
I =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=

53. CHAPTER 3. GEOMETRY
43
=
=
Figure 24.
=
234. °
=
γ
+
β
+
α PSM
O =
=
235. ( )
Ä
~
O
i +
= =
=
236.
O
Ç
Ç
p O
N
= =
=
=
=
3.14 Cyclic Quadrilateral
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= O
N Ç
I
Ç =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
fåíÉêå~ä=~åÖäÉëW= δ
γ
β
α I
I
I =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=

54. CHAPTER 3. GEOMETRY
44
=
=
Figure 25.
=
237. °
=
δ
+
β
=
γ
+
α NUM =
=
238. míçäÉãó∞ë=qÜÉçêÉã=
O
NÇ
Ç
ÄÇ
~Å =
+ =
=
239. Ç
Å
Ä
~
i +
+
+
= =
=
240.
( )( )( )
( )( )( )( )
Ç
é
Å
é
Ä
é
~
é
ÅÇ
~Ä
ÄÅ
~Ç
ÄÇ
~Å
Q
N
o
−
−
−
−
+
+
+
= I==
ïÜÉêÉ=
O
i
é = K=
=
241. ϕ
= ëáå
Ç
Ç
O
N
p O
N I==
( )( )( )( )
Ç
é
Å
é
Ä
é
~
é
p −
−
−
−
= I==
ïÜÉêÉ=
O
i
é = K=
=
=
=

55. CHAPTER 3. GEOMETRY
45
3.15 Tangential Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= O
N Ç
I
Ç =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
=
=
Figure 26.
=
242. Ç
Ä
Å
~ +
=
+ =
=
243. ( ) ( )
Ç
Ä
O
Å
~
O
Ç
Å
Ä
~
i +
=
+
=
+
+
+
= =
=
244.
( ) ( )
é
O
é
Ä
~
Ä
~
Ç
Ç
ê
O
O
O
O
O
N −
+
−
−
= I==
ïÜÉêÉ=
O
i
é = K==
=

56. CHAPTER 3. GEOMETRY
46
245. ϕ
=
= ëáå
Ç
Ç
O
N
éê
p O
N =
=
=
=
3.16 General Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= O
N Ç
I
Ç =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
fåíÉêå~ä=~åÖäÉëW= δ
γ
β
α I
I
I =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
======= =
=
Figure 27.
=
246. °
=
δ
+
γ
+
β
+
α PSM =
=
247. Ç
Å
Ä
~
i +
+
+
= =
=

57. CHAPTER 3. GEOMETRY
47
248. ϕ
= ëáå
Ç
Ç
O
N
p O
N =
=
=
=
3.17 Regular Hexagon
=
páÇÉW=~=
fåíÉêå~ä=~åÖäÉW=α =
pä~åí=ÜÉáÖÜíW=ã=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
=
=
Figure 28.
=
249. °
=
α NOM =
=
250.
O
P
~
ã
ê =
= =

58. CHAPTER 3. GEOMETRY
48
251. ~
o = =
=
252. ~
S
i = =
=
253.
O
P
P
~
éê
p
O
=
= I==
ïÜÉêÉ=
O
i
é = K=
=
=
=
3.18 Regular Polygon
=
páÇÉW=~=
kìãÄÉê=çÑ=ëáÇÉëW=å=
fåíÉêå~ä=~åÖäÉW=α =
pä~åí=ÜÉáÖÜíW=ã=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=

59. CHAPTER 3. GEOMETRY
49
=
=
Figure 29.
=
254. °
⋅
−
=
α NUM
O
O
å
=
=
255. °
⋅
−
=
α NUM
O
O
å
=
=
256.
å
ëáå
O
~
o
π
= =
=
257.
Q
~
o
å
í~å
O
~
ã
ê
O
O
−
=
π
=
= =
=
258. å~
i = =
=
259.
å
O
ëáå
O
åo
p
O
π
= I==
Q
~
o
é
éê
p
O
O
−
=
= I==

60. CHAPTER 3. GEOMETRY
50
ïÜÉêÉ=
O
i
é = K==
=
=
=
3.19 Circle
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
`ÜçêÇW=~=
pÉÅ~åí=ëÉÖãÉåíëW=ÉI=Ñ=
q~åÖÉåí=ëÉÖãÉåíW=Ö=
`Éåíê~ä=~åÖäÉW=α =
fåëÅêáÄÉÇ=~åÖäÉW=β =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
260.
O
ëáå
o
O
~
α
= =
=
=
=
Figure 30.
=

61. CHAPTER 3. GEOMETRY
51
261. O
N
O
N Ä
Ä
~
~ = =
=
=
=
Figure 31.
=
262. N
N ÑÑ
ÉÉ = =
=
===== =
=
Figure 32.
=
263. N
O
ÑÑ
Ö = =
=

62. CHAPTER 3. GEOMETRY
52
===== =
=
Figure 33.
=
264.
O
α
=
β =
=
=
=
Figure 34.
=
265. Ç
o
O
i π
=
π
= =
=
266.
O
io
Q
Ç
o
p
O
O
=
π
=
π
= ==
=

63. CHAPTER 3. GEOMETRY
53
3.20 Sector of a Circle
=
o~Çáìë=çÑ=~=ÅáêÅäÉW=o=
^êÅ=äÉåÖíÜW=ë=
`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=
`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW=α=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 35.
=
267. oñ
ë = =
=
268.
°
α
π
=
NUM
o
ë =
=
269. o
O
ë
i +
= =
=
270.
°
α
π
=
=
=
PSM
o
O
ñ
o
O
oë
p
O
O
==
=
=

64. CHAPTER 3. GEOMETRY
54
3.21 Segment of a Circle
=
o~Çáìë=çÑ=~=ÅáêÅäÉW=o=
^êÅ=äÉåÖíÜW=ë=
`ÜçêÇW=~=
`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=
`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW=α=
eÉáÖÜí=çÑ=íÜÉ=ëÉÖãÉåíW=Ü=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 36.
=
271. O
Ü
Üo
O
O
~ −
= =
=
272. O
O
~
o
Q
O
N
o
Ü −
−
= I= o
Ü < =
=
273. ~
ë
i +
= =
=

65. CHAPTER 3. GEOMETRY
55
274. ( )
[ ] ( )
ñ
ëáå
ñ
O
o
ëáå
NUM
O
o
Ü
o
~
ëo
O
N
p
O
O
−
=






α
−
°
απ
=
−
−
= I==
Ü~
P
O
p ≈ K=
=
=
=
3.22 Cube
=
bÇÖÉW=~==
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ëéÜÉêÉW=ê=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=== =
=
Figure 37.
=
275. P
~
Ç = =
=
276.
O
~
ê = =
=

66. CHAPTER 3. GEOMETRY
56
277.
O
P
~
o = =
=
278. O
~
S
p = =
=
279. P
~
s = ==
=
=
=
3.23 Rectangular Parallelepiped
=
bÇÖÉëW=~I=ÄI=Å==
aá~Öçå~äW=Ç=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
===== =
=
Figure 38.
=
280. O
O
O
Å
Ä
~
Ç +
+
= =
=
281. ( )
ÄÅ
~Å
~Ä
O
p +
+
= =
=
282. ~ÄÅ
s = ==

67. CHAPTER 3. GEOMETRY
57
3.24 Prism
=
i~íÉê~ä=ÉÇÖÉW=ä=
eÉáÖÜíW=Ü=
i~íÉê~ä=~êÉ~W= i
p =
^êÉ~=çÑ=Ä~ëÉW= _
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
===== =
=
Figure 39.
=
283. _
i p
O
p
p +
= K==
=
284. i~íÉê~ä=^êÉ~=çÑ=~=oáÖÜí=mêáëã=
( )ä
~
~
~
~
p å
P
O
N
i +
+
+
+
= K =
=
285. i~íÉê~ä=^êÉ~=çÑ=~å=lÄäáèìÉ=mêáëã=
éä
pi = I==
ïÜÉêÉ=é=áë=íÜÉ=éÉêáãÉíÉê=çÑ=íÜÉ=Åêçëë=ëÉÅíáçåK=
=

68. CHAPTER 3. GEOMETRY
58
286. Ü
p
s _
= =
=
287. `~î~äáÉêáDë=mêáåÅáéäÉ==
dáîÉå=íïç=ëçäáÇë=áåÅäìÇÉÇ=ÄÉíïÉÉå=é~ê~ääÉä=éä~åÉëK=fÑ=ÉîÉêó=
éä~åÉ=Åêçëë=ëÉÅíáçå=é~ê~ääÉä=íç=íÜÉ=ÖáîÉå=éä~åÉë=Ü~ë=íÜÉ=ë~ãÉ=
~êÉ~=áå=ÄçíÜ=ëçäáÇëI=íÜÉå=íÜÉ=îçäìãÉë=çÑ=íÜÉ=ëçäáÇë=~êÉ=Éèì~äK=
=
=
=
3.25 Regular Tetrahedron
=
qêá~åÖäÉ=ëáÇÉ=äÉåÖíÜW=~=
eÉáÖÜíW=Ü=
^êÉ~=çÑ=Ä~ëÉW= _
p =
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 40.
=
288. ~
P
O
Ü = =
=

69. CHAPTER 3. GEOMETRY
59
289.
Q
~
P
p
O
_ = =
=
290. O
~
P
p = =
=
291.
O
S
~
Ü
p
P
N
s
P
_ =
= K==
=
=
=
3.26 Regular Pyramid
=
páÇÉ=çÑ=Ä~ëÉW=~=
i~íÉê~ä=ÉÇÖÉW=Ä=
eÉáÖÜíW=Ü=
pä~åí=ÜÉáÖÜíW=ã==
kìãÄÉê=çÑ=ëáÇÉëW=å==
pÉãáéÉêáãÉíÉê=çÑ=Ä~ëÉW=é=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉ=çÑ=Ä~ëÉW=ê=
^êÉ~=çÑ=Ä~ëÉW= _
p =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=

70. CHAPTER 3. GEOMETRY
60
=
=
Figure 41.
=
292.
Q
~
Ä
ã
O
O
−
= =
=
293.
å
ëáå
O
~
å
ëáå
Ä
Q
Ü
O
O
O
π
−
π
= =
=
294. éã
~
Ä
Q
å~
Q
N
å~ã
O
N
p O
O
i =
−
=
= =
=
295. éê
p_ = =
=
296. i
_ p
p
p +
= =
=
297. éêÜ
P
N
Ü
p
P
N
s _ =
= ==
=
=
=

71. CHAPTER 3. GEOMETRY
61
3.27 Frustum of a Regular Pyramid
=
_~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW=



å
P
O
N
å
P
O
N
Ä
I
I
Ä
I
Ä
I
Ä
~
I
I
~
I
~
I
~
K
K
=
eÉáÖÜíW=Ü=
pä~åí=ÜÉáÖÜíW=ã==
^êÉ~=çÑ=Ä~ëÉëW= N
p I= O
p =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
mÉêáãÉíÉê=çÑ=Ä~ëÉëW= N
m I= O
m =
pÅ~äÉ=Ñ~ÅíçêW=â=
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 42.
=
298. â
~
Ä
~
Ä
~
Ä
~
Ä
~
Ä
å
å
P
P
O
O
N
N
=
=
=
=
=
= K =
=

72. CHAPTER 3. GEOMETRY
62
299. O
N
O
â
p
p
= =
=
300.
( )
O
m
m
ã
p O
N
i
+
= =
=
301. O
N
i p
p
p
p +
+
= =
=
302. ( )
O
O
N
N p
p
p
p
P
Ü
s +
+
= =
=
303. [ ]
O
N
O
N
â
â
N
P
Üp
~
Ä
~
Ä
N
P
Üp
s +
+
=














+
+
= =
=
=
=
3.28 Rectangular Right Wedge
=
páÇÉë=çÑ=Ä~ëÉW=~I=Ä=
qçé=ÉÇÖÉW=Å=
eÉáÖÜíW=Ü=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
^êÉ~=çÑ=Ä~ëÉW= _
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=

73. CHAPTER 3. GEOMETRY
63
=
=
Figure 43.
=
304. ( ) ( )O
O
O
O
i Å
~
Ü
Ä
Ä
Ü
Q
Å
~
O
N
p −
+
+
+
+
= =
=
305. ~Ä
p_ = =
=
306. i
_ p
p
p +
= =
=
307. ( )
Å
~
O
S
ÄÜ
s +
= =
=
=
=
3.29 Platonic Solids
=
bÇÖÉW=~=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=

74. CHAPTER 3. GEOMETRY
64
308. cáîÉ=mä~íçåáÅ=pçäáÇë=
qÜÉ= éä~íçåáÅ= ëçäáÇë= ~êÉ= ÅçåîÉñ= éçäóÜÉÇê~= ïáíÜ= Éèìáî~äÉåí=
Ñ~ÅÉë=ÅçãéçëÉÇ=çÑ=ÅçåÖêìÉåí=ÅçåîÉñ=êÉÖìä~ê=éçäóÖçåëK==
=
pçäáÇ= kìãÄÉê=
çÑ=sÉêíáÅÉë
kìãÄÉê=
çÑ=bÇÖÉë=
kìãÄÉê=
çÑ=c~ÅÉë=
pÉÅíáçå=
qÉíê~ÜÉÇêçå== Q= S= Q= PKOR=
`ìÄÉ= U= NO= S= PKOO=
lÅí~ÜÉÇêçå= S= NO= U= PKOT=
fÅçë~ÜÉÇêçå= NO= PM= OM= PKOT=
açÇÉÅ~ÜÉÇêçå= OM= PM= NO= PKOT=
=
=
Octahedron
=
=
=
Figure 44.
=
309.
S
S
~
ê = =
=
310.
O
O
~
o = =
=

75. CHAPTER 3. GEOMETRY
65
311. P
~
O
p O
= =
=
312.
P
O
~
s
P
= =
=
=
Icosahedron
=
=
=
Figure 45.
=
313.
( )
NO
R
P
P
~
ê
+
= =
=
314. ( )
R
R
O
Q
~
o +
= =
=
315. P
~
R
p O
= =
=
316.
( )
NO
R
P
~
R
s
P
+
= =
=
=

76. CHAPTER 3. GEOMETRY
66
Dodecahedron
=
=
=
Figure 46.
=
317.
( )
O
R
NN
OR
NM
~
ê
+
= =
=
318.
( )
Q
R
N
P
~
o
+
= =
=
319. ( )
R
O
R
R
~
P
p O
+
= =
=
320.
( )
Q
R
T
NR
~
s
P
+
= =
=
=
=
3.30 Right Circular Cylinder
=
o~Çáìë=çÑ=Ä~ëÉW=o=
aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=

77. CHAPTER 3. GEOMETRY
67
eÉáÖÜíW=e=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
^êÉ~=çÑ=Ä~ëÉW= _
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
===== =
=
Figure 47.
=
321. oe
O
pi π
= =
=
322. ( ) 





+
π
=
+
π
=
+
=
O
Ç
e
Ç
o
e
o
O
p
O
p
p _
i =
=
323. e
o
e
p
s O
_ π
=
= =
=
=
=

78. CHAPTER 3. GEOMETRY
68
3.31 Right Circular Cylinder with
an Oblique Plane Face
=
o~Çáìë=çÑ=Ä~ëÉW=o=
qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= N
Ü =
qÜÉ=ëÜçêíÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= O
Ü =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= _
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 48.
=
324. ( )
O
N
i Ü
Ü
o
p +
π
= =
=
325.
O
O
N
O
O
_
O
Ü
Ü
o
o
o
p 




 −
+
π
+
π
= =
=

79. CHAPTER 3. GEOMETRY
69
326.













 −
+
+
+
+
π
=
+
=
O
O
N
O
O
N
_
i
O
Ü
Ü
o
o
Ü
Ü
o
p
p
p =
=
327. ( )
O
N
O
Ü
Ü
O
o
s +
π
= =
=
=
=
3.32 Right Circular Cone
o~Çáìë=çÑ=Ä~ëÉW=o=
aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=
eÉáÖÜíW=e=
pä~åí=ÜÉáÖÜíW=ã=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
^êÉ~=çÑ=Ä~ëÉW= _
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 49.

80. CHAPTER 3. GEOMETRY
70
328. O
O
o
ã
e −
= =
=
329.
O
ãÇ
oã
pi
π
=
π
= =
=
330. O
_ o
p π
= =
=
331. ( ) 





+
π
=
+
π
=
+
=
O
Ç
ã
Ç
O
N
o
ã
o
p
p
p _
i =
=
332. e
o
P
N
e
p
P
N
s O
_ π
=
= =
=
=
=
3.33 Frustum of a Right Circular Cone
=
o~Çáìë=çÑ=Ä~ëÉëW=oI=ê=
eÉáÖÜíW=e=
pä~åí=ÜÉáÖÜíW=ã=
pÅ~äÉ=Ñ~ÅíçêW=â=
^êÉ~=çÑ=Ä~ëÉëW= N
p I= O
p =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=

81. CHAPTER 3. GEOMETRY
71
=
=
Figure 50.
=
333. ( )O
O
ê
o
ã
e −
−
= =
=
334. â
ê
o
= =
=
335. O
O
O
N
O
â
ê
o
p
p
=
= =
=
336. ( )
ê
o
ã
pi +
π
= =
=
337. ( )
[ ]
ê
o
ã
ê
o
p
p
p
p O
O
i
O
N +
+
+
π
=
+
+
= =
=
338. ( )
O
O
N
N p
p
p
p
P
Ü
s +
+
= =
=
339. [ ]
O
N
O
N
â
â
N
P
Üp
ê
o
ê
o
N
P
Üp
s +
+
=














+
+
= =
=
=
=

82. CHAPTER 3. GEOMETRY
72
3.34 Sphere
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
Figure 51.
=
340. O
o
Q
p π
= =
=
341. po
P
N
Ç
S
N
e
o
P
Q
s P
P
=
π
=
π
= =
=
=
=
3.35 Spherical Cap
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉW=ê=
eÉáÖÜíW=Ü=
^êÉ~=çÑ=éä~åÉ=Ñ~ÅÉW= _
p =
^êÉ~=çÑ=ëéÜÉêáÅ~ä=Å~éW= `
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=

83. CHAPTER 3. GEOMETRY
73
=
=
Figure 52.
=
342.
Ü
O
Ü
ê
o
O
O
+
= =
=
343. O
_ ê
p π
= =
=
344. ( )
O
O
` ê
Ü
p +
π
= =
=
345. ( ) ( )
O
O
O
`
_ ê
oÜ
O
ê
O
Ü
p
p
p +
π
=
+
π
=
+
= =
=
346. ( ) ( )
O
O
O
Ü
ê
P
Ü
S
Ü
o
P
Ü
S
s +
π
=
−
π
= =
=
=
=
3.36 Spherical Sector
=
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉ=çÑ=ëéÜÉêáÅ~ä=Å~éW=ê=
eÉáÖÜíW=Ü=
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=

84. CHAPTER 3. GEOMETRY
74
====== === =
=
Figure 53.
=
347. ( )
ê
Ü
O
o
p +
π
= =
=
348. Ü
o
P
O
s O
π
= =
=
kçíÉW=qÜÉ=ÖáîÉå=Ñçêãìä~ë=~êÉ=ÅçêêÉÅí=ÄçíÜ=Ñçê=±çéÉå≤=~åÇ=
±ÅäçëÉÇ≤=ëéÜÉêáÅ~ä=ëÉÅíçêK=
=
=
=
3.37 Spherical Segment
=
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉëW= N
ê I= O
ê =
eÉáÖÜíW=Ü=
^êÉ~=çÑ=ëéÜÉêáÅ~ä=ëìêÑ~ÅÉW= p
p =
^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= N
p I= O
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=

85. CHAPTER 3. GEOMETRY
75
===== =
=
Figure 54.
=
349. oÜ
O
pp π
= =
=
350. ( )
O
O
O
N
O
N
p ê
ê
oÜ
O
p
p
p
p +
+
π
=
+
+
= =
=
351. ( )
O
O
O
O
N Ü
ê
P
ê
P
Ü
S
N
s +
+
π
= =
=
=
=
3.38 Spherical Wedge
=
o~ÇáìëW=o=
aáÜÉÇê~ä=~åÖäÉ=áå=ÇÉÖêÉÉëW=ñ=
aáÜÉÇê~ä=~åÖäÉ=áå=ê~Çá~åëW=α=
^êÉ~=çÑ=ëéÜÉêáÅ~ä=äìåÉW= i
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=

86. CHAPTER 3. GEOMETRY
76
=
=
Figure 55.
=
352. ñ
o
O
VM
o
p O
O
i =
α
π
= =
=
353. ñ
o
O
o
VM
o
o
p O
O
O
O
+
π
=
α
π
+
π
= =
=
354. ñ
o
P
O
OTM
o
s P
P
=
α
π
= =
=
=
=
3.39 Ellipsoid
=
pÉãá-~ñÉëW=~I=ÄI=Å=
sçäìãÉW=s=

87. CHAPTER 3. GEOMETRY
77
======= =
=
Figure 56.
=
355. ~ÄÅ
P
Q
s π
= =
=
=
=
Prolate Spheroid
=
pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä
~ > F=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
356. 





+
π
=
É
É
~êÅëáå
~
Ä
Ä
O
p I==
ïÜÉêÉ=
~
Ä
~
É
O
O
−
= K=
=
357. ~
Ä
P
Q
s O
π
= =
=

88. CHAPTER 3. GEOMETRY
78
Oblate Spheroid
=
pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä
~ < F=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
358.


















+
π
=
~
L
ÄÉ
~
ÄÉ
~êÅëáåÜ
~
Ä
Ä
O
p I==
ïÜÉêÉ=
Ä
~
Ä
É
O
O
−
= K=
=
359. ~
Ä
P
Q
s O
π
= =
=
=
=
3.40 Circular Torus
=
j~àçê=ê~ÇáìëW=o=
jáåçê=ê~ÇáìëW=ê=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=

89. CHAPTER 3. GEOMETRY
79
== =
Picture 57.
=
360. oê
Q
p O
π
= =
=
361. O
O
oê
O
s π
= =
=
=

90. 80
Chapter 4
Trigonometry
=
=
=
=
^åÖäÉëW=α I=β =
oÉ~ä=åìãÄÉêë=EÅççêÇáå~íÉë=çÑ=~=éçáåíFW=ñI=ó==
tÜçäÉ=åìãÄÉêW=â=
=
=
4.1 Radian and Degree Measures of Angles
=
362. ?
QR
D
NT
RT
NUM
ê~Ç
N °
≈
π
°
= =
=
363. ê~Ç
MNTQRP
K
M
ê~Ç
NUM
N ≈
π
=
° =
=
364. ê~Ç
MMMOVN
K
M
ê~Ç
SM
NUM
D
N ≈
⋅
π
= =
=
365. ê~Ç
MMMMMR
K
M
ê~Ç
PSMM
NUM
?
N ≈
⋅
π
= =
=
366. =
=
^åÖäÉ=
EÇÉÖêÉÉëF=
M= PM= QR= SM= VM= NUM= OTM= PSM=
^åÖäÉ=
Eê~Çá~åëF= M=
S
π
=
Q
π
=
P
π
=
O
π
= π=
O
Pπ
= π
O =
=
=
=

91. CHAPTER 4. TRIGONOMETRY
81
4.2 Definitions and Graphs of Trigonometric
Functions
=
= =
=
Figure 58.
=
367.
ê
ó
ëáå =
α =
=
368.
ê
ñ
Åçë =
α =
=
369.
ñ
ó
í~å =
α =
=
370.
ó
ñ
Åçí =
α =
=

92. CHAPTER 4. TRIGONOMETRY
82
371.
ñ
ê
ëÉÅ =
α =
=
372.
ó
ê
ÅçëÉÅ =
α =
=
373. páåÉ=cìåÅíáçå=
ñ
ëáå
ó = I= N
ñ
ëáå
N ≤
≤
− K=
=
=
Figure 59.
=
374. `çëáåÉ=cìåÅíáçå==
ñ
Åçë
ó = I= N
ñ
Åçë
N ≤
≤
− K=

93. CHAPTER 4. TRIGONOMETRY
83
=
=
Figure 60.
=
375. q~åÖÉåí=cìåÅíáçå=
ñ
í~å
ó = I= ( )
O
N
â
O
ñ
π
+
≠ I= K
ñ
í~å ∞
≤
≤
∞
− =
=
=
=
Figure 61.
=

94. CHAPTER 4. TRIGONOMETRY
84
376. `çí~åÖÉåí=cìåÅíáçå==
ñ
Åçí
ó = I= π
≠ â
ñ I== ∞
≤
≤
∞
− ñ
Åçí K=
=
=
=
Figure 62.
=
377. pÉÅ~åí=cìåÅíáçå=
ñ
ëÉÅ
ó = I= ( )
O
N
â
O
ñ
π
+
≠ K=
==

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

96. CHAPTER 4. TRIGONOMETRY
86
4.3. Signs of Trigonometric Functions
379. =
=
nì~Çê~åí=
páå
α =
`çë
α =
q~å
α =
`çí
α =
pÉÅ
α =
`çëÉÅ=
α =
f= H= H= H= H= H= H=
ff= H= = = = = H=
fff= = = H= H= = =
fs= = H= = = H= =
=
=
=
380. =
=
=
Figure 65.
=
=
=
=
=
=
=
=
=
=

97. CHAPTER 4. TRIGONOMETRY
87
4.4 Trigonometric Functions of Common
Angles
381. =
°
α = ê~Ç
α = α
ëáå = α
Åçë = α
í~å = α
Åçí α
ëÉÅ = α
ÅçëÉÅ =
M= M= M= N= M= ∞= N= ∞=
PM=
S
π
=
O
N
=
O
P
=
P
N
= P =
P
O
= O=
QR=
Q
π
=
O
O
=
O
O
= N= N= O = O =
SM=
P
π
=
O
P
=
O
N
= P =
P
N
= O=
P
O
=
VM=
O
π
= N= M= ∞ = M= ∞ = N=
NOM=
P
Oπ
=
O
P
=
O
N
− = P
− =
P
N
− O
− =
P
O
=
NUM= π= M= N
− = M= ∞ = N
− = ∞ =
OTM=
O
Pπ
= N
− = M= ∞= M= ∞= N
− =
PSM= π
O = M= N= M= ∞ = N= ∞ =
=
=
=
=
=
=
=
=
=
=
=
=
=

98. CHAPTER 4. TRIGONOMETRY
88
382. =
°
α = ê~Ç
α = α
ëáå = α
Åçë = α
í~å = α
Åçí =
NR=
NO
π
=
Q
O
S −
=
Q
O
S +
= P
O− = P
O+ =
NU=
NM
π
=
Q
N
R −
=
Q
R
O
NM +
R
R
O
R−
= R
O
R+ =
PS=
R
π
=
Q
R
O
NM −
Q
N
R +
=
N
R
R
O
NM
+
−
R
O
NM
N
R
−
+
=
RQ=
NM
Pπ
=
Q
N
R +
=
Q
R
O
NM −
R
O
NM
N
R
−
+
N
R
R
O
NM
+
−
=
TO=
R
Oπ
=
Q
R
O
NM +
Q
N
R −
= R
O
R+ = R
R
O
R−
=
TR=
NO
Rπ
=
Q
O
S +
=
Q
O
S −
= P
O+ = P
O− =
=
=
=
4.5 Most Important Formulas
=
383. N
Åçë
ëáå O
O
=
α
+
α =
=
384. N
í~å
ëÉÅ O
O
=
α
−
α =
=
385. N
Åçí
ÅëÅ O
O
=
α
−
α =
=
386.
α
α
=
α
Åçë
ëáå
í~å =

99. CHAPTER 4. TRIGONOMETRY
89
387.
α
α
=
α
ëáå
Åçë
Åçí =
=
388. N
Åçí
í~å =
α
⋅
α =
=
389.
α
=
α
Åçë
N
ëÉÅ =
=
390.
α
=
α
ëáå
N
ÅçëÉÅ =
=
=
=
4.6 Reduction Formulas
=
391. =
=
β = β
ëáå = β
Åçë = β
í~å = β
Åçí =
α
− = α
− ëáå = α
+ Åçë = α
− í~å = α
− Åçí =
α
−
°
VM = α
+ Åçë = α
+ ëáå = α
+ Åçí = α
+ í~å =
α
+
°
VM = α
+ Åçë = α
− ëáå = α
− Åçí = α
− í~å =
α
−
°
NUM α
+ ëáå = α
− Åçë = α
− í~å = α
− Åçí =
α
+
°
NUM α
− ëáå = α
− Åçë = α
+ í~å = α
+ Åçí =
α
−
°
OTM α
− Åçë = α
− ëáå = α
+ Åçí = α
+ í~å =
α
+
°
OTM α
− Åçë = α
+ ëáå = α
− Åçí = α
− í~å =
α
−
°
PSM α
− ëáå = α
+ Åçë = α
− í~å = α
− Åçí =
α
+
°
PSM α
+ ëáå = α
+ Åçë = α
+ í~å = α
+ Åçí =
=
=
=
=
=
=

100. CHAPTER 4. TRIGONOMETRY
90
4.7 Periodicity of Trigonometric Functions
=
392. ( ) α
=
π
±
α ëáå
å
O
ëáå I=éÉêáçÇ= π
O =çê= °
PSM K=
=
393. ( ) α
=
π
±
α Åçë
å
O
Åçë I=éÉêáçÇ= π
O =çê= °
PSM K=
=
394. ( ) α
=
π
±
α í~å
å
í~å I=éÉêáçÇ=π=çê= °
NUM K=
=
395. ( ) α
=
π
±
α Åçí
å
Åçí I=éÉêáçÇ=π=çê= °
NUM K=
=
=
=
4.8 Relations between Trigonometric
Functions
=
396. ( ) N
Q
O
Åçë
O
O
Åçë
N
O
N
Åçë
N
ëáå O
O
−





 π
−
α
=
α
−
±
=
α
−
±
=
α =
=
O
í~å
N
O
í~å
O
O α
+
α
= =
=
397. ( ) N
O
Åçë
O
O
Åçë
N
O
N
ëáå
N
Åçë O
O
−
α
=
α
+
±
=
α
−
±
=
α =
=
O
í~å
N
O
í~å
N
O
O
α
+
α
−
= =
=
398.
α
α
−
=
α
+
α
=
−
α
±
=
α
α
=
α
O
ëáå
O
Åçë
N
O
Åçë
N
O
ëáå
N
ëÉÅ
Åçë
ëáå
í~å O
=

101. CHAPTER 4. TRIGONOMETRY
91
=
O
í~å
N
O
í~å
O
O
Åçë
N
O
Åçë
N
O α
+
α
=
α
+
α
−
±
= =
=
399.
α
−
α
=
α
α
+
=
−
α
±
=
α
α
=
α
O
Åçë
N
O
ëáå
O
ëáå
O
Åçë
N
N
ÅëÅ
ëáå
Åçë
Åçí O
=
=
O
í~å
O
O
í~å
N
O
Åçë
N
O
Åçë
N
O
α
α
−
=
α
−
α
+
±
= =
=
400.
O
í~å
N
O
í~å
N
í~å
N
Åçë
N
ëÉÅ
O
O
O
α
−
α
+
=
α
+
±
=
α
=
α =
=
401.
O
í~å
O
O
í~å
N
Åçí
N
ëáå
N
ÅëÅ
O
O
α
α
+
=
α
+
±
=
α
=
α =
=
=
=
4.9 Addition and Subtraction Formulas
=
402. ( ) α
β
+
β
α
=
β
+
α Åçë
ëáå
Åçë
ëáå
ëáå =
=
403. ( ) α
β
−
β
α
=
−
α Åçë
ëáå
Åçë
ëáå
ó
ëáå =
=
404. ( ) β
α
−
β
α
=
β
+
α ëáå
ëáå
Åçë
Åçë
Åçë =
=
405. ( ) β
α
+
β
α
=
β
−
α ëáå
ëáå
Åçë
Åçë
Åçë =

102. CHAPTER 4. TRIGONOMETRY
92
406. ( )
β
α
−
β
+
α
=
β
+
α
í~å
í~å
N
í~å
í~å
í~å =
=
407. ( )
β
α
+
β
−
α
=
β
−
α
í~å
í~å
N
í~å
í~å
í~å =
=
408. ( )
β
+
α
β
α
−
=
β
+
α
í~å
í~å
í~å
í~å
N
Åçí =
=
409. ( )
β
−
α
β
α
+
=
β
−
α
í~å
í~å
í~å
í~å
N
Åçí =
=
=
=
4.10 Double Angle Formulas
=
410. α
⋅
α
=
α Åçë
ëáå
O
O
ëáå =
=
411. N
Åçë
O
ëáå
O
N
ëáå
Åçë
O
Åçë O
O
O
O
−
α
=
α
−
=
α
−
α
=
α =
=
412.
α
−
α
=
α
−
α
=
α
í~å
Åçí
O
í~å
N
í~å
O
O
í~å O
=
=
413.
O
í~å
Åçí
Åçí
O
N
Åçí
O
Åçí
O
α
−
α
=
α
−
α
=
α =
=
=
=
=
=
=

103. CHAPTER 4. TRIGONOMETRY
93
4.11 Multiple Angle Formulas
=
414. α
−
α
⋅
α
=
α
−
α
=
α P
O
P
ëáå
ëáå
Åçë
P
ëáå
Q
ëáå
P
P
ëáå =
=
415. α
⋅
α
−
α
⋅
α
=
α Åçë
ëáå
U
Åçë
ëáå
Q
Q
ëáå P
=
=
416. α
+
α
−
α
=
α R
P
ëáå
NS
ëáå
OM
ëáå
R
R
ëáå =
=
417. α
⋅
α
−
α
=
α
−
α
=
α O
P
P
ëáå
Åçë
P
Åçë
Åçë
P
Åçë
Q
P
Åçë =
=
418. N
Åçë
U
Åçë
U
Q
Åçë O
Q
+
α
−
α
=
α =
=
419. α
+
α
−
α
=
α Åçë
R
Åçë
OM
Åçë
NS
R
Åçë P
R
=
=
420.
α
−
α
−
α
=
α O
P
í~å
P
N
í~å
í~å
P
P
í~å =
=
421.
α
+
α
−
α
−
α
=
α Q
O
P
í~å
í~å
S
N
í~å
Q
í~å
Q
Q
í~å =
=
422.
α
+
α
−
α
+
α
−
α
=
α Q
O
P
R
í~å
R
í~å
NM
N
í~å
R
í~å
NM
í~å
R
í~å =
=
423.
N
Åçí
P
Åçí
P
Åçí
P
Åçí O
P
−
α
α
−
α
=
α =
=
424.
α
−
α
α
+
α
−
=
α P
Q
O
í~å
Q
í~å
Q
í~å
í~å
S
N
Q
Åçí ==
=

104. CHAPTER 4. TRIGONOMETRY
94
425.
α
+
α
−
α
α
+
α
−
=
α
í~å
R
í~å
NM
í~å
í~å
R
í~å
NM
N
R
Åçí P
R
Q
O
=
=
=
=
4.12 Half Angle Formulas
=
426.
O
Åçë
N
O
ëáå
α
−
±
=
α
=
=
427.
O
Åçë
N
O
Åçë
α
+
±
=
α
=
=
428. α
−
α
=
α
α
−
=
α
+
α
=
α
+
α
−
±
=
α
Åçí
ÅëÅ
ëáå
Åçë
N
Åçë
N
ëáå
Åçë
N
Åçë
N
O
í~å =
=
429. α
+
α
=
α
α
+
=
α
−
α
=
α
−
α
+
±
=
α
Åçí
ÅëÅ
ëáå
Åçë
N
Åçë
N
ëáå
Åçë
N
Åçë
N
O
Åçí =
=
=
=
4.13 Half Angle Tangent Identities
=
430.
O
í~å
N
O
í~å
O
ëáå
O α
+
α
=
α =
=

105. CHAPTER 4. TRIGONOMETRY
95
431.
O
í~å
N
O
í~å
N
Åçë
O
O
α
+
α
−
=
α =
=
432.
O
í~å
N
O
í~å
O
í~å
O α
−
α
=
α =
=
433.
O
í~å
O
O
í~å
N
Åçí
O
α
α
−
=
α =
=
=
=
4.14 Transforming of Trigonometric
Expressions to Product
=
434.
O
Åçë
O
ëáå
O
ëáå
ëáå
β
−
α
β
+
α
=
β
+
α =
=
435.
O
ëáå
O
Åçë
O
ëáå
ëáå
β
−
α
β
+
α
=
β
−
α =
=
436.
O
Åçë
O
Åçë
O
Åçë
Åçë
β
−
α
β
+
α
=
β
+
α =
=
437.
O
ëáå
O
ëáå
O
Åçë
Åçë
β
−
α
β
+
α
−
=
β
−
α =
=

106. CHAPTER 4. TRIGONOMETRY
96
438.
( )
β
⋅
α
β
+
α
=
β
+
α
Åçë
Åçë
ëáå
í~å
í~å =
=
439.
( )
β
⋅
α
β
−
α
=
β
−
α
Åçë
Åçë
ëáå
í~å
í~å =
=
440.
( )
β
⋅
α
α
+
β
=
β
+
α
ëáå
ëáå
ëáå
Åçí
Åçí =
=
441.
( )
β
⋅
α
α
−
β
=
β
−
α
ëáå
ëáå
ëáå
Åçí
Åçí =
=
442. 





α
+
π
=






α
−
π
=
α
+
α
Q
ëáå
O
Q
Åçë
O
ëáå
Åçë =
=
443. 





α
+
π
=






α
−
π
=
α
−
α
Q
Åçë
O
Q
ëáå
O
ëáå
Åçë =
=
444.
( )
β
⋅
α
β
−
α
=
β
+
α
ëáå
Åçë
Åçë
Åçí
í~å =
=
445.
( )
β
⋅
α
β
+
α
−
=
β
−
α
ëáå
Åçë
Åçë
Åçí
í~å =
=
446.
O
Åçë
O
Åçë
N O α
=
α
+ =
=
447.
O
ëáå
O
Åçë
N O α
=
α
− =
=

107. CHAPTER 4. TRIGONOMETRY
97
448. 




 α
−
π
=
α
+
O
Q
Åçë
O
ëáå
N O
=
=
449. 




 α
−
π
=
α
−
O
Q
ëáå
O
ëáå
N O
=
=
=
=
4.15 Transforming of Trigonometric
Expressions to Sum
=
450.
( ) ( )
O
Åçë
Åçë
ëáå
ëáå
β
+
α
−
β
−
α
=
β
⋅
α =
=
451.
( ) ( )
O
Åçë
Åçë
Åçë
Åçë
β
+
α
+
β
−
α
=
β
⋅
α =
=
452.
( ) ( )
O
ëáå
ëáå
Åçë
ëáå
β
+
α
+
β
−
α
=
β
⋅
α =
=
453.
β
+
α
β
+
α
=
β
⋅
α
Åçí
Åçí
í~å
í~å
í~å
í~å =
=
454.
β
+
α
β
+
α
=
β
⋅
α
í~å
í~å
Åçí
Åçí
Åçí
Åçí =
=
455.
β
+
α
β
+
α
=
β
⋅
α
í~å
Åçí
Åçí
í~å
Åçí
í~å =
=
=
=

108. CHAPTER 4. TRIGONOMETRY
98
4.16 Powers of Trigonometric Functions
=
456.
O
O
Åçë
N
ëáåO α
−
=
α =
=
457.
Q
P
ëáå
ëáå
P
ëáåP α
−
α
=
α =
=
458.
U
P
O
Åçë
Q
Q
Åçë
ëáåQ +
α
−
α
=
α =
=
459.
NS
R
ëáå
P
ëáå
R
ëáå
NM
ëáåR α
+
α
−
α
=
α =
=
460.
PO
S
Åçë
Q
Åçë
S
O
Åçë
NR
NM
ëáåS α
−
α
+
α
−
=
α =
=
461.
O
O
Åçë
N
ÅçëO α
+
=
α =
=
462.
Q
P
Åçë
Åçë
P
ÅçëP α
+
α
=
α =
=
463.
U
P
O
Åçë
Q
Q
Åçë
ÅçëQ +
α
+
α
=
α =
=
464.
NS
R
Åçë
P
ëáå
R
Åçë
NM
ÅçëR α
+
α
+
α
=
α =
=
465.
PO
S
Åçë
Q
Åçë
S
O
Åçë
NR
NM
ÅçëS α
+
α
+
α
+
=
α =
=

109. CHAPTER 4. TRIGONOMETRY
99
4.17 Graphs of Inverse Trigonometric
Functions
=
466. fåîÉêëÉ=páåÉ=cìåÅíáçå==
ñ
~êÅëáå
ó = I= N
ñ
N ≤
≤
− I=
O
ñ
~êÅëáå
O
π
≤
≤
π
− K=
=
=
=
Figure 66.
=
467. fåîÉêëÉ=`çëáåÉ=cìåÅíáçå==
ñ
~êÅÅçë
ó = I= N
ñ
N ≤
≤
− I= π
≤
≤ ñ
~êÅÅçë
M K=
=

110. CHAPTER 4. TRIGONOMETRY
100
=
=
Figure 67.
=
468. fåîÉêëÉ=q~åÖÉåí=cìåÅíáçå==
ñ
~êÅí~å
ó = I= ∞
≤
≤
∞
− ñ I=
O
ñ
~êÅí~å
O
π
<
<
π
− K=
=
===== =
=
Figure 68.

111. CHAPTER 4. TRIGONOMETRY
101
469. fåîÉêëÉ=`çí~åÖÉåí=cìåÅíáçå==
ñ
Åçí
~êÅ
ó = I= ∞
≤
≤
∞
− ñ I= π
<
< ñ
Åçí
~êÅ
M K=
===== =
Figure 69.
=
470. fåîÉêëÉ=pÉÅ~åí=cìåÅíáçå==
( ] [ ) K
I
O
O
I
M
ñ
ëÉÅ
~êÅ
I
I
N
N
I
ñ
I
ñ
=
~êÅëÉÅ
ó 





π
π
∪





 π
∈
∞
∪
−
∞
−
∈
=
=
Figure 70.

112. CHAPTER 4. TRIGONOMETRY
102
471. fåîÉêëÉ=`çëÉÅ~åí=cìåÅíáçå==
( ] [ ) K
O
I
M
M
I
O
ñ
ÅëÅ
~êÅ
I
I
N
N
I
ñ
I
ñ
~êÅÅëÅ
ó 




 π
∪





 π
−
∈
∞
∪
−
∞
−
∈
=
=
=
Figure 71.
=
=
4.18 Principal Values of Inverse
Trigonometric Functions
472.
ñ = M=
O
N
=
O
O
=
O
P
N=
ñ
~êÅëáå = °
M = °
PM = °
QR = °
SM °
VM
ñ
~êÅÅçë = °
VM °
SM = °
QR = °
PM °
M =
ñ = O
N
−
O
O
−
O
P
− N
− = =
ñ
~êÅëáå =
°
−PM
=
°
− QR °
− SM
°
− VM
=
=
ñ
~êÅÅçë =
°
NOM
=
°
NPR = °
NRM =
°
NUM
=
=

113. CHAPTER 4. TRIGONOMETRY
103
473.
ñ = M=
P
P
N= P =
P
P
− N
− = P
− =
ñ
~êÅí~å = °
M = °
PM °
QR °
SM °
−PM
°
− QR
=
°
− SM =
ñ
Åçí
~êÅ = °
VM °
SM °
QR °
PM °
NOM =
°
NPR
=
°
NRM =
=
=
=
4.19 Relations between Inverse
Trigonometric Functions
=
474. ( ) ñ
~êÅëáå
ñ
~êÅëáå −
=
− =
=
475. ñ
~êÅÅçë
O
ñ
~êÅëáå −
π
= =
=
476. O
ñ
N
~êÅÅçë
ñ
~êÅëáå −
= I= N
ñ
M ≤
≤ K=
=
477. O
ñ
N
~êÅÅçë
ñ
~êÅëáå −
−
= I= M
ñ
N ≤
≤
− K=
=
478.
O
ñ
N
ñ
~êÅí~å
ñ
~êÅëáå
−
= I= N
ñO
< K=
=
479.
ñ
ñ
N
Åçí
~êÅ
ñ
~êÅëáå
O
−
= I= N
ñ
M ≤
< K=
=
480. π
−
−
=
ñ
ñ
N
Åçí
~êÅ
ñ
~êÅëáå
O
I= M
ñ
N <
≤
− K=
=
481. ( ) ñ
~êÅÅçë
ñ
~êÅÅçë −
π
=
− =

114. CHAPTER 4. TRIGONOMETRY
104
482. ñ
~êÅëáå
O
ñ
~êÅÅçë −
π
= =
=
483. O
ñ
N
~êÅëáå
ñ
~êÅÅçë −
= I= N
ñ
M ≤
≤ K=
=
484. O
ñ
N
~êÅëáå
ñ
~êÅÅçë −
−
π
= I= M
ñ
N ≤
≤
− K=
=
485.
ñ
ñ
N
~êÅí~å
ñ
~êÅÅçë
O
−
= I= N
ñ
M ≤
< K=
=
486.
ñ
ñ
N
~êÅí~å
ñ
~êÅÅçë
O
−
+
π
= I= M
ñ
N <
≤
− K=
=
487.
O
ñ
N
ñ
Åçí
~êÅ
ñ
~êÅÅçë
−
= I= N
ñ
N ≤
≤
− K=
=
488. ( ) ñ
~êÅí~å
ñ
~êÅí~å −
=
− =
=
489. ñ
Åçí
~êÅ
O
ñ
~êÅí~å −
π
= =
=
490.
O
ñ
N
ñ
~êÅëáå
ñ
~êÅí~å
+
= =
=
491.
O
ñ
N
N
~êÅÅçë
ñ
~êÅí~å
+
= I= M
ñ ≥ K=
=
492.
O
ñ
N
N
~êÅÅçë
ñ
~êÅí~å
+
−
= I= M
ñ ≤ K=
=

115. CHAPTER 4. TRIGONOMETRY
105
493.
ñ
N
~êÅí~å
O
ñ
~êÅí~å −
π
= I= M
ñ > K=
=
494.
ñ
N
~êÅí~å
O
ñ
~êÅí~å −
π
−
= I= M
ñ < K=
=
495.
ñ
N
Åçí
~êÅ
ñ
~êÅí~å = I= M
ñ > K=
=
496. π
−
=
ñ
N
Åçí
~êÅ
ñ
~êÅí~å I= M
ñ < K=
=
497. ( ) ñ
Åçí
~êÅ
ñ
Åçí
~êÅ −
π
=
− =
=
498. ñ
~êÅí~å
O
ñ
Åçí
~êÅ −
π
= =
=
499.
O
ñ
N
N
~êÅëáå
ñ
Åçí
~êÅ
+
= I= M
ñ > K=
=
500.
O
ñ
N
N
~êÅëáå
ñ
Åçí
~êÅ
+
−
π
= I= M
ñ < K=
=
501.
O
ñ
N
ñ
~êÅÅçë
ñ
Åçí
~êÅ
+
= =
=
502.
ñ
N
~êÅí~å
ñ
Åçí
~êÅ = I= M
ñ > K=
=
503.
ñ
N
~êÅí~å
ñ
Åçí
~êÅ +
π
= I= M
ñ < K=
=
=

116. CHAPTER 4. TRIGONOMETRY
106
4.20 Trigonometric Equations
=
tÜçäÉ=åìãÄÉêW=å=
=
=
504. ~
ñ
ëáå = I= ( ) å
~
~êÅëáå
N
ñ
å
π
+
−
= =
=
505. ~
ñ
Åçë = I= å
O
~
~êÅÅçë
ñ π
+
±
= =
=
506. ~
ñ
í~å = I= å
~
~êÅí~å
ñ π
+
= =
=
507. ~
ñ
Åçí = I= å
~
Åçí
~êÅ
ñ π
+
= =
=
=
=
4.21 Relations to Hyperbolic Functions
=
fã~Öáå~êó=ìåáíW=á=
=
=
508. ( ) ñ
ëáåÜ
á
áñ
ëáå = =
=
509. ( ) ñ
í~åÜ
á
áñ
í~å = =
=
510. ( ) ñ
ÅçíÜ
á
áñ
Åçí −
= =
=
511. ( ) ñ
ëÉÅÜ
áñ
ëÉÅ = =
=
512. ( ) ñ
ÅëÅÜ
á
áñ
ÅëÅ −
= =
=
=
=

117. 107
Chapter 5
Matrices and Determinants
=
=
=
=
j~íêáÅÉëW=^I=_I=`=
bäÉãÉåíë=çÑ=~=ã~íêáñW= á
~ I= á
Ä I= áà
~ I= áà
Ä I= áà
Å =
aÉíÉêãáå~åí=çÑ=~=ã~íêáñW= ^
ÇÉí =
jáåçê=çÑ=~å=ÉäÉãÉåí= áà
~ W= áà
j =
`çÑ~Åíçê=çÑ=~å=ÉäÉãÉåí= áà
~ W= áà
` =
qê~åëéçëÉ=çÑ=~=ã~íêáñW= q
^ I= ^
ú
=
^Çàçáåí=çÑ=~=ã~íêáñW= ^
~Çà =
qê~ÅÉ=çÑ=~=ã~íêáñW= ^
íê =
fåîÉêëÉ=çÑ=~=ã~íêáñW= N
^−
=
oÉ~ä=åìãÄÉêW=â=
oÉ~ä=î~êá~ÄäÉëW= á
ñ =
k~íìê~ä=åìãÄÉêëW=ãI=å===
=
=
5.1 Determinants
=
513. pÉÅçåÇ=lêÇÉê=aÉíÉêãáå~åí=
N
O
O
N
O
O
N
N
Ä
~
Ä
~
Ä
~
Ä
~
^
ÇÉí −
=
= =
=
=
=
=
=

118. CHAPTER 5. MATRICES AND DETERMINANTS
108
514. qÜáêÇ=lêÇÉê=aÉíÉêãáå~åí=
−
+
+
=
= PO
ON
NP
PN
OP
NO
PP
OO
NN
PP
PO
PN
OP
OO
ON
NP
NO
NN
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
^
ÇÉí =
PN
OO
NP
PP
ON
NO
PO
OP
NN ~
~
~
~
~
~
~
~
~ −
−
− =
=
515. p~êêìë=oìäÉ=E^êêçï=oìäÉF=
=
=
Figure 72.
=
516. k-íÜ=lêÇÉê=aÉíÉêãáå~åí=
åå
åà
O
å
N
å
áå
áà
O
á
N
á
å
O
à
O
OO
ON
å
N
à
N
NO
NN
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
^
ÇÉí
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
= =
=
517. jáåçê=
qÜÉ=ãáåçê= áà
j =~ëëçÅá~íÉÇ=ïáíÜ=íÜÉ=ÉäÉãÉåí= áà
~ =çÑ=å-íÜ=çêÇÉê=
ã~íêáñ=^=áë=íÜÉ= ( )
N
å − -íÜ=çêÇÉê=ÇÉíÉêãáå~åí=ÇÉêáîÉÇ=Ñêçã=
íÜÉ=ã~íêáñ=^=Äó=ÇÉäÉíáçå=çÑ=áíë=á-íÜ=êçï=~åÇ=à-íÜ=ÅçäìãåK===
=

119. CHAPTER 5. MATRICES AND DETERMINANTS
109
518. `çÑ~Åíçê=
( ) áà
à
á
áà j
N
`
+
−
= =
=
519. i~éä~ÅÉ=bñé~åëáçå=çÑ=å-íÜ=lêÇÉê=aÉíÉêãáå~åí=
i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=á-íÜ=êçï=
∑
=
=
å
N
à
áà
áà`
~
^
ÇÉí I= å
I
I
O
I
N
á K
= K=
i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=à-íÜ=Åçäìãå=
∑
=
=
å
N
á
áà
áà`
~
^
ÇÉí I= å
I
I
O
I
N
à K
= K==
=
=
=
5.2 Properties of Determinants
=
520. qÜÉ==î~äìÉ==çÑ=~=ÇÉíÉêãáå~åí=êÉã~áåë==ìåÅÜ~åÖÉÇ=áÑ=êçïë=~êÉ=
ÅÜ~åÖÉÇ=íç=Åçäìãåë=~åÇ=Åçäìãåë=íç=êçïëK=
=
O
O
N
N
O
N
O
N
Ä
~
Ä
~
Ä
Ä
~
~
= ==
=
521. fÑ=íïç==êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áåíÉêÅÜ~åÖÉÇI=íÜÉ=ëáÖå=çÑ=
íÜÉ=ÇÉíÉêãáå~åí=áë=ÅÜ~åÖÉÇK=
N
N
O
O
O
O
N
N
Ä
~
Ä
~
Ä
~
Ä
~
−
= =
=
522. fÑ=íïç=êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áÇÉåíáÅ~äI=íÜÉ=î~äìÉ=çÑ=íÜÉ=
ÇÉíÉêãáå~åí=áë=òÉêçK=
M
~
~
~
~
O
O
N
N
= =
=

120. CHAPTER 5. MATRICES AND DETERMINANTS
110
523. fÑ==íÜÉ===ÉäÉãÉåíë==çÑ==~åó=êçï==Eçê=ÅçäìãåF=~êÉ=ãìäíáéäáÉÇ=Äó=====
~==Åçããçå==Ñ~ÅíçêI==íÜÉ==ÇÉíÉêãáå~åí==áë==ãìäíáéäáÉÇ==Äó==íÜ~í=
Ñ~ÅíçêK=
O
O
N
N
O
O
N
N
Ä
~
Ä
~
â
Ä
~
âÄ
â~
= =
=
524. fÑ==íÜÉ==ÉäÉãÉåíë==çÑ==~åó==êçï==Eçê==ÅçäìãåF=~êÉ=áåÅêÉ~ëÉÇ=Eçê=
ÇÉÅêÉ~ëÉÇFÄó=Éèì~ä=ãìäíáéäÉë=çÑ=íÜÉ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=
çÑ=~åó=çíÜÉê=êçï==Eçê=ÅçäìãåFI==íÜÉ=î~äìÉ=çÑ=íÜÉ=ÇÉíÉêãáå~åí=
áë=ìåÅÜ~åÖÉÇK=
O
O
N
N
O
O
O
N
N
N
Ä
~
Ä
~
Ä
âÄ
~
Ä
âÄ
~
=
+
+
=
=
=
=
5.3 Matrices
=
525. aÉÑáåáíáçå=
^å= å
ã× =ã~íêáñ=^=áë=~=êÉÅí~åÖìä~ê=~êê~ó=çÑ=ÉäÉãÉåíë=Eåìã-
ÄÉêë=çê=ÑìåÅíáçåëF=ïáíÜ=ã=êçïë=~åÇ=å=ÅçäìãåëK==
[ ]












=
=
ãå
O
ã
N
ã
å
O
OO
ON
å
N
NO
NN
áà
~
~
~
~
~
~
~
~
~
~
^
K
M
M
M
K
K
==
=
526. pèì~êÉ=ã~íêáñ=áë=~=ã~íêáñ=çÑ=çêÇÉê= å
å× K==
=
527. ^=ëèì~êÉ=ã~íêáñ==[ ]
áà
~ ==áë==ëóããÉíêáÅ==áÑ== àá
áà ~
~ = I==áKÉK==áí==áë=
ëóããÉíêáÅ=~Äçìí=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK==
=
528. ^=ëèì~êÉ=ã~íêáñ=[ ]
áà
~ =áë=ëâÉï-ëóããÉíêáÅ=áÑ= àá
áà ~
~ −
= K==
=

121. CHAPTER 5. MATRICES AND DETERMINANTS
111
529. aá~Öçå~ä=ã~íêáñ==áë==~=ëèì~êÉ==ã~íêáñ=ïáíÜ=~ää==ÉäÉãÉåíë==òÉêç=
ÉñÅÉéí=íÜçëÉ=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK==
=
530. råáí=ã~íêáñ==áë==~=Çá~Öçå~ä==ã~íêáñ==áå=ïÜáÅÜ=íÜÉ=ÉäÉãÉåíë=çå=
íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~ä=~êÉ=~ää=ìåáíóK=qÜÉ=ìåáí=ã~íêáñ=áë===========
ÇÉåçíÉÇ=Äó=fK==
=
531. ^=åìää=ã~íêáñ=áë=çåÉ=ïÜçëÉ=ÉäÉãÉåíë=~êÉ=~ää=òÉêçK=
=
=
=
5.4 Operations with Matrices
=
532. qïç=ã~íêáÅÉë=^=~åÇ=_=~êÉ=Éèì~ä=áÑI=~åÇ=çåäó=áÑI=íÜÉó=~êÉ=ÄçíÜ=
çÑ==íÜÉ==ë~ãÉ==ëÜ~éÉ== å
ã× ==~åÇ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=~êÉ=
Éèì~äK=
=
533. qïç=ã~íêáÅÉë==^=~åÇ=_==Å~å=ÄÉ=~ÇÇÉÇ=Eçê=ëìÄíê~ÅíÉÇF=çÑI=~åÇ=
çåäó=áÑI=íÜÉó=Ü~îÉ=íÜÉ=ë~ãÉ=ëÜ~éÉ= å
ã× K=fÑ==
[ ]
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=
=
ãå
O
ã
N
ã
å
O
OO
ON
å
N
NO
NN
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~
~
~
~
~
~
~
~
~
~
^
K
M
M
M
K
K
I==
[ ]
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=
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O
ã
N
ã
å
O
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å
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Ä
Ä
Ä
Ä
Ä
Ä
Ä
Ä
_
K
M
M
M
K
K
I==
=
=
=
=
=

122. CHAPTER 5. MATRICES AND DETERMINANTS
112
íÜÉå==












+
+
+
+
+
+
+
+
+
=
+
ãå
ãå
O
ã
O
ã
N
ã
N
ã
å
O
å
O
OO
OO
ON
ON
å
N
å
N
NO
NO
NN
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Ä
~
Ä
~
Ä
~
Ä
~
Ä
~
Ä
~
Ä
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Ä
~
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_
^
K
M
M
M
K
K
K=
=
534. fÑ=â=áë=~=ëÅ~ä~êI=~åÇ= [ ]
áà
~
^ = =áë=~=ã~íêáñI=íÜÉå=
[ ]
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â~
â~
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K=
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535. jìäíáéäáÅ~íáçå=çÑ=qïç=j~íêáÅÉë=
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=
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123. CHAPTER 5. MATRICES AND DETERMINANTS
113
íÜÉå==


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
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Å
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Å
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ïÜÉêÉ==
∑
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λ
λ
λ
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+
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å
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à
á
åà
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à
O
O
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à
N
N
á
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~
Ä
~
Ä
~
Ä
~
Å K =
E ã
I
I
O
I
N
á K
= X â
I
I
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à K
= FK==
=
qÜìë=áÑ=
[ ] 
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536. qê~åëéçëÉ=çÑ=~=j~íêáñ=
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fÑ=^=áë=íÜÉ=çêáÖáå~ä=ã~íêáñI=áíë=íê~åëéçëÉ=áë=ÇÉåçíÉÇ= q
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538. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå==
( ) q
q
q
^
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^_ = K=
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124. CHAPTER 5. MATRICES AND DETERMINANTS
114
539. ^Çàçáåí=çÑ=j~íêáñ=
fÑ=^=áë=~=ëèì~êÉ= å
å× ã~íêáñI=áíë=~ÇàçáåíI=ÇÉåçíÉÇ=Äó= ^
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` =çÑ=^W=
[ ]q
áà
`
^
~Çà = K==
=
540. qê~ÅÉ=çÑ=~=j~íêáñ=
fÑ=^=áë=~=ëèì~êÉ= å
å× ã~íêáñI=áíë=íê~ÅÉI=ÇÉåçíÉÇ=Äó= ^
íê I=áë=
ÇÉÑáåÉÇ=íç=ÄÉ==íÜÉ=ëìã=çÑ==íÜÉ=íÉêãë=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äW=
åå
OO
NN ~
~
~
^
íê +
+
+
= K K=
=
541. fåîÉêëÉ=çÑ=~=j~íêáñ=
fÑ=^=áë=~=ëèì~êÉ= å
å× ã~íêáñ=ïáíÜ=~=åçåëáåÖìä~ê=ÇÉíÉêãáå~åí=
^
ÇÉí I=íÜÉå=áíë=áåîÉêëÉ= N
^−
=áë=ÖáîÉå=Äó=
^
ÇÉí
^
~Çà
^ N
=
−
K=
=
542. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå==
( ) N
N
N
^
_
^_ −
−
−
= K=
=
543. fÑ==^==áë=~=ëèì~êÉ=== å
å× ==ã~íêáñI==íÜÉ==ÉáÖÉåîÉÅíçêë==u===ë~íáëÑó=
íÜÉ=Éèì~íáçå=
u
^u λ
= I==
ïÜáäÉ=íÜÉ=ÉáÖÉåî~äìÉë=λ =ë~íáëÑó=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå=
M
f
^ =
λ
− K===
=
=
=
5.5 Systems of Linear Equations
=
=
s~êá~ÄäÉëW=ñI=óI=òI= N
ñ I= K
I
ñO =
oÉ~ä=åìãÄÉêëW= K
I
~
I
~
I
Ä
I
~
I
~
I
~ NO
NN
N
P
O
N =

125. CHAPTER 5. MATRICES AND DETERMINANTS
115
aÉíÉêãáå~åíëW=aI= ñ
a I= ó
a I= ò
a ==
j~íêáÅÉëW=^I=_I=u=
=
=
544.



=
+
=
+
O
O
O
N
N
N
Ç
ó
Ä
ñ
~
Ç
ó
Ä
ñ
~
I==
a
a
ñ ñ
= I=
a
a
ó
ó
= =E`ê~ãÉê∞ë=êìäÉFI==
ïÜÉêÉ==
N
O
O
N
O
O
N
N
Ä
~
Ä
~
Ä
~
Ä
~
a −
=
= I==
N
O
O
N
O
O
N
N
ñ Ä
Ç
Ä
Ç
Ä
Ç
Ä
Ç
a −
=
= I==
N
O
O
N
O
O
N
N
ó Ç
~
Ç
~
Ç
~
Ç
~
a −
=
= K==
=
545. fÑ= M
a ≠ I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==
a
a
ñ ñ
= I=
a
a
ó
ó
= K=
fÑ= M
a = =~åÇ= M
añ ≠ Eçê= M
aó ≠ FI=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë==åç==
ëçäìíáçåK=
fÑ= M
a
a
a ó
ñ =
=
= I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = áåÑáåáíÉäó= = ã~åó==
ëçäìíáçåëK=
=
546.





=
+
+
=
+
+
=
+
+
P
P
P
P
O
O
O
O
N
N
N
N
Ç
ò
Å
ó
Ä
ñ
~
Ç
ò
Å
ó
Ä
ñ
~
=
Ç
ò
Å
ó
Ä
ñ
~
I==
a
a
ñ ñ
= I=
a
a
ó
ó
= I=
a
a
ò ò
= =E`ê~ãÉê∞ë=êìäÉFI==
=

126. CHAPTER 5. MATRICES AND DETERMINANTS
116
ïÜÉêÉ==
P
P
P
O
O
O
N
N
N
Å
Ä
~
Å
Ä
~
Å
Ä
~
a = I=
P
P
P
O
O
O
N
N
N
ñ
Å
Ä
Ç
Å
Ä
Ç
Å
Ä
Ç
a = I=
P
P
P
O
O
O
N
N
N
ó
Å
Ç
~
Å
Ç
~
Å
Ç
~
a = I=
P
P
P
O
O
O
N
N
N
ò
Ç
Ä
~
Ç
Ä
~
Ç
Ä
~
a = K==
=
547. fÑ= M
a ≠ I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==
a
a
ñ ñ
= I=
a
a
ó
ó
= I=
a
a
ò ò
= K=
fÑ= M
a = =~åÇ= M
añ ≠ Eçê= M
aó ≠ =çê= M
aò ≠ FI=íÜÉå=íÜÉ=ëóëíÉã=
Ü~ë=åç=ëçäìíáçåK=
fÑ= M
a
a
a
a ò
ó
ñ =
=
=
= I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= áåÑáåáíÉäó=
ã~åó=ëçäìíáçåëK=
=
548. j~íêáñ=cçêã=çÑ=~=póëíÉã=çÑ=å=iáåÉ~ê=bèì~íáçåë=áå=================
å=råâåçïåë=
qÜÉ=ëÉí=çÑ=äáåÉ~ê=Éèì~íáçåë==







=
+
+
+
=
+
+
+
=
+
+
+
å
å
åå
O
O
å
N
N
å
O
å
å
O
O
OO
N
ON
N
å
å
N
O
NO
N
NN
Ä
ñ
~
ñ
~
ñ
~
Ä
ñ
~
ñ
~
ñ
~
Ä
ñ
~
ñ
~
ñ
~
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
=
Å~å=ÄÉ=ïêáííÉå=áå=ã~íêáñ=Ñçêã=














=














⋅














å
O
N
å
O
N
åå
O
å
N
å
å
O
OO
ON
å
N
NO
NN
Ä
Ä
Ä
ñ
ñ
ñ
~
~
~
~
~
~
~
~
~
M
M
K
M
M
M
K
K
I==
áKÉK==
_
u
^ =
⋅ I==

127. CHAPTER 5. MATRICES AND DETERMINANTS
117
ïÜÉêÉ==














=
åå
O
å
N
å
å
O
OO
ON
å
N
NO
NN
~
~
~
~
~
~
~
~
~
^
K
M
M
M
K
K
I=














=
å
O
N
ñ
ñ
ñ
u
M
I=














=
å
O
N
Ä
Ä
Ä
_
M
K==
=
549. pçäìíáçå=çÑ=~=pÉí=çÑ=iáåÉ~ê=bèì~íáçåë= å
å× =
_
^
u N
⋅
= −
I==
ïÜÉêÉ= N
^−
=áë=íÜÉ=áåîÉêëÉ=çÑ=^K=
=
=

128. 118
Chapter 6
Vectors
=
=
=
=
sÉÅíçêëW=ì
r
I= î
r
I= ï
r
I= ê
r
I=
→
^_ I=£=
sÉÅíçê=äÉåÖíÜW= ì
r
I= î
r
I=£=
råáí=îÉÅíçêëW= á
r
I= à
r
I=â
r
=
kìää=îÉÅíçêW=M
r
=
`ççêÇáå~íÉë=çÑ=îÉÅíçê=ì
r
W= N
N
N w
I
v
I
u =
`ççêÇáå~íÉë=çÑ=îÉÅíçê= î
r
W= O
O
O w
I
v
I
u =
pÅ~ä~êëW=λ Iµ=
aáêÉÅíáçå=ÅçëáåÉëW= α
Åçë I= β
Åçë I= γ
Åçë =
^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW=θ =
=
=
6.1 Vector Coordinates
=
550. råáí=sÉÅíçêë=
( )
M
I
M
I
N
á =
r
I=
( )
M
I
N
I
M
à =
r
I=
( )
N
I
M
I
M
â =
r
I=
N
â
à
á =
=
=
r
r
r
K=
=
551. ( ) ( ) ( )â
ò
ò
à
ó
ó
á
ñ
ñ
^_
ê M
N
M
N
M
N
r
r
r
r
−
+
−
+
−
=
=
→
=
=

129. CHAPTER 6. VECTORS
119
======= =
=
Figure 73.
=
552. ( ) ( ) ( )O
M
N
O
M
N
O
M
N ò
ò
ó
ó
ñ
ñ
^_
ê −
+
−
+
−
=
=
→
r
=
=
553. fÑ= ê
^_
r
=
→
I=íÜÉå= ê
_^
r
−
=
→
K=
=
=
=
Figure 74.
=
554. α
= Åçë
ê
u
r
I=
β
= Åçë
ê
v
r
I=
γ
= Åçë
ê
w
r
K=

130. CHAPTER 6. VECTORS
120
===== =
=
Figure 75.
=
555. fÑ= ( ) ( )
N
N
N
N w
I
v
I
u
ê
w
I
v
I
u
ê
r
r
= I=íÜÉå==
N
u
u = I= N
v
v = I= N
w
w = K==
==
=
6.2 Vector Addition
=
556. î
ì
ï
r
r
r
+
= =
=
== =
=
Figure 76.

131. CHAPTER 6. VECTORS
121
== =
=
Figure 77.
=
557. å
P
O
N ì
ì
ì
ì
ï
r
K
r
r
r
r
+
+
+
+
= =
=
== =
=
Figure 78.
=
558. `çããìí~íáîÉ=i~ï=
ì
î
î
ì
r
r
r
r
+
=
+ =
=
559. ^ëëçÅá~íáîÉ=i~ï=
( ) ( )
ï
î
ì
ï
î
ì
r
r
r
r
r
r
+
+
=
+
+ =
=
560. ( )
O
N
O
N
O
N w
w
I
v
v
I
u
u
î
ì +
+
+
=
+
r
r
=
=
=
=
=
=
=

132. CHAPTER 6. VECTORS
122
6.3 Vector Subtraction
=
561. î
ì
ï
r
r
r
−
= =áÑ= ì
ï
î
r
r
r
=
+ K=
=
=
=
Figure 79.
=
== =
=
Figure 80.
=
562. ( )
î
ì
î
ì
r
r
r
r
−
+
=
− =
=
563. ( )
M
I
M
I
M
M
ì
ì =
=
−
r
r
r
=
=
564. M
M =
r
=
=
565. ( )
O
N
O
N
O
N w
w
I
v
v
I
u
u
î
ì −
−
−
=
−
r
r
I==
=
=
=
6.4 Scaling Vectors
=
566. ì
ï
r
r
λ
= =

133. CHAPTER 6. VECTORS
123
=
=
Figure 81.
=
567. ì
ï
r
r
⋅
λ
= =
=
568. ( )
w
I
v
I
u
ì λ
λ
λ
=
λ
r
=
=
569. λ
=
λ ì
ì
r
r
=
=
570. ( ) ì
ì
ì
r
r
r
µ
+
λ
=
µ
+
λ =
=
571. ( ) ( ) ( )ì
ì
ì
r
r
r
λµ
=
λ
µ
=
µ
λ =
=
572. ( ) î
ì
î
ì
r
r
r
r
λ
+
λ
=
+
λ =
=
=
=
6.5 Scalar Product
=
573. pÅ~ä~ê=mêçÇìÅí=çÑ=sÉÅíçêë=ì
r
=~åÇ î
r
=
θ
⋅
⋅
=
⋅ Åçë
î
ì
î
ì
r
r
r
r
I==
ïÜÉêÉ=θ =áë=íÜÉ=~åÖäÉ=ÄÉíïÉÉå=îÉÅíçêë=ì
r
=~åÇ î
r
K====
=

134. CHAPTER 6. VECTORS
124
= =
=
Figure 82.
=
574. pÅ~ä~ê=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=
fÑ= ( )
N
N
N w
I
v
I
u
ì =
r
I= ( )
O
O
O w
I
v
I
u
î =
r
I=íÜÉå==
O
N
O
N
O
N w
w
v
v
u
u
î
ì +
+
=
⋅
r
r
K=
=
575. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë==
fÑ= ( )
N
N
N w
I
v
I
u
ì =
r
I= ( )
O
O
O w
I
v
I
u
î =
r
I=íÜÉå==
O
O
O
O
O
O
O
N
O
N
O
N
O
N
O
N
O
N
w
v
u
w
v
u
w
w
v
v
u
u
Åçë
+
+
+
+
+
+
=
θ K=
=
576. `çããìí~íáîÉ=mêçéÉêíó=
ì
î
î
ì
r
r
r
r
⋅
=
⋅ =
=
577. ^ëëçÅá~íáîÉ=mêçéÉêíó=
( ) ( ) î
ì
î
ì
r
r
r
r
⋅
λµ
=
µ
⋅
λ =
=
578. aáëíêáÄìíáîÉ=mêçéÉêíó=
( ) ï
ì
î
ì
ï
î
ì
r
r
r
r
r
r
r
⋅
+
⋅
=
+
⋅ =
=
579. M
î
ì =
⋅
r
r
=áÑ=ì
r
I î
r
=~êÉ=çêíÜçÖçå~ä=E
O
π
=
θ FK=
=
580. M
î
ì >
⋅
r
r
=áÑ=
O
M
π
<
θ
< K=
=

135. CHAPTER 6. VECTORS
125
581. M
î
ì <
⋅
r
r
=áÑ= π
<
θ
<
π
O
K=
=
582. î
ì
î
ì
r
r
r
r
⋅
≤
⋅ =
=
583. î
ì
î
ì
r
r
r
r
⋅
=
⋅ =áÑ=ì
r
I î
r
=~êÉ=é~ê~ääÉä=E M
=
θ FK=
=
584. fÑ= ( )
N
N
N w
I
v
I
u
ì =
r
I=íÜÉå==
O
N
O
N
O
N
O
O
w
v
u
ì
ì
ì
ì +
+
=
=
=
⋅
r
r
r
r
K=
=
585. N
â
â
à
à
á
á =
⋅
=
⋅
=
⋅
r
r
r
r
r
r
=
=
586. M
á
â
â
à
à
á =
⋅
=
⋅
=
⋅
r
r
r
r
r
r
=
=
=
=
6.6 Vector Product
=
587. sÉÅíçê=mêçÇìÅí=çÑ=sÉÅíçêë=ì
r
=~åÇ î
r
=
ï
î
ì
r
r
r
=
× I=ïÜÉêÉ==
• θ
⋅
⋅
= ëáå
î
ì
ï
r
r
r
I=ïÜÉêÉ=
O
M
π
≤
θ
≤ X=
• ì
ï
r
r
⊥ = ~åÇ= î
ï
r
r
⊥ X=
• =sÉÅíçêë=ì
r
I= î
r
I= ï
r
=Ñçêã=~=êáÖÜí-Ü~åÇÉÇ=ëÅêÉïK=
=

136. CHAPTER 6. VECTORS
126
======= =
=
Figure 83.
=
588.
O
O
O
N
N
N
w
v
u
w
v
u
â
à
á
î
ì
ï
r
r
r
r
r
r
=
×
= =
=
589. 







−
=
×
=
O
O
N
N
O
O
N
N
O
O
N
N
v
u
v
u
I
w
u
w
u
I
w
v
w
v
î
ì
ï
r
r
r
=
=
590. θ
⋅
⋅
=
×
= ëáå
î
ì
î
ì
p
r
r
r
r
=EcáÖKUPF=
=
591. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë=EcáÖKUPF=
î
ì
î
ì
ëáå r
r
r
r
⋅
×
=
θ =
=
592. kçåÅçããìí~íáîÉ=mêçéÉêíó=
( )
ì
î
î
ì
r
r
r
r
×
−
=
× ==
=
593. ^ëëçÅá~íáîÉ=mêçéÉêíó=
( ) ( ) î
ì
î
ì
r
r
r
r
×
λµ
=
µ
×
λ =
=
=

137. CHAPTER 6. VECTORS
127
594. aáëíêáÄìíáîÉ=mêçéÉêíó=
( ) ï
ì
î
ì
ï
î
ì
r
r
r
r
r
r
r
×
+
×
=
+
× =
=
595. M
î
ì
r
r
r
=
× =áÑ=ì
r
=~åÇ= î
r
=~êÉ=é~ê~ääÉä=E M
=
θ FK=
=
596. M
â
â
à
à
á
á
r
r
r
r
r
r
r
=
×
=
×
=
× =
=
597. â
à
á
r
r
r
=
× I= á
â
à
r
r
r
=
× I= à
á
â
r
r
r
=
× =
=
=
=
6.7 Triple Product
=
598. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=
[ ] ( ) ( ) ( )
î
ì
ï
ì
ï
î
ï
î
ì
ï
î
ì
r
r
r
r
r
r
r
r
r
r
r
r
×
⋅
=
×
⋅
=
×
⋅
= =
=
599. [ ] [ ] [ ] [ ] [ ] [ ]
î
ï
ì
ì
î
ï
ï
ì
î
ì
ï
î
î
ì
ï
ï
î
ì
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
−
=
−
=
−
=
=
= =
=
600. ( ) [ ]
ï
î
ì
â
ï
î
ì
â
r
r
r
r
r
r
=
×
⋅ =
=
601. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=
( )
P
P
P
O
O
O
N
N
N
w
v
u
w
v
u
w
v
u
ï
î
ì =
×
⋅
r
r
r
I==
ïÜÉêÉ==
( )
N
N
N w
I
v
I
u
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r
I= ( )
O
O
O w
I
v
I
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r
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P
P
P w
I
v
I
u
ï =
r
K==
=
602. sçäìãÉ=çÑ=m~ê~ääÉäÉéáéÉÇ=
( )
ï
î
ì
s
r
r
r
×
⋅
= =
=

138. CHAPTER 6. VECTORS
128
============ =
=
Figure 84.
=
603. sçäìãÉ=çÑ=móê~ãáÇ=
( )
ï
î
ì
S
N
s
r
r
r
×
⋅
= =
=
=
=
Figure 85.
=
604. fÑ== ( ) M
ï
î
ì =
×
⋅
r
r
r
I=íÜÉå=íÜÉ=îÉÅíçêë==ì
r
I= î
r
I=~åÇ= ï
r
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ÇÉéÉåÇÉåí=I=ëç= î
ì
ï
r
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+
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= =Ñçê=ëçãÉ=ëÅ~ä~êë=λ =~åÇ=µK==
=
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ï
î
ì ≠
×
⋅
r
r
r
I=íÜÉå=íÜÉ=îÉÅíçêë==ì
r
I= î
r
I=~åÇ= ï
r
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áåÇÉéÉåÇÉåíK=
=

139. CHAPTER 6. VECTORS
129
606. sÉÅíçê=qêáéäÉ=mêçÇìÅí=
( ) ( ) ( )ï
î
ì
î
ï
ì
ï
î
ì
r
r
r
r
r
r
r
r
r
⋅
−
⋅
=
×
× ==
=
=
=
=
=
=
=
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140. 130
Chapter 7
Analytic Geometry
=
=
=
=
7.1 One-Dimensional Coordinate System
=
mçáåí=ÅççêÇáå~íÉëW= M
ñ I= N
ñ I= O
ñ I= M
ó I= N
ó I= O
ó =
oÉ~ä=åìãÄÉêW=λ ==
aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç=
=
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O
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Figure 86.
=
608. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç=λ =
λ
+
λ
+
=
N
ñ
ñ
ñ O
N
M I=
`_
^`
=
λ I= N
−
≠
λ K=
=
======== =
=
Figure 87.

141. CHAPTER 7. ANALYTIC GEOMETRY
131
609. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
O
ñ
ñ
ñ O
N
M
+
= I= N
=
λ K=
=
=
=
7.2 Two-Dimensional Coordinate System
=
mçáåí=ÅççêÇáå~íÉëW= M
ñ I= N
ñ I= O
ñ I= M
ó I= N
ó I= O
ó =
mçä~ê=ÅççêÇáå~íÉëW= ϕ
I
ê =
oÉ~ä=åìãÄÉêW=λ ==
mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI==
aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç=
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610. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=
( ) ( )O
N
O
O
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O ó
ó
ñ
ñ
^_
Ç −
+
−
=
= =
=
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=
Figure 88.

142. CHAPTER 7. ANALYTIC GEOMETRY
132
611. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç=λ =
λ
+
λ
+
=
N
ñ
ñ
ñ O
N
M I=
λ
+
λ
+
=
N
ó
ó
ó O
N
M I==
`_
^`
=
λ I= N
−
≠
λ K=
=
======= =
=
Figure 89.
=
=

143. CHAPTER 7. ANALYTIC GEOMETRY
133
======= =
=
Figure 90.
=
612. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
O
ñ
ñ
ñ O
N
M
+
= I=
O
ó
ó
ó O
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M
+
= I= N
=
λ K=
=
613. `ÉåíêçáÇ=EfåíÉêëÉÅíáçå=çÑ=jÉÇá~åëF=çÑ=~=qêá~åÖäÉ=
P
ñ
ñ
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+
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N ó
I
ñ
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O ó
I
ñ
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P
P ó
I
ñ
` ==~êÉ=îÉêíáÅÉë=çÑ=
íÜÉ=íêá~åÖäÉ= ^_` K= =
=

144. CHAPTER 7. ANALYTIC GEOMETRY
134
========= =
=
Figure 91.
=
614. fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
Å
Ä
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+
+
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+
+
+
+
= I==
ïÜÉêÉ= _`
~ = I= `^
Ä = I= ^_
Å = K==
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Figure 92.

145. CHAPTER 7. ANALYTIC GEOMETRY
135
615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê======================
_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
N
ó
ñ
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ó
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O
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ó
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ñ
P
P
O
O
N
N
P
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P
O
P
O
O
O
O
O
N
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+
+
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ó
ñ
N
ó
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ó
ñ
ñ
N
ó
ñ
ñ
N
ó
ñ
ñ
ó
P
P
O
O
N
N
O
P
O
P
P
O
O
O
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O
O
N
O
N
N
M
+
+
+
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=
======== =
==
Figure 93.
=
=
=
=
=
=
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146. CHAPTER 7. ANALYTIC GEOMETRY
136
616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ=
N
ó
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ó
ñ
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ó
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ñ
P
P
O
O
N
N
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M
+
+
+
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N
ó
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N
ó
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N
ñ
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ó
ñ
N
ñ
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ñ
ó
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P
P
O
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N
P
O
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O
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M
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+
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====== =
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Figure 94.
=
617. ^êÉ~=çÑ=~=qêá~åÖäÉ=
( ) ( )
N
P
N
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N
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P
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O
O
N
N
ó
ó
ñ
ñ
ó
ó
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ñ
O
N
N
ó
ñ
N
ó
ñ
N
ó
ñ
O
N
p
−
−
−
−
±
=
±
= =
=
=
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147. CHAPTER 7. ANALYTIC GEOMETRY
137
618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä=
( ) ( )( ) ( )( )
[ +
+
−
+
+
−
±
= P
O
P
O
O
N
O
N ó
ó
ñ
ñ
ó
ó
ñ
ñ
O
N
p =
( )( ) ( )( )]
N
Q
N
Q
Q
P
Q
P ó
ó
ñ
ñ
ó
ó
ñ
ñ +
−
+
+
−
+ =
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Figure 95.
=
kçíÉW=få=Ñçêãìä~ë=SNTI=SNU=ïÉ=ÅÜççëÉ=íÜÉ=ëáÖå=EHF=çê=E¥F=ëç=
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=
619. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=áå=mçä~ê=`ççêÇáå~íÉë=
( )
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ê
ê
O
ê
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−
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148. CHAPTER 7. ANALYTIC GEOMETRY
138
=
=
Figure 96.
=
620. `çåîÉêíáåÖ=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=íç=mçä~ê=`ççêÇáå~íÉë=
ϕ
= Åçë
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ê
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Figure 97.
=
621. `çåîÉêíáåÖ=mçä~ê=`ççêÇáå~íÉë=íç=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=
O
O
ó
ñ
ê +
= I=
ñ
ó
í~å =
ϕ K=

149. CHAPTER 7. ANALYTIC GEOMETRY
139
7.3 Straight Line in Plane
=
mçáåí=ÅççêÇáå~íÉëW=uI=vI=ñI= M
ñ I= N
ñ I== M
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M
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qÜÉ=îÉÅíçê= ( )
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624. bñéäáÅáí=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=EpäçéÉ-fåíÉêÅÉéí=cçêãF=
Ä
âñ
ó +
= K==

150. CHAPTER 7. ANALYTIC GEOMETRY
140
qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= α
= í~å
â K=
=
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Figure 99.
=
625. dê~ÇáÉåí=çÑ=~=iáåÉ==
N
O
N
O
ñ
ñ
ó
ó
í~å
â
−
−
=
α
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=
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Figure 100.

151. CHAPTER 7. ANALYTIC GEOMETRY
141
626. bèì~íáçå=çÑ=~=iáåÉ=dáîÉå=~=mçáåí=~åÇ=íÜÉ=dê~ÇáÉåí=
( )
M
M ñ
ñ
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+
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M
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Figure 101.
=
627. bèì~íáçå=çÑ=~=iáåÉ=qÜ~í=m~ëëÉë=qÜêçìÖÜ=qïç=mçáåíë=
N
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ñ
ñ
ñ
ñ
ó
ó
ó
ó
−
−
=
−
−
==
çê=
M
N
ó
ñ
N
ó
ñ
N
ó
ñ
O
O
N
N = K=
=

152. CHAPTER 7. ANALYTIC GEOMETRY
142
=
=
Figure 102.
=
628. fåíÉêÅÉéí=cçêã=
N
Ä
ó
~
ñ
=
+ =
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Figure 103.
=
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153. CHAPTER 7. ANALYTIC GEOMETRY
143
629. kçêã~ä=cçêã=
M
é
ëáå
ó
Åçë
ñ =
−
β
+
β =
=
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Figure 104.
=
630. mçáåí=aáêÉÅíáçå=cçêã=
v
ó
ó
u
ñ
ñ N
N −
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I==
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v
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u =áë=íÜÉ=ÇáêÉÅíáçå=çÑ=íÜÉ=äáåÉ=~åÇ= ( )
N
N
N ó
I
ñ
m =äáÉë=
çå=íÜÉ=äáåÉK=
=

154. CHAPTER 7. ANALYTIC GEOMETRY
144
=
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Figure 105.
=
631. sÉêíáÅ~ä=iáåÉ=
~
ñ = =
=
632. eçêáòçåí~ä=iáåÉ=
Ä
ó = =
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→
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155. CHAPTER 7. ANALYTIC GEOMETRY
145
=
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Figure 106.
=
634. píê~áÖÜí=iáåÉ=áå=m~ê~ãÉíêáÅ=cçêã=
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156. CHAPTER 7. ANALYTIC GEOMETRY
146
=
Figure 107.
=
635. aáëí~åÅÉ=cêçã=~=mçáåí=qç=~=iáåÉ=
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Figure 108.

157. CHAPTER 7. ANALYTIC GEOMETRY
147
636. m~ê~ääÉä=iáåÉë=
qïç=äáåÉë= N
N Ä
ñ
â
ó +
= =~åÇ= O
O Ä
ñ
â
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O
N â
â = K=
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^ N
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+
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ó
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ñ
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O
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Figure 109.
=
637. mÉêéÉåÇáÅìä~ê=iáåÉë=
qïç=äáåÉë= N
N Ä
ñ
â
ó +
= =~åÇ= O
O Ä
ñ
â
ó +
= =~êÉ=éÉêéÉåÇáÅìä~ê=áÑ==
N
O
â
N
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â
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+
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ñ
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+
+ = ~êÉ=
éÉêéÉåÇáÅìä~ê=áÑ=
M
_
_
^
^ O
N
O
N =
+ K=
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158. CHAPTER 7. ANALYTIC GEOMETRY
148
=
=
Figure 110.
=
638. ^åÖäÉ=_ÉíïÉÉå=qïç=iáåÉë=
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N
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^
_
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+
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+
+
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159. CHAPTER 7. ANALYTIC GEOMETRY
149
=
=
Figure 111.
=
639. fåíÉêëÉÅíáçå=çÑ=qïç=iáåÉë=
fÑ=íïç=äáåÉë= M
`
ó
_
ñ
^ N
N
N =
+
+ =~åÇ= M
`
ó
_
ñ
^ O
O
O =
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+ =áåíÉê-
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7.4 Circle
=
o~ÇáìëW=o=
`ÉåíÉê=çÑ=ÅáêÅäÉW=( )
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ñ I= N
ó I=£=
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160. CHAPTER 7. ANALYTIC GEOMETRY
150
640. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ=
cçêãF=
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Figure 112.
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+
−
Figure 113.

161. CHAPTER 7. ANALYTIC GEOMETRY
151
642. qÜêÉÉ=mçáåí=cçêã
M
N
ó
ñ
ó
ñ
N
ó
ñ
ó
ñ
N
ó
ñ
ó
ñ
N
ó
ñ
ó
ñ
P
P
O
P
O
P
O
O
O
O
O
O
N
N
O
N
O
N
O
O
=
+
+
+
+
=
=
=
Figure 114.
=
643. m~ê~ãÉíêáÅ=cçêã



=
=
í
ëáå
o
ó
í
Åçë
o
ñ
I= π
≤
≤ O
í
M K
=
644. dÉåÉê~ä=cçêã
M
c
bó
añ
^ó
^ñ O
O
=
+
+
+
+ =E^=åçåòÉêçI= ^c
Q
b
a O
O
>
+ FK==
qÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ÅáêÅäÉ=Ü~ë=ÅççêÇáå~íÉë=( )
Ä
I
~ I=ïÜÉêÉ==
^
O
a
~ −
= I=
^
O
b
Ä −
= K=
qÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅäÉ=áë

162. CHAPTER 7. ANALYTIC GEOMETRY
152
^
O
^c
Q
b
a
o
O
O
−
+
= K
=
=
=
7.5 Ellipse
=
pÉãáã~àçê=~ñáëW=~=
pÉãáãáåçê=~ñáëW=Ä=
cçÅáW= ( )
M
I
Å
cN − I= ( )
M
I
Å
cO =
aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= =
bÅÅÉåíêáÅáíóW=É==
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
645. bèì~íáçå=çÑ=~å=bääáéëÉ=Epí~åÇ~êÇ=cçêãF
N
Ä
ó
~
ñ
O
O
O
O
=
+
=
=
Figure 115.

163. CHAPTER 7. ANALYTIC GEOMETRY
153
646. ~
O
ê
ê O
N =
+ I=
ïÜÉêÉ== N
ê I== O
ê ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó==éçáåí== ( )
ó
I
ñ
m ==çå=
íÜÉ=ÉääáéëÉ=íç=íÜÉ=íïç=ÑçÅáK=
=
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Figure 116.
=
647. O
O
O
Å
Ä
~ +
=
=
648. bÅÅÉåíêáÅáíó
N
~
Å
É <
= =
=
649. bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë
Å
~
É
~
ñ
O
±
=
±
= =
=
650. m~ê~ãÉíêáÅ=cçêã



=
=
í
ëáå
Ä
ó
í
Åçë
~
ñ
I= π
≤
≤ O
í
M K
=
=

164. CHAPTER 7. ANALYTIC GEOMETRY
154
651. dÉåÉê~ä=cçêã
M
c
bó
añ
`ó
_ñó
^ñ O
O
=
+
+
+
+
+ I==
ïÜÉêÉ= M
^`
Q
_O
<
− K=
=
652. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë
M
c
bó
añ
`ó
^ñ O
O
=
+
+
+
+ I==
ïÜÉêÉ= M
^` > K
=
653. `áêÅìãÑÉêÉåÅÉ
( )
É
~b
Q
i = I==
ïÜÉêÉ==íÜÉ==ÑìåÅíáçå=b==áë==íÜÉ=ÅçãéäÉíÉ==ÉääáéíáÅ=áåíÉÖê~ä==çÑ=
íÜÉ=ëÉÅçåÇ=âáåÇK==
=
654. ^ééêçñáã~íÉ=cçêãìä~ë=çÑ=íÜÉ=`áêÅìãÑÉêÉåÅÉ
( )
( )
~Ä
Ä
~
R
K
N
i −
+
π
= I==
( )
O
O
Ä
~
O
i +
π
= K=
=
655. ~Ä
p π
= =
=
=
=
7.6 Hyperbola
=
qê~åëîÉêëÉ=~ñáëW=~=
`çåàìÖ~íÉ=~ñáëW=Ä=
cçÅáW= ( )
M
I
Å
cN − I= ( )
M
I
Å
cO =
aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= =
bÅÅÉåíêáÅáíóW=É==
^ëóãéíçíÉëW=ëI=í=
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=íI=â=
=
=
=

165. CHAPTER 7. ANALYTIC GEOMETRY
155
656. bèì~íáçå=çÑ=~=eóéÉêÄçä~=Epí~åÇ~êÇ=cçêãF=
N
Ä
ó
~
ñ
O
O
O
O
=
− =
=
=
=
Figure 117.
=
657. ~
O
ê
ê O
N =
− I=
ïÜÉêÉ== N
ê I== O
ê ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó=éçáåí== ( )
ó
I
ñ
m ==çå=
íÜÉ=ÜóéÉêÄçä~=íç=íÜÉ=íïç=ÑçÅáK=
=

166. CHAPTER 7. ANALYTIC GEOMETRY
156
=
=
Figure 118.
=
658. bèì~íáçåë=çÑ=^ëóãéíçíÉë=
ñ
~
Ä
ó ±
= =
=
659. O
O
O
Ä
~
Å +
= =
=
660. bÅÅÉåíêáÅáíó
N
~
Å
É >
= =
=
661. bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë
Å
~
É
~
ñ
O
±
=
±
= =
=
=
=

167. CHAPTER 7. ANALYTIC GEOMETRY
157
662. m~ê~ãÉíêáÅ=bèì~íáçåë=çÑ=íÜÉ=oáÖÜí=_ê~åÅÜ=çÑ=~=eóéÉêÄçä~=



=
=
í
ëáåÜ
Ä
ó
í
ÅçëÜ
~
ñ
I= π
≤
≤ O
í
M K
=
663. dÉåÉê~ä=cçêã
M
c
bó
añ
`ó
_ñó
^ñ O
O
=
+
+
+
+
+ I==
ïÜÉêÉ= M
^`
Q
_O
>
− K=
=
664. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë
M
c
bó
añ
`ó
^ñ O
O
=
+
+
+
+ I==
ïÜÉêÉ= M
^` < K=
665. ^ëóãéíçíáÅ=cçêã=
Q
É
ñó
O
= I==
çê==
ñ
â
ó = I=ïÜÉêÉ=
Q
É
â
O
= K=
få= íÜáë= Å~ëÉ= I= íÜÉ= ~ëóãéíçíÉë= Ü~îÉ= Éèì~íáçåë= M
ñ = = ~åÇ=
M
ó = K==
=

168. CHAPTER 7. ANALYTIC GEOMETRY
158
=
=
Figure 119.
=
=
=
7.7 Parabola
=
cçÅ~ä=é~ê~ãÉíÉêW=é=
cçÅìëW=c=
sÉêíÉñW= ( )
M
M ó
I
ñ
j =
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=éI=~I=ÄI=Å=
=
=
666. bèì~íáçå=çÑ=~=m~ê~Äçä~=Epí~åÇ~êÇ=cçêãF
éñ
O
óO
=
=

169. CHAPTER 7. ANALYTIC GEOMETRY
159
=
=
Figure 120.
=
bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ
O
é
ñ −
= I=
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=






M
I
O
é
c I=
`ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ=
( )
M
I
M
j K=
=
667. dÉåÉê~ä=cçêã
M
c
bó
añ
`ó
_ñó
^ñ O
O
=
+
+
+
+
+ I==
ïÜÉêÉ= M
^`
Q
_O
=
− K=
=
668. O
~ñ
ó = I=
~
O
N
é = K=
bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ

170. CHAPTER 7. ANALYTIC GEOMETRY
160
O
é
ó −
= I=
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=






O
é
I
M
c I=
`ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ=
( )
M
I
M
j K=
=
=
=
Figure 121.
=
669. dÉåÉê~ä=cçêãI=^ñáë=m~ê~ääÉä=íç=íÜÉ=ó-~ñáë==
M
c
bó
añ
^ñO
=
+
+
+ =E^I=b=åçåòÉêçFI==
Å
Äñ
~ñ
ó O
+
+
= I=
~
O
N
é = K==
bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ
O
é
ó
ó M −
= I=
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=

171. CHAPTER 7. ANALYTIC GEOMETRY
161






+
O
é
ó
I
ñ
c M
M I=
`ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ=
~
O
Ä
ñM −
= I=
~
Q
Ä
~Å
Q
Å
Äñ
~ñ
ó
O
M
O
M
M
−
=
+
+
= K=
=
=
=
Figure 122.
=
=
=
7.8 Three-Dimensional Coordinate System
=
mçáåí=ÅççêÇáå~íÉëW= M
ñ I= M
ó I= M
ò I= N
ñ I= N
ó I= N
ò I=£=
oÉ~ä=åìãÄÉêW=λ ==
aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç=
^êÉ~W=p=
sçäìãÉW=s=
=