1300 Math Formulas

1300 Math Formulas ==============================
fp_k= =VVQVNMTTQN==
`==OMMQ=^KpK=^=o=oK=

i
=q=~==~==~K=

ii
Preface ====q= ~= = ~= = = = = -= ~= K= f= ~= = = = =~==~==~~=~~==I=I=~=I=~=~~K=q==~= = = ~I= ~I= ~= = =k= pI= ^~I= dI= qI= j~=~= a~I= sI= ^~= dI= `~I=a~=b~I=pI=~=m~=qK==q= = ~= = I= I= ~= ~= ~== = ~= ~= = ~= ~I= = =~===~=~=~===K===
==

iii
Contents ====
1 krj_bo=pbqp= NKN= p=f==1= NKO= p==k==5= NKP= _~=f==7= NKQ= `=k==8= =2 ^idb_o^= OKN= c~=c~==12= OKO= m=c~==13= OKP= m==14= OKQ= o==15= OKR= i~==16= OKS= b~==18= OKT= f~==19= OKU= `=f=c~==22= =3 dbljbqov= PKN= o=q~==24= PKO= f=q~==27= PKP= b~~=q~==28= PKQ= p~=q~==29= PKR= p~==33= PKS= o~==34= PKT= m~~~==35= PKU= o==36= PKV= q~==37= PKNM= f=q~==38= PKNN= f=q~==f=`==40= PKNO= q~==f=`==41=

iv
PKNP= h==42= PKNQ= `=n~~~==43= PKNR= q~~=n~~~==45= PKNS= d~=n~~~==46= PKNT= o~=e~==47= PKNU= o~=m==48= PKNV= `==50= PKOM= p==~=`==53= PKON= p==~=`==54= PKOO= `==55= PKOP= o~~=m~~==56= PKOQ= m==57= PKOR= o~=q~==58= PKOS= o~=m~==59= PKOT= c==~=o~=m~==61= PKOU= o~~=o=t==62= PKOV= m~=p==63= PKPM= o=`~=`==66= PKPN= o=`~=`==~=l=m~=c~==68= PKPO= o=`~=`==69= PKPP= c==~=o=`~=`==70= PKPQ= p==72= PKPR= p~=`~==72= PKPS= p~=p==73= PKPT= p~=p==74= PKPU= p~=t==75= PKPV= b==76= PKQM= `~=q==78= = =4 qofdlkljbqov= QKN= o~~=~=a=j~==^==80= QKO= a=~=d~==q=c==81= QKP= p==q=c==86= QKQ= q=c==`=^==87= QKR= j=f~=c~==88=

v
QKS= o=c~==89= QKT= m==q=c==90= QKU= o~==q=c==90= QKV= ^=~=p~=c~==91= QKNM= a=^=c~==92= QKNN= j=^=c~==93= QKNO= e~=^=c~==94= QKNP= e~=^=q~=f==94= QKNQ= q~==q=b==m==95= QKNR= q~==q=b==p==97=== QKNS= m==q=c==98= QKNT= d~==f=q=c==99= QKNU= m~=s~==f=q=c==102= QKNV= o~==f=q=c==103= QKOM= q=b~==106= QKON= o~==e=c==106= = =5 j^qof`bp=^ka=abqbojfk^kqp= RKN= a~==107= RKO= m==a~==109= RKP= j~==110= RKQ= l~==j~==111= RKR= p==i~=b~==114= = =6 sb`qlop= SKN= s=`~==118= SKO= s=^==120= SKP= s=p~==122= SKQ= p~=s==122= SKR= p~~=m==123= SKS= s=m==125= SKT= q=m=127= = =7 ^k^ivqf`=dbljbqov= TKN= l=-a~=`~=p==130=

vi
TKO= q=-a~=`~=p==131= TKP= p~=i==m~==139= TKQ= `==149= TKR= b==152= TKS= e~==154= TKT= m~~~==158= TKU= q=-a~=`~=p==161= TKV= m~==165= TKNM= p~=i==p~==175= TKNN= n~=p~==180= TKNO= p==189= = =8 afccbobkqfî=`î`rirp= UKN= c=~=q=d~==191= UKO= i==c==208= UKP= a=~=m===a~==209= UKQ= q~==a~==211= UKR= e=l=a~==215= UKS= ^~==a~==217= UKT= a~==221= UKU= j~~=c==222= UKV= a~=l~==225= = =9 fkqbdoî=`î`rirp= VKN= f=f~==227= VKO= f~==o~~=c==228= VKP= f~==f~~=c==231= VKQ= f~==q=c==237= VKR= f~==e=c==241= VKS= f~==b~=~=i~=c==242= VKT= o=c~==243= VKU= a=f~==247= VKV= f=f~==253= VKNM= a=f~==257= VKNN= q=f~==269=

vii
VKNO= i=f~==275= VKNP= p~=f~==285= = =10 afccbobkqf^i=bnr^qflkp= NMKN= c=l=l~=a~=b~==295= NMKO= p=l=l~=a~=b~==298= NMKP= p=m~~=a~=b~==302= = =11 pbofbp= NNKN= ^=p==304= NNKO= d=p==305= NNKP= p=c=p==305= NNKQ= f=p==307= NNKR= m==`=p==307= NNKS= `=q==308= NNKT= ^~=p==310= NNKU= m=p==311= NNKV= a~=~=f~==m=p==312= NNKNM= q~=~=j~~=p==313= NNKNN= m=p=b~==p=c==314= NNKNO= _~=p==316= NNKNP= c=p==316= = =12 mol_^_fifqv= NOKN= m~=~=`~==318= NOKO= m~=c~==319= = = =
==

viii
=q=~==~==~K=
=

1
Chapter 1
Number Sets ====
1.1 Set Identities =pW=^I=_I=`=r~=W=f=`=W= ^ =m=W= _^ ==b=W= =r==W= _^ =f==W= _^ =a==W= _y^ ==
=1. f^ =
=2. ^^ =
=3. _^ = == _^ =~= ^_ .=
=4. b=p=
^ ==
5. r==p=={ }_^_^` == =
=

CHAPTER 1. NUMBER SETS
2
===== ==
Figure 1. =
6. `~=^__^ = =
=7. ^~=
( ) ( ) `_^`_^ = ==8. f==p=
{ }_~^_^` == = ==
===== ==
Figure 2. =
9. `~=^__^ = =
=10. ^~=
( ) ( ) `_^`_^ = ==

3
11. a=( ) ( ) ( )`^_^`_^ = I=( ) ( ) ( )`^_^`_^ = K=
=12. f=
^^^ = I==^^^ = =
=13. a~=
=^ I=ff^ = =
=14. f=
^^ = I==^f^ =
=15. `=
{ }^f^ = =
16. `==f=~=r f^^ = I===^^ =
=17. a=j~=i~
( ) _^_^ = I==( ) _^_^ = ==
18. a==p { }^~_^y_` == =
=

4
===== ==
Figure 3. =
19. ( )_^y_^y_ = =
20. ^_^y_ = =
21. =^y^ =
22. ^_y^ = == =_^ . =
===== ==
Figure 4. =
23. ( ) ( ) ( )`_y`^`_y^ = 24. ^yf^ = 25. `~~=m
( ){ }_~^I_^` == ==

5
1.2 Sets of Numbers =k~~=W=k=t=W= Mk =
fW=w=m=W= +w =k~=W= w =o~~=W=n=o~=W=o==`=W=`====
26. k~~=k `=W { }KIPIOINk = K=
27. t=k
`==~=W= { }KIPIOINIMkM = K==28. f
t==~===~=W={ }KIPIOINkw ==+ I=
{ }NIOIPIw = K I={ } { }KK IPIOINIMINIOIPIwMww == + K=
=29. o~~=k
o~==~=~W==
== M~w~w~~
~n K=
=30. f~~=k
k~=~=~=~K =

6
31. o~=k==r==~~=~=~~=W=oK=
=32. `=k
{ }o~o` += I======~~=K
=33. `onwk =
=
=== ==
Figure 5. ======

7
1.3 Basic Identities =o~=W=~I=I==
==
34. ^=f=~M~ =+ =
=35. ^=f=
( ) M~~ =+ ==
36. `~==^=~~ +=+ =
=37. ^~==^=
( ) ( )~~ ++=++ ==38. a==p~=
( )~~ += ==
39. j~=f=~N~ = =
=40. j~=f=
N~
N~ = I= M~
=41. j~=q=M
MM~ = =
42. `~==j~=~~ =
==

8
43. ^~==j~=( ) ( )~~ =
=44. a=i~=
( ) ~~~ +=+ ==45. a==a=
N~
~= =
===
1.4 Complex Numbers =k~~=W==f~~=W==`=W==o~=~W=~I==f~~=~W=I==j==~==W=I= N I= O =^==~==W= I= N I= O =
==
N = = R = = NQ =+ =NO = = NS = = N OQ =+ =P = = T = = PQ =+ =
46.
NQ = = NU = = N Q = ==47. ~ += ==48. `=m~=
=

9
===== ==
Figure 6. =
49. ( ) ( ) ( ) ( )~~ +++=+++ ==50. ( ) ( ) ( ) ( )~~ +=++ ==51. ( )( ) ( ) ( )~~~ ++=++ ==
52.
~
~
~OOOO
+
+++
=++
=
=53. `~=`=k=
~~|||||||
=+ ==54. = ~ I= = ==
=

10
==
Figure 7. =
55. m~=m~==`=k=( )+=+ ~ =
=56. j=~=^==~=`=k=
f= ~ + ==~==I==OO ~ += =EFI==
~
~~= =E~FK=
=57. m==m~=o~=
( ) ( )OOONNNON ++= =( ) ( )[ ]ONONON +++= =
=58. `~=k==m~=o~=
( ) ( ) ( )[ ]+=+ |||||||||||||||||||||
==59. f==~=`=k==m~=o~=
( ) ( ) ( )[ ]+=+ N
N=

11
60. n==m~=o~=( )( ) ( ) ( )[ ]ONONO
N
OOO
NNN
O
N
+=
++
= =
=61. m==~=`=k=
( )[ ] ( ) ( )[ ]+=+= ==62. c~=a=j=
( ) ( ) ( )+=+ ==63. k=o==~=`=k=
( )
++
+=+=
O
O I==
==NIIOINIM = K K==
=64. b=c~=
+= ===

12
Chapter 2
Algebra ====
2.1 Factoring Formulas =o~=W=~I=I===k~~=W==
==
65. ( )( )~~~ OO += ==
66. ( )( )OOPP ~~~~ ++= ==67. ( )( )OOPP ~~~~ ++=+ ==68. ( )( ) ( )( )( )OOOOOOQQ ~~~~~~ ++=+= ==
69. ( )( )QPOOPQRR ~~~~~~ ++++= ==70. ( )( )QPOOPQRR ~~~~~~ +++=+ ==71. f===I==
( )( )NOOPON ~~~~~~ +++=+ K K===72. f===I===
( )( )NOOPON ~~~~~~ +++++= K I==

CHAPTER 2. ALGEBRA
13
( )( )NOOPON ~~~~~~ +++=+ K K====
2.2 Product Formulas o~=W=~I=I===t=W=I====
73. ( ) OOO ~O~~ += ==
74. ( ) OOO ~O~~ ++=+ ==
75. ( ) POOPP ~P~P~~ += ==
76. ( ) POOPP ~P~P~~ +++=+ ==
77. ( ) QPOOPQQ ~Q~S~Q~~ ++= ==
78. ( ) QPOOPQQ ~Q~S~Q~~ ++++=+ ==79. _~=c~=
( ) I`~`~`~`~`~ NNOOONNM +++++=+ K
= ( )>>>
`
= =~==~=K=
=
80. ( ) O~O~O~~ OOOO +++++=++ ==
81. ( ) ++++++=+++++ OOOOOO ~~ KK =( )~~~~O +++++++++++ KKK =

CHAPTER 2. ALGEBRA
14
2.3 Powers =_~=E=~=FW=~I===m=E~~=FW=I==
==
82. ~~~ += ==
83.
~~
~ = =
=
84. ( ) ~~ = ==
85.
~
~=
=
=
86. ( ) ~~ = ==87. N~M = I= M~ ==88. N~N = ==
89.
~
N~ = =
=
90.
~~ = ======

CHAPTER 2. ALGEBRA
15
2.4 Roots =_~W=~I===m=E~~=FW=I==
MI~ ====E O = I= k F===
91. ~~ = ==
92. ~~ = ==
93.
~
~= I= M =
=
94.
~
~
~== I= M K=
=
95. ( ) ~~ = ==
96. ( ) ~~ = ==
97. ~~ = ==
98.
~~ = ==
99. ~~ = ==
100. ( ) ~~ = ==

CHAPTER 2. ALGEBRA
16
101. ~
~
~
N N
= I= M~ K=
=
102. O
~~
O
~~~
OO
+= =
=
103. ~
~
~
N
=
m
=
===
2.5 Logarithms =m=~=W=I=I=~I=I==k~~=W=====
104. a==i~= ~= ==~===
~ = I= M~ > I= N~ K==
105. MN~ = ==
106. N~~ = ==
107.
=N~
N~M~ =
=108. ( ) ~~~ += =
=
109.
~~~ = =

CHAPTER 2. ALGEBRA
17
110. ( ) ~~ = ==
111.
N ~
~ = =
=
112. ~
~
~ == I= M > I= N K=
=
113. ~
N
~ = =
=114. ~~ = =
=115. i~==_~=NM=
NM = ==
116. k~~=i~= = I==
= KTNUOUNUOUKO
NN
=
+=
=
=
117. QPQOVQKMNM
N == =
=
118. PMORURKO
N == =
=====

CHAPTER 2. ALGEBRA
18
2.6 Equations =o~=W=~I=I=I=I=I=I==pW= N I= O I= N I= O I= P =
==
119. i~=b~==l=s~~=
M~ =+ I=~
= K==
=120. n~~=b~=
M~O =++ I=~O
~Q
O
OIN
= K=
=121. a~=
~Qa O = ==
122. s=c~=f= MO =++ I===
==+
ON
ON K=
=
123. M~O =+ I= MN = I= ~
O = K=
=
124. M~ O =+ I=~
OIN = K=
=125. `=b~K=`~~=c~K==
MP =++ I==

CHAPTER 2. ALGEBRA
19
N += I= ( ) ( ) OP
O
N PIO ++= I==
==
P
OO
P
O
O
+
+= I= P
OO
P
O
O
+
= K==
==
2.7 Inequalities s~~W=I=I==
o~=W=
PON ~II~I~I~
III~
KI=I==
a~W=aI= a I= a I= a ==
==
126. f~I=f~=k~=~=d~===
f~= f~=k~= d~=~ = [ ]I~ =
=~ < = ( ]I~ =
=~

CHAPTER 2. ALGEBRA
20
127. f= ~ > I== ~ < K==128. f= ~ > I== M~ > == M~ I== ~ +>+ K==130. f= ~ > I== ~ > K==131. f= ~ > =~= > I== ~ +>+ K==132. f= ~ > =~= > I== ~ > K==133. f= ~ > =~= M > I== ~ > K==
134. f= ~ > =~= M > I==
~> K=
=135. f= ~ > =~= M < I== ~ < K==
136. f= ~ > =~= M < I==
~< K=
=137. f= ~M I== ~ < K==138. f= ~M K==
139. f= ~M =I= M > X=~=~==~=== ~ = K===
141. O~
N~ + I== M~ > X=~=~=~=~==~= N~ = K=

CHAPTER 2. ALGEBRA
21
142.
~~~~~~ ON ON
+++
KK I== M~II~I~ ON >K K=
=
143. f= M~ >+ =~= M~ > I==~
> K=
=
144. f= M~ >+ =~= M~ < I==~
< K==
=145. M~ O >++ ==
= M~ > = M~ < ====
Ma> =
=
=
N < I= O > =
=
==
ON =
== =
===
Ma< =
=
=

CHAPTER 2. ALGEBRA
22
146. ~~ ++ ==147. f= ~ < I== ~~ I== ~ < =~= ~ > I== M~ > K==
149. f= ~O < I== ~ < I== M~ > K==
150. f= ~O > I== ~ > I== M~ > K==
151. f= ( )( ) M
> I==( ) ( )( )
>
M
MK=
=
152. ( )( ) M
< I==( ) ( )( )

CHAPTER 2. ALGEBRA
23
154. p=`=f=c~=f======~I====~==W=
( )N`^ += K==155. `=`=f=
f====~=E FI===`^ = K=
==

24
Chapter 3
Geometry ====
3.1 Right Triangle =i==~==~W=~I==eW==^W==j~W= ~ I= I= =
^W= I =o~===W=o=o~===W==^~W=p===
==
Figure 8. =
156. =+ VM ==

CHAPTER 3. GEOMETRY
25
157. ==
~ =
=
158. ==
=
=
159. ==
~~ =
=
160. == ~~
=
=
161. ==
=
=
162. == ~
=
=163. m~~=q=
OOO ~ =+ ==
164. ~O = I= O = I=== = ~= = ~= = = = = ~= ~= I= -I====K==
===== ==
Figure 9. =

CHAPTER 3. GEOMETRY
26
165. O = I=======~====~K==
=
166. Q
~
OOO
~ = I= Q
~O
OO = I===
= ~ =~= =~==~====~=~=K==
=
==
Figure 10. =
167. O
= I==
= ===~====K=
=
168. O
o == =
=
169. ~
~
O
~
++=
+= =
=170. ~ = =
==

CHAPTER 3. GEOMETRY
27
171. O
O
~p == =
===
3.2 Isosceles Triangle =_~W=~=iW==_~=~W= =s=~W= =^===~W==mW=i=^~W=p=
==
==
Figure 11. =
172. O
VM
= =
=
173. Q
~
OOO = =

CHAPTER 3. GEOMETRY
28
174. O~i += ==
175. == O
O
~p
O
=
===
3.3 Equilateral Triangle =p==~=~~=~W=~=^W==o~===W=o=o~===W==mW=i=^~W=p===
==
Figure 12. =
176. O
P~ = =
=

CHAPTER 3. GEOMETRY
29
177. P
P~
P
Oo == =
=
178. O
o
S
P~
P
N === =
=179. ~Pi = =
=
180. Q
P~
O
~p
O
== =
===
3.4 Scalene Triangle E^=~=====~F=
==p==~=~W=~I=I==
pW=O
~
++= ==
^==~=~W= II =^====~I=I=W= ~ II =
j~====~I=I=W= ~ II =
_===~= II W= ~ II =o~===W=o=o~===W==^~W=p===

CHAPTER 3. GEOMETRY
30
===== ==
Figure 13. =
181. =++ NUM ==
182. ~ >+ I==~ >+ I==~ >+ K=
=183. ~

CHAPTER 3. GEOMETRY
31
185. i~==`=+= O~ OOO I=+= ~O~ OOO I=+= ~O~ OOO K=
=186. i~==p=
oO
~=
=
=
I==
=o===~====K===
187. pQ
~
O
~
O
~
O
O
O
O
~o
~
====
=
=
= =
=
188. ( )( )( )
~O
= I==
~
N
N
N
N++= K=
=
189. ( )( )
O
=
I=
( )
~
O
=
I=
( )( )( )~
O~
=
K=
=
190. ( )( )( )~~
O~ = I=
( )( )( )~
O = I=
( )( )( )~
O = K=

CHAPTER 3. GEOMETRY
32
191. == ~ I=== ~ I=== ~ K=
=
192. Q
~
O
OOOO~
+= I==
Q
O
~
OOOO
+= I==
Q
O
~
OOOO
+= K=
=
===== ==
Figure 15. =
193. ~PO
^j = I= PO
_j = I= PO
`j = =EcKNRFK=
=
194. ( )( )O
O~
~Q
+
= I==
( )( )O
O
~
~Q
+
= I==
( )( )O
O
~
~Q
+
= K=
=

CHAPTER 3. GEOMETRY
33
195. O
O
O
~p ~ === I==
O
O
~
O
~p
=
=
= I==
( )( )( )~p = =Ee=c~FI=p = I==
oQ
~p = I=
= oOp O I=
O~
O~
O~p O
= K=
===
3.5 Square p==~=~W=~=a~~W==o~===W=o=o~===W==mW=i=^~W=p==
==
Figure 16.

CHAPTER 3. GEOMETRY
34
196. O~ = ===
197. O
O~
O
o == =
=
198. O
~ = =
=199. ~Qi = =
=
200. O~p = ====
3.6 Rectangle =p==~=~W=~I==a~~W==o~===W=o=mW=i=^~W=p===
==
Figure 17. =
201. OO ~ += ==

CHAPTER 3. GEOMETRY
35
202. O
o = =
=203. ( )~Oi += =
=204. ~p = =
===
3.7 Parallelogram =p==~=~~~W=~I==a~~W= ON I =`=~W= I =^===~~W= =^W===mW=i=^~W=p===
===== ==
Figure 18. =
205. =+ NUM ==
206. ( )OOOOON ~O +=+ ==

CHAPTER 3. GEOMETRY
36
207. == ==
208. ( )~Oi += ==
209. == ~~p I==
= O
Np ON K=
===
3.8 Rhombus =p==~=W=~=a~~W= ON I =`=~W= I =^W=e=o~===W==mW=i=^~W=p===
===== ==
Figure 19. =

CHAPTER 3. GEOMETRY
37
210. =+ NUM ==
211. OOOON ~Q =+ ==
212. ~O
~ ON== =
=
213. O
~
~Q
O
ON
=== =
=214. ~Qi = =
=
215. == ~~p O I==
ONO
Np = K=
===
3.9 Trapezoid =_~==~=~W=~I==jW==^W==^~W=p===

CHAPTER 3. GEOMETRY
38
==
Figure 20. =
216. O
~
+= =
=
217. O
~p =
+= =
===
3.10 Isosceles Trapezoid =_~==~=~W=~I==iW==jW==^W==a~~W==o~===W=o=^~W=p===

CHAPTER 3. GEOMETRY
39
==
Figure 21. =
218. O
~
+= =
=
219. O~ += ==
220. ( )OO ~Q
N = =
=
221. ( )( )~O~O
~o
O
+++
= =
=
222. O
~p =
+= =
======

CHAPTER 3. GEOMETRY
40
3.11 Isosceles Trapezoid with Inscribed Circle
=_~==~=~W=~I==iW==jW==^W==a~~W==o~===W=o=o~===W==mW=i=^~W=p===
==
Figure 22. =
223. O~ =+ ==
224. O
~ =
+= =
=225. OOO += =
=

CHAPTER 3. GEOMETRY
41
226. O
~
O
== =
=
227. ~
S
~
U
~
O
~
N
O
Q
O
o OO
O
+++
=+=+=== =
=228. ( ) Q~Oi =+= =
=
229. ( )O
i
O
~~
O
~p ===
+=
+= ==
===
3.12 Trapezoid with Inscribed Circle =_~==~=~W=~I==i~~=W=I==jW==^W==a~~W= ON I =^===~~W= =o~===W==o~===W=o=mW=i=^~W=p==

CHAPTER 3. GEOMETRY
42
==
Figure 23. =
230. ~ +=+ ==
231. O
O
~
+=
+= =
=232. ( ) ( )O~Oi +=+= =
=
233. O
O
~p =
+=
+= I==
= O
Np ON K=
===
3.13 Kite =p==~=W=~I==a~~W= ON I =^W= II =mW=i=^~W=p===

CHAPTER 3. GEOMETRY
43
==
Figure 24. =
234. =++ PSMO ==
235. ( )~Oi += ==
236. O
p ON= =
===
3.14 Cyclic Quadrilateral p==~=~~~W=~I=I=I==a~~W= ON I =^===~~W= =f~=~W= III =o~===W=o=mW=i=pW===^~W=p=

CHAPTER 3. GEOMETRY
44
==
Figure 25. =
237. =+=+ NUM ==
238. m=q=ON~ =+ =
=239. ~i +++= =
=
240. ( )( )( )( )( )( )( )~~~~
Q
No
+++
= I==
=O
i = K=
=
241. = O
Np ON I==
( )( )( )( )~p = I==
=O
i = K=
===

CHAPTER 3. GEOMETRY
45
3.15 Tangential Quadrilateral =p==~=~~~W=~I=I=I==a~~W= ON I =^===~~W= =o~===W==mW=i=pW===^~W=p===
==
Figure 26. =
242. ~ +=+ ==
243. ( ) ( )O~O~i +=+=+++= ==
244. ( ) ( )
O
~~
OOOO
ON += I==
=O
i = K==
=

CHAPTER 3. GEOMETRY
46
245. == O
Np ON =
===
3.16 General Quadrilateral =p==~=~~~W=~I=I=I==a~~W= ON I =^===~~W= =f~=~W= III =mW=i=^~W=p===
======= ==
Figure 27. =
246. =+++ PSM ==
247. ~i +++= ==

CHAPTER 3. GEOMETRY
47
248. = O
Np ON =
===
3.17 Regular Hexagon =pW=~=f~=~W= =p~=W==o~===W==o~===W=o=mW=i=pW===^~W=p===
==
Figure 28. =
249. = NOM ==
250. O
P~ == =

CHAPTER 3. GEOMETRY
48
251. ~o = ==
252. ~Si = ==
253. O
PP~p
O
== I==
=O
i = K=
===
3.18 Regular Polygon =pW=~=k==W==f~=~W= =p~=W==o~===W==o~===W=o=mW=i=pW===^~W=p===

CHAPTER 3. GEOMETRY
49
==
Figure 29. =
254. = NUMO
O=
=
255. = NUMO
O=
=
256.
O
~o
= =
=
257. Q
~o
~O
~
OO =
== =
=258. ~i = =
=
259.
O
O
op
O = I==
Q
~op
OO == I==

CHAPTER 3. GEOMETRY
50
=O
i = K==
===
3.19 Circle =o~W=o=a~W==`W=~=p~=W=I==q~=W==`~=~W= =f=~W= =mW=i=^~W=p===
260. O
oO~
= =
=
==
Figure 30. =

CHAPTER 3. GEOMETRY
51
261. ONON ~~ = ==
==
Figure 31. =
262. NN = ==
===== ==
Figure 32. =
263. NO = ==

CHAPTER 3. GEOMETRY
52
===== ==
Figure 33. =
264. O
= =
=
==
Figure 34. =
265. oOi == ==
266. O
io
Q
op
OO =
== ==
=

CHAPTER 3. GEOMETRY
53
3.20 Sector of a Circle =o~==~=W=o=^=W==`~=~=E=~~FW==`~=~=E=FW= =mW=i=^~W=p===
==
Figure 35. =
267. o = ==
268.
=NUM
o =
=269. oOi += =
=
270.
===PSM
o
O
o
O
op
OO
==
==

CHAPTER 3. GEOMETRY
54
3.21 Segment of a Circle =o~==~=W=o=^=W==`W=~=`~=~=E=~~FW==`~=~=E=FW= =e===W==mW=i=^~W=p===
==
Figure 36. =
271. OoOO~ = ==
272. OO ~oQO
No = I= o < =
=273. ~i += =
=

CHAPTER 3. GEOMETRY
55
274. ( )[ ] ( )O
o
NUMO
oo~o
O
Np
OO
=
== I==
~P
Op K=
===
3.22 Cube =bW=~==a~~W==o~===W==o~===W==p~=~~W=p=sW=s===
=== ==
Figure 37. =
275. P~ = ==
276. O
~ = =
=

CHAPTER 3. GEOMETRY
56
277. O
P~o = =
=278. O~Sp = =
=279. P~s = ==
===
3.23 Rectangular Parallelepiped =bW=~I=I===a~~W==p~=~~W=p=sW=s===
===== ==
Figure 38. =
280. OOO ~ ++= ==
281. ( )~~Op ++= ==
282. ~s = ==

CHAPTER 3. GEOMETRY
57
3.24 Prism =i~~=W==eW==i~~=~~W= ip =^~==~W= _p =q~=~=~~W=p=sW=s===
===== ==
Figure 39. =
283. _i pOpp += K===
284. i~~=^~==~=o=m=( )~~~~p PONi ++++= K =
=285. i~~=^~==~=l=m=
pi = I==========K=
=

CHAPTER 3. GEOMETRY
58
286. ps _= ==
287. `~~D=m==d=== ==~~=~K=f==~===~~====~=~==~=~~===I=======~=~K====
3.25 Regular Tetrahedron =q~==W=~=eW==^~==~W= _p =p~=~~W=p=sW=s===
==
Figure 40. =
288. ~P
O = =
=

CHAPTER 3. GEOMETRY
59
289. Q
~Pp
O
_ = =
=
290. O~Pp = ==
291. OS
~p
P
Ns
P
_ == K==
===
3.26 Regular Pyramid =p==~W=~=i~~=W==eW==p~=W===k==W===p==~W==o~=====~W==^~==~W= _p =i~~=~=~~W= ip =q~=~=~~W=p=sW=s===

CHAPTER 3. GEOMETRY
60
==
Figure 41. =
292. Q
~
OO = =
=
293.
O
~
Q
OOO
= =
=
294. ~Q~Q
N~
O
Np OOi === =
=295. p_ = =
=296. i_ ppp += =
=
297. P
Np
P
Ns _ == ==
===

CHAPTER 3. GEOMETRY
61
3.27 Frustum of a Regular Pyramid =
_~=~===W=
PON
PON
IIII
~II~I~I~
K
K=
eW==p~=W===^~==~W= Np I= Op =i~~=~=~~W= ip =m==~W= Nm I= Om =p~=~W==q~=~=~~W=p=sW=s===
==
Figure 42. =
298. ~
~
~
~
~
P
P
O
O
N
N ====== K =
=

CHAPTER 3. GEOMETRY
62
299. ON
O p
p= =
=
300. ( )O
mmp ONi
+= =
=301. ONi pppp ++= =
=
302. ( )OONN ppppP
s ++= =
=
303. [ ]ONO
N NP
p
~
~
N
P
ps ++=
++= =
===
3.28 Rectangular Right Wedge =p==~W=~I==q=W==eW==i~~=~=~~W= ip =^~==~W= _p =q~=~=~~W=p=sW=s===

CHAPTER 3. GEOMETRY
63
==
Figure 43. =
304. ( ) ( )OOOOi ~Q~ON
p ++++= =
=305. ~p_ = =
=306. i_ ppp += =
=
307. ( )~OS
s += =
===
3.29 Platonic Solids =bW=~=o~===W==o~===W=o=p~=~~W=p=sW=s===

CHAPTER 3. GEOMETRY
64
308. c=m~=p=q= ~= = ~= = ~= = ~=~=====~=K==
=p= k=
=sk==b=
k==c~=
p=
q~== Q= S= Q= PKOR=`= U= NO= S= PKOO=
l~= S= NO= U= PKOT=f~= NO= PM= OM= PKOT=
a~= OM= PM= NO= PKOT===
Octahedron =
==
Figure 44. =
309. S
S~ = =
=
310. O
O~o = =
=

CHAPTER 3. GEOMETRY
65
311. P~Op O= ==
312. P
O~s
P
= =
==
Icosahedron =
==
Figure 45. =
313. ( )NO
RPP~
+= =
=
314. ( )RROQ
~o += =
=
315. P~Rp O= ==
316. ( )NO
RP~Rs
P += =
==

CHAPTER 3. GEOMETRY
66
Dodecahedron =
==
Figure 46. =
317. ( )O
RNNORNM~
+= =
=
318. ( )Q
RNP~o
+= =
=
319. ( )RORR~Pp O += ==
320. ( )Q
RTNR~s
P += =
===
3.30 Right Circular Cylinder =o~==~W=o=a~==~W==

CHAPTER 3. GEOMETRY
67
eW=e=i~~=~=~~W= ip =^~==~W= _p =q~=~=~~W=p=sW=s===
===== ==
Figure 47. =
321. oeOpi = ==
322. ( )
+=+=+=
O
eoeoOpOpp _i =
=
323. eoeps O_ == ====

CHAPTER 3. GEOMETRY
68
3.31 Right Circular Cylinder with an Oblique Plane Face
=o~==~W=o=q=~===~=W= N =q====~=W= O =i~~=~=~~W= ip =^~==~==~W= _p =q~=~=~~W=p=sW=s===
==
Figure 48. =
324. ( )ONi op += ==
325. O
ONOO_ O
ooop
++= =
=

CHAPTER 3. GEOMETRY
69
326.
++++=+=
O
ONOON_i O
oooppp =
=
327. ( )ONO
O
os +
= =
===
3.32 Right Circular Cone o~==~W=o=a~==~W==eW=e=p~=W==i~~=~=~~W= ip =^~==~W= _p =q~=~=~~W=p=sW=s===
==
Figure 49.

CHAPTER 3. GEOMETRY
70
328. OO oe = ==
329. O
opi
== =
=
330. O_ op = ==
331. ( )
+=+=+=
O
O
Nooppp _i =
=
332. eoP
Nep
P
Ns O_ == =
===
3.33 Frustum of a Right Circular Cone =o~==~W=oI==eW=e=p~=W==p~=~W==^~==~W= Np I= Op =i~~=~=~~W= ip =q~=~=~~W=p=sW=s===

CHAPTER 3. GEOMETRY
71
==
Figure 50. =
333. ( )OO oe = ==
334.
o= =
=
335. OO
O
N
O
o
p
p== =
=336. ( )opi += ==337. ( )[ ]oopppp OOiON +++=++= ==
338. ( )OONN ppppP
s ++= =
=
339. [ ]ONO
N NP
p
o
oN
P
ps ++=
++= =
===

CHAPTER 3. GEOMETRY
72
3.34 Sphere =o~W=o=a~W==p~=~~W=p=sW=s==
==
Figure 51. =
340. OoQp = ==
341. poP
N
S
Neo
P
Qs PP === =
===
3.35 Spherical Cap o~==W=o=o~==~W==eW==^~==~=~W= _p =^~==~=~W= `p =
q~=~=~~W=p=sW=s=

CHAPTER 3. GEOMETRY
73
==
Figure 52. =
342. O
o
OO += =
=343. O_ p = ==344. ( )OO` p += ==345. ( ) ( )OOO`_ oOOppp +=+=+= ==
346. ( ) ( )OOO PS
oPS
s +
=
= =
===
3.36 Spherical Sector =o~==W=o=o~==~==~=~W==eW==q~=~=~~W=p=sW=s==

CHAPTER 3. GEOMETRY
74
====== === ==
Figure 53. =
347. ( )Oop += ==
348. oP
Os O= =
=kW=q== ~=~=== == ~==~=K====
3.37 Spherical Segment =o~==W=o=o~==~W= N I= O =eW==^~==~=~W= pp =
^~==~==~W= Np I= Op =q~=~=~~W=p=sW=s==

CHAPTER 3. GEOMETRY
75
===== ==
Figure 54. =
349. oOpp = ==350. ( )OOONONp oOpppp ++=++= ==
351. ( )OOOON PPSN
s ++= =
===
3.38 Spherical Wedge =o~W=o=a~=~==W==a~=~==~~W= =^~==~=W= ip =q~=~=~~W=p=sW=s===

CHAPTER 3. GEOMETRY
76
==
Figure 55. =
352. oOVM
op O
O
i =
= =
=
353. oOoVM
oop OO
OO +=
+= =
=
354. oP
O
OTM
os P
P
=
= =
===
3.39 Ellipsoid =p-~W=~I=I==sW=s=

CHAPTER 3. GEOMETRY
77
======= ==
Figure 56. =
355. ~P
Qs = =
===
Prolate Spheroid =p-~W=~I=I==E ~ > F=p~=~~W=p=sW=s===
356.
+=
~~Op I==
=~
~
OO = K=
=
357. ~P
Qs O= =
=

CHAPTER 3. GEOMETRY
78
Oblate Spheroid =p-~W=~I=I==E ~ < F=p~=~~W=p=sW=s===
358.
+=~L
~
~~
Op I==
=
~
OO = K=
=
359. ~P
Qs O= =
===
3.40 Circular Torus =j~=~W=o=j=~W==p~=~~W=p=sW=s==

CHAPTER 3. GEOMETRY
79
== =Picture 57.
=360. oQp O= ==361. OOoOs = =
==

80
Chapter 4
Trigonometry ====^W= I= =o~==E~==~=FW=I===t=W====
4.1 Radian and Degree Measures of Angles =
362. ?QRDNTRTNUM~N
= =
=
363. ~MNTQRPKM~NUM
N
= =
=
364. ~MMMOVNKM~SMNUM
DN
= =
=
365. ~MMMMMRKM~PSMMNUM
?N
= =
=366. ==
^=EF=
M= PM= QR= SM= VM= NUM= OTM= PSM=
^=E~~F= M= S
=
Q
=
P
=
O
= =
O
P= O =
===

CHAPTER 4. TRIGONOMETRY
81
4.2 Definitions and Graphs of Trigonometric Functions
=
= ==
Figure 58. =
367.
= =
=
368.
= =
=
369.
~ = =
=
370.
= =
=

82
371.
= =
=
372.
= =
=373. p=c=
= I= NN K==
=
Figure 59. =
374. `=c== = I= NN K=

83
==
Figure 60. =
375. q~=c=
~ = I= ( )O
NO
+ I= K~ =
=
==
Figure 61. =

84
376. `~=c== = I= I== K=
=
==
Figure 62. =
377. p~=c=
= I= ( )O
NO
+ K=
==

85
==
Figure 63. =
378. `~=c== = I= K=
=
Figure 64.

86
4.3. Signs of Trigonometric Functions 379. ==
n~~=p =
` =
q~ =
` =
p =
`= =
f= H= H= H= H= H= H=ff= H= = = = = H=fff= = = H= H= = =fs= = H= = = H= ==
==
380. ==
=
Figure 65.
==========

87
4.4 Trigonometric Functions of Common Angles 381. = = ~ = = = ~ = = =
M= M= M= N= M= = N= =
PM=S
=
O
N=
O
P=
P
N= P =
P
O= O=
QR=Q
=
O
O=
O
O= N= N= O = O =
SM=P
=
O
P=
O
N= P =
P
N= O=
P
O=
VM=O
= N= M= = M= = N=
NOM=P
O=
O
P=
O
N = P =
P
N O =
P
O=
NUM= = M= N = M= = N = =
OTM=O
P= N = M= = M= = N =
PSM= O = M= N= M= = N= ==============

88
382. = = ~ = = = ~ = =
NR=NO
=
Q
OS =
Q
OS += PO = PO+ =
NU=NM
=
Q
NR =
Q
RONM+R
ROR= ROR+ =
PS=R
=
Q
RONMQ
NR +=
NR
RONM
+
RONM
NR
+
=
RQ=NM
P=
Q
NR +=
Q
RONMRONM
NR
+NR
RONM
+
=
TO=R
O=
Q
RONM+Q
NR = ROR+ = R
ROR
=
TR=NO
R=
Q
OS +=
Q
OS = PO+ = PO =
===
4.5 Most Important Formulas =
383. N OO =+ ==
384. N~ OO = ==
385. N OO = ==
386.
=
~ =

89
387.
=
=
=388. N~ = =
=
389.
=
N =
=
390.
=
N =
===
4.6 Reduction Formulas =
391. ==
= = = ~ = = = = + = ~ = =VM = + = + = + = + ~ =+VM = + = = = ~ =NUM + = = ~ = =+NUM = = + ~ = + =OTM = = + = + ~ =+OTM = + = = ~ =PSM = + = ~ = =+PSM + = + = + ~ = + ==
=====

90
4.7 Periodicity of Trigonometric Functions =
392. ( ) = O I== O == PSM K==
393. ( ) = O I== O == PSM K==
394. ( ) = ~~ I== == NUM K==
395. ( ) = I== == NUM K====
4.8 Relations between Trigonometric Functions
=
396. ( ) NQO
OONO
NN OO
=== =
=
O~N
O~O
O +
= =
=
397. ( ) NO
OONO
NN OO
=+== =
=
O~N
O~N
O
O
+
= =
=
398.
=+
==
=O
ON
ON
ON
~ O =

91
=
O~N
O~O
ON
ON
O +
=+
= =
=
399.
=
+
==
=ON
O
O
ONN
O =
=
O~O
O~N
ON
ONO
=+
= =
=
400.
O~N
O~N
~N
N
O
O
O
+
=+=
= =
=
401.
O~O
O~N
N
N
O
O
+
=+=
= =
===
4.9 Addition and Subtraction Formulas =
402. ( ) +=+ ==
403. ( ) = ==
404. ( ) =+ ==
405. ( ) += =

92
406. ( )+
=+~~N
~~~ =
=
407. ( )+
=~~N
~~~ =
=
408. ( )+
=+~~
~~N =
=
409. ( )+
=~~
~~N =
===
4.10 Double Angle Formulas =
410. = OO ==
411. NOONO OOOO === ==
412.
=
=~
O
~N
~OO~
O=
=
413. O
~
O
NO
O =
= =
======

93
4.11 Multiple Angle Formulas =
414. == POP PQPP ==
415. = UQQ P ==
416. += RP NSOMRR ==
417. == OPP PPQP ==
418. NUUQ OQ += ==
419. += ROMNSR PR ==
420.
=O
P
~PN
~~PP~ =
=
421. +
=QO
P
~~SN
~Q~QQ~ =
=
422. ++
=QO
PR
~R~NMN
~R~NM~R~ =
=
423. NP
PP
O
P
= =
=
424. +
=P
QO
~Q~Q
~~SNQ ==
=

94
425. +
+=
~R~NM~
~R~NMNR
PR
QO
=
===
4.12 Half Angle Formulas =
426. O
N
O
=
=
=
427. O
N
O
+=
=
=
428. =
=+
=
+
=
N
N
N
N
O~ =
=
429. +=+
=
=
+
=
N
N
N
N
O =
===
4.13 Half Angle Tangent Identities =
430.
O~N
O~O
O +
= =
=

95
431.
O~N
O~N
O
O
+
= =
=
432.
O~N
O~O
~O
= =
=
433.
O~O
O~N
O
= =
===
4.14 Transforming of Trigonometric Expressions to Product
=
434. O
O
O+
=+ =
=
435. O
O
O+
= =
=
436. O
O
O+
=+ =
=
437. O
O
O+
= =
=

96
438. ( )
+=+
~~ =
=
439. ( )
=
~~ =
=
440. ( )
+=+
=
=
441. ( )
=
=
=
442.
+
=
=+Q
OQ
O =
=
443.
+
=
=Q
OQ
O =
=
444. ( )
=+
~ =
=
445. ( )
+=
~ =
=
446. O
ON O
=+ =
=
447. O
ON O
= =
=

97
448.
=+OQ
ON O =
=
449.
=OQ
ON O =
===
4.15 Transforming of Trigonometric Expressions to Sum
=
450. ( ) ( )O
+= =
=
451. ( ) ( )O
++= =
=
452. ( ) ( )O
++= =
=
453. ++
=
~~~~ =
=
454. ++
=~~
=
=
455. ++
=~
~~ =
===

98
4.16 Powers of Trigonometric Functions =
456. O
ONO
= =
=
457. Q
PPP
= =
=
458. U
POQQQ
+= =
=
459. NS
RPRNMR
+= =
=
460. PO
SQSONRNMS
+= =
=
461. O
ONO
+= =
=
462. Q
PPP
+= =
=
463. U
POQQQ
++= =
=
464. NS
RPRNMR
++= =
=
465. PO
SQSONRNMS
+++= =
=

99
4.17 Graphs of Inverse Trigonometric Functions
=466. f=p=c==
~ = I= NN I=O
~O
K=
=
==
Figure 66. =
467. f=`=c==~ = I= NN I= ~M K=
=

100
==
Figure 67. =
468. f=q~=c==
~~ = I= I=O
~~O

101
469. f=`~=c==~ = I= I=

102
471. f=`~=c==
( ] [ ) KO
IMMIO
~IINNII~
=
==
Figure 71. ==
4.18 Principal Values of Inverse Trigonometric Functions 472. = M= O
N=
O
O=
O
PN=
~ = M = PM = QR = SM VM~ = VM SM = QR = PM M =
= O
N
O
O
O
P N = =
~ =PM
= QR SM VM
==
~ =NOM
=NPR = NRM = NUM
==

103
473. = M=
P
PN= P =
P
P N = P =
~~ = M = PM QR SM PM QR=
SM =
~ = VM SM QR PM NOM = NPR=
NRM = ===
4.19 Relations between Inverse Trigonometric Functions
=474. ( ) ~~ = =
=
475. ~O
~
= =
=
476. ON~~ = I= NM K==
477. ON~~ = I= MN K==
478. ON
~~~
= I= N
O < K=
=
479.
N~~
O= I= NM < K=
=
480. =
N~~
O
I= MN

104
482. ~O
~
= =
=
483. ON~~ = I= NM K==
484. ON~~ = I= MN K==
485.
N~~~
O= I= NM < K=
=
486.
N~~~
O+= I= MN

105
493.
N~~
O~~
= I= M > K=
=
494.
N~~
O~~
= I= M < K=
=
495.
N~~~ = I= M > K=
=
496. =
N~~~ I= M < K=
=497. ( ) ~~ = =
=
498. ~~O
~
= =
=
499. ON
N~~
+= I= M > K=
=
500. ON
N~~
+= I= M < K=
=
501. ON
~~
+= =
=
502.
N~~~ = I= M > K=
=
503.
N~~~ += I= M < K=
==

106
4.20 Trigonometric Equations =t=W====
504. ~ = I= ( ) ~~N += ==
505. ~ = I= O~~ += ==
506. ~~ = I= ~~~ += ==
507. ~ = I= ~~ += ====
4.21 Relations to Hyperbolic Functions =f~~=W====
508. ( ) = ==
509. ( ) ~~ = ==
510. ( ) = ==
511. ( ) = ==
512. ( ) = ====

107
Chapter 5
Matrices and Determinants ====j~W=^I=_I=`=b==~=~W= ~ I= I= ~ I= I= =
a~==~=~W= ^ =j==~== ~ W= j =
`~==~== ~ W= ` =
q~==~=~W= q^ I= ^
=^==~=~W= ^~ =q~==~=~W= ^ =
f==~=~W= N^ =o~=W==o~=~~W= =k~~=W=I======
5.1 Determinants =
513. p=l=a~=
NOONOO
NN ~~~
~^ == =
=====

CHAPTER 5. MATRICES AND DETERMINANTS
108
514. q=l=a~=
++== POONNPPNOPNOPPOONNPPPOPN
OPOOON
NPNONN
~~~~~~~~~
~~~
~~~
~~~
^ =
PNOONPPPONNOPOOPNN ~~~~~~~~~ ==
515. p~=o=E^=oF=
==
Figure 72. =
516. k-=l=a~=
ON
ON
OOOOON
NNNONN
~~~~
~~~~
~~~~
~~~~
^
KK
KKKKKK
KK
KKKKKK
KK
KK
= =
=517. j=
q== j =~~==== ~ ==-==~= ^= = = ( )N -= = ~= = ==~=^=====-==~=-=K====

109
518. `~=( ) jN` += =
=519. i~~=b~==-=l=a~=
i~~=~=====-==
=
=
N`~^ I= IIOIN K= K=
i~~=~=====-==
=
=
N`~^ I= IIOIN K= K==
===
5.2 Properties of Determinants =
520. q==~===~=~=~==~===~=~===~===K=
=OO
NN
ON
ON
~
~
~~= ==
=521. f=====E==F=~==~I====
=~==~K=
NN
OO
OO
NN
~
~
~
~= =
=522. f====E==F=~==~I==~===
~==K=
M~~
~~
OO
NN = =
=

110
523. f=========~===E=F=~=======~====~I====~========~=~K=
OO
NN
OO
NN
~
~
~
~= =
=524. f========~====E==F=~=~=E=
~F=~=======~====E=FI===~===~==~K=
OO
NN
OOO
NNN
~
~
~
~=
++
=
===
5.3 Matrices =
525. a=^= =~=^==~=~~=~~===E-==F====~==K==
[ ]
==
ON
OOOON
NNONN
~~~
~~~
~~~
~^
K
MMM
K
K
==
=526. p~=~==~=~=== K==
=527. ^=~=~== [ ]~ ======== ~~ = I==KK=====
=~==~=~~K===
528. ^=~=~= [ ]~ ==-== ~~ = K===

111
529. a~~=~====~=~==~==~=========~=~~K===
530. r=~====~=~~==~========~=~~=~=~=K=q==~==============fK===
531. ^==~=====~=~=K====
5.4 Operations with Matrices =
532. q=~=^=~=_=~=~=I=~==I==~======~==~== ==~===~=~K==
533. q=~==^=~=_==~==~=E=~F=I=~==I==~==~=~= K=f==
[ ]
==
ON
OOOON
NNONN
~~~
~~~
~~~
~^
K
MMM
K
K
I==
[ ]
==
ON
OOOON
NNONN
_
K
MMM
K
K
I==
=====

112
==
+++
++++++
=+
OONN
OOOOOOONON
NNNONONNNN
~~~
~~~
~~~
_^
K
MMM
K
K
K=
=534. f===~=~~I=~= [ ]~^ = ==~=~I==
[ ]
==
ON
OOOON
NNONN
~~~
~~~
~~~
~^
K
MMM
K
K
K=
=535. j~==q=j~=
q= ~= ~= = = = = = == = = = = = = ~= = = = ====K===f=
[ ]
==
ON
OOOON
NNONN
~~~
~~~
~~~
~^
K
MMM
K
K
I==
[ ]
==
ON
OOOON
NNONN
_
K
MMM
K
K
I=
=====

113
==
==
ON
OOOON
NNONN
`^_
K
MMM
K
K
I==
==
=
=+++=
NOONN ~~~~ K =
E IIOIN K= X IIOIN K= FK===q==
[ ]
==
OPOOON
NPNONN ~~~
~~~~^ I= [ ]
==
P
O
N
_ I==
==
=
=
POPOOONON
PNPONONNN
P
O
N
OPOOON
NPNONN
~~~
~~~
~~~
~~~^_ K==
=536. q~==~=j~=
f===~===~=~=~=~I====~==~==~===~=~K===f= ^= = = ~= ~I= = ~= = = q^ = =
^
K===
537. q=~=^==~== f^^q = K===
538. f==~==^_==I===( ) qqq ^_^_ = K===

114
539. ^==j~=f=^==~=~= ~I==~I=== ^~ I===~===~==~= ` ==^W=
[ ]q`^~ = K===
540. q~==~=j~=f= ^= = ~= ~= ~I= = ~I= = = ^ I= =============~=~~W=
OONN ~~~^ +++= K K==541. f==~=j~=
f=^==~=~= ~==~=~=~=^ I==== N^ ====
^
^~^ N = K=
=542. f==~==^_==I===
( ) NNN ^_^_ = K==
543. f==^===~=~=== ==~I======u===~==~=
u^u = I====~= =~==~~=~=
Mf^ = K======
5.5 Systems of Linear Equations ==s~~W=I=I=I= N I= KIO =o~=W= KI~I~II~I~I~ NONNNPON =

115
a~W=aI= a I= a I= a ==
j~W=^I=_I=u===
544.
=+=+
OOO
NNN
~
~I==
a
a = I=
a
a = =E`~=FI==
==
NOONOO
NN ~~~
~a == I==
NOONOO
NN
a == I==
NOONOO
NN ~~~
~a == K==
=545. f= Ma I====~=~==W==
a
a = I=
a
a = K=
f= Ma= = ~= Ma E= Ma FI= = = = ~= = ==K=f= Maaa === I= = = = ~= = = = ~==K==
546.
=++=++=++
PPPP
OOOO
NNNN
~
~
=~
I==
a
a = I=
a
a = I=
a
a = =E`~=FI==
=

116
==
PPP
OOO
NNN
~
~
~
a= I=
PPP
OOO
NNN
a = I=
PPP
OOO
NNN
~
~
~
a = I=
PPP
OOO
NNN
~
~
~
a = K==
=547. f= Ma I====~=~==W==
a
a = I=
a
a = I=
a
a = K=
f= Ma= =~= Ma E= Ma == Ma FI====~==K=f= Maaaa ==== I= = = = ~= =~=K==
548. j~=c==~=p===i~=b~===================r=q===~=~==
=+++
=+++=+++
OONN
OOOOONON
NNONONNN
~~~
~~~
~~~
K
KKKKKKKKKKKK
K
K
=
~====~==
=
O
N
O
N
ON
OOOON
NNONN
~~~
~~~
~~~
MM
K
MMM
K
K
I==
KK==_u^ = I==

117
==
=
ON
OOOON
NNONN
~~~
~~~
~~~
^
K
MMM
K
K
I=
=
O
N
uM
I=
=
O
N
_M
K==
=549. p==~=p==i~=b~= =
_^u N = I=== N^ =====^K===

118
Chapter 6
Vectors ====
sW= r
I= r
I= r
I= r
I=
^_ I==s=W=
rI= r
I==
r=W= r
I= r
I= r
=
k=W= Mr
=`~===
rW= NNN wIvIu =
`~=== r
W= OOO wIvIu =p~~W= I =a=W= I= I= =^===W= ===
6.1 Vector Coordinates =
550. r=s=( )MIMIN =
rI=
( )MINIM =r
I=
( )NIMIM =r
I=
N ===rrr
K=
=
551. ( ) ( ) ( )^_ MNMNMNrrrr
++==
=
=

CHAPTER 6. VECTORS
119
======= ==
Figure 73. =
552. ( ) ( ) ( )OMNOMNOMN ^_ ++==r
=
=
553. f= ^_ r=
I== _^r
=
K==
==
Figure 74. =
554. = u r I== v
rI=
= wr
K=

CHAPTER 6. VECTORS
120
===== ==
Figure 75. =
555. f= ( ) ( )NNNN wIvIuwIvIurr
= I===
Nuu = I= Nvv = I= Nww = K=====
6.2 Vector Addition =
556. rrr += ==
== ==
Figure 76.

CHAPTER 6. VECTORS
121
== ==
Figure 77. =
557. PON r
Krrrr
++++= ==
== ==
Figure 78. =
558. `~=i~=rrrr
+=+ ==
559. ^~=i~=( ) ( ) rrrrrr ++=++ ==
560. ( )ONONON wwIvvIuu +++=+rr
=======

CHAPTER 6. VECTORS
122
6.3 Vector Subtraction =
561. rrr = == rrr =+ K==
==
Figure 79. =
== ==
Figure 80. =
562. ( ) rrrr += ==
563. ( )MIMIMM ==rrr
==
564. MM =r
=
=565. ( )ONONON wwIvvIuu =
rrI==
===
6.4 Scaling Vectors =
566. rr = =

CHAPTER 6. VECTORS
123
==
Figure 81. =
567. rr = ==
568. ( )wIvIu =r ==
569. = rr ==
570. ( ) rrr +=+ ==
571. ( ) ( ) ( ) rrr == ==
572. ( ) rrrr +=+ ====
6.5 Scalar Product =
573. p~~=m==s= r =~ r ==
rrrrI==
= ===~=== r
=~ r
K=====

CHAPTER 6. VECTORS
124
= ==
Figure 82. =
574. p~~=m==`~=c=f= ( )NNN wIvIu =r
I= ( )OOO wIvIu =r
I===
ONONON wwvvuu ++=rr
K==
575. ^=_=q=s==f= ( )NNN wIvIu =r
I= ( )OOO wIvIu =r
I===
OO
OO
OO
ON
ON
ON
ONONON
wvuwvu
wwvvuu
++++
++= K=
=576. `~=m=
rrrr= =
=577. ^~=m=
( ) ( ) rrrr = ==
578. a=m=( ) rrrrrrr +=+ =
=
579. M = rr == r I r =~=~=EO
= FK=
=
580. M > rr ==O
M

CHAPTER 6. VECTORS
125
581. M

CHAPTER 6. VECTORS
126
======= ==
Figure 83. =
588. OOO
NNN
wvu
wvu
rrr
rrr== =
=
589.
==
OO
NN
OO
NN
OO
NN
vu
vuI
wu
wuI
wv
wvrrr
=
=590. == p rrrr =EcKUPF=
=591. ^=_=q=s=EcKUPF=
rr
rr
= =
=592. k~=m=
( ) rrrr = ===
593. ^~=m=( ) ( ) rrrr = ===

CHAPTER 6. VECTORS
127
594. a=m=( ) rrrrrrr +=+ =
=595. M
rrr= ==
r=~=
r=~=~~=E M= FK=
=
596. Mrrrrrrr
=== ==
597. rrr
= I= rrr
= I= rrr
= ====
6.7 Triple Product =
598. p~~=q=m=[ ] ( ) ( ) ( ) rrrrrrrrrrrr === ==
599. [ ] [ ] [ ] [ ] [ ] [ ] rrrrrrrrrrrrrrrrrr ===== ==
600. ( ) [ ] rrrrrr = ==
601. p~~=q=m==`~=c=
( )PPP
OOO
NNN
wvu
wvu
wvu
=rrr
I==
==( )NNN wIvIu =
rI= ( )OOO wIvIu =r
I= ( )PPP wIvIu =r
K==
=602. s==m~~=
( )s rrr = ==

CHAPTER 6. VECTORS
128
============ ==
Figure 84. =
603. s==m~=
( )S
Ns
rrr= =
=
==
Figure 85. =
604. f== ( ) M = rrr I===== r I= r I=~= r =~=~==I==
rrr+= ===~~= =~= K==
=605. f== ( ) M rrr I===== r I= r I=~= r =~=~=
K==

CHAPTER 6. VECTORS
129
606. s=q=m=( ) ( ) ( ) rrrrrrrrr = ==
========

130
Chapter 7
Analytic Geometry ====
7.1 One-Dimensional Coordinate System =m=~W= M I= N I= O I= M I= N I= O =
o~=W= ==a~===W====
607. a~=_=q=m=ONNO ^_ === =
=
==
Figure 86. =
608. a=~=i=p===o~= =
++
=N
ONM I= `_
^`= I= N K=
=
======== ==
Figure 87.

CHAPTER 7. ANALYTIC GEOMETRY
131
609. j==~=i=p=
O
ONM
+= I= N= K=
===
7.2 Two-Dimensional Coordinate System =m=~W= M I= N I= O I= M I= N I= O =
m~=~W= I =o~=W= ==m=~=W=~I=I=I==a~===W==^~W=p===
610. a~=_=q=m=( ) ( )ONOONO ^_ +== =
=
==
Figure 88.

132
611. a=~=i=p===o~= =
++
=N
ONM I= +
+=
N
ONM I==
`_
^`= I= N K=
=
======= ==
Figure 89. ==

133
======= ==
Figure 90. =
612. j==~=i=p=
O
ONM
+= I=
O
ONM
+= I= N= K=
=613. `=Ef==j~F==~=q~=
P
PONM
++= I=
P
PONM
++= I==
== ( )NN I^ I== ( )OO I_ I==~== ( )PP I` ==~====~= ^_` K= ==

134
========= ==
Figure 91. =
614. f=Ef==^=_F==~=q~=
~
~ PONM ++
++= I=
~
~ PONM ++
++= I==
= _`~ = I= `^ = I= ^_ = K===
======== ==
Figure 92.

135
615. `=Ef===p=m~======================_F==~=q~=
N
N
N
O
N
N
N
PP
OO
NN
POP
OP
OOO
OO
NON
ON
M
+++
= I=
N
N
N
O
N
N
N
PP
OO
NN
OP
OPP
OO
OOO
ON
ONN
M
+++
= =
=
======== ===
Figure 93. =======

136
616. l=Ef==^F==~=q~=
N
N
N
N
N
N
PP
OO
NN
OPONP
OONPO
ONPON
M
+++
= I=
N
N
N
N
N
N
PP
OO
NN
PONOP
ONPOO
NPOON
M
+++
= =
=
====== ==
Figure 94. =
617. ^~==~=q~=
( ) ( )NPNP
NONO
PP
OO
NN
O
N
N
N
N
O
Np
== =
===

137
618. ^~==~=n~~~=
( ) ( )( ) ( )( )[ ++++= POPOONON ON
p =
( )( ) ( )( )]NQNQQPQP ++++ ==
=== ==
Figure 95. =kW=f=~=SNTI=SNU=====EHF==EF==~===~==~==~~K==
=619. a~=_=q=m==m~=`~=
( )NOONOOON O^_ +== ==

138
==
Figure 96. =
620. `=o~~=`~==m~=`~== I= = K=
=
==
Figure 97. =
621. `=m~=`~==o~~=`~=OO += I=
~ = K=

139
7.3 Straight Line in Plane =m=~W=uI=vI=I= M I= N I== M I= N I= N~ I= O~ I===
o~=W=I=~I=I=I=I=^I=_I=`I= N^ I= O^ I==^W= I= =^===W= =k~=W=
r=
m=W= r
I= ~r
I= r
===
622. d~=b~==~=p~=i=M`_^ =++ =
=623. k~=s==~=p~=i=
q== ( )_I^r ==~==== M`_^ =++ K==
==
Figure 98. =
624. b=b~==~=p~=i=Ep-f=cF= += K==

140
q=~===== = ~ K==
==
Figure 99. =
625. d~==~=i==
NO
NO
~
== =
=
==
Figure 100.

141
626. b~==~=i=d=~=m=~==d~=( )MM += I==
====~I= ( )MM Im ==~====K==
==
Figure 101. =
627. b~==~=i=q~=m~=q=q=m=
NO
N
NO
N
=
==
=
M
N
N
N
OO
NN = K=
=

142
==
Figure 102. =
628. f=c=
N
~
=+ =
=
==
Figure 103. ==

143
629. k~=c=M =+ =
=
==
Figure 104. =
630. m=a=c=
v
u
NN =
I==
= ( )vIu = = = = = = = ~= ( )NNN Im = ===K==

144
==
Figure 105. =
631. s~=i=~ = =
=632. e~=i=
= ==
633. s=b~==~=p~=i=~rrr
+= I====l======~I=u==~=~~====I==~r
======~===^====I=
r
==~====I=~~===I====~=~~I==
=lur
======~==u===K===

145
==
Figure 106. =
634. p~=i==m~~=c=
+=+=
OO
NN
~
~I==
==( )I ~==~==~=====I==( )ON ~I~ =~==~==~=====I==( )ON I =~==~==~==~~===I====~=~~K==

146
=Figure 107.
=635. a~=c=~=m=q=~=i=
q=~==== ( )I~m ====M`_^ =++ ===
OO _^
`_^~
+
++= K=
=
==
Figure 108.

147
636. m~~=i=q== NN += =~= OO += =~=~~===
ON = K=q= = M`_^ NNN =++ = ~= M`_^ OOO =++ = ~=~~==
O
N
O
N
_
_
^
^= K=
=
==
Figure 109. =
637. m~=i=q== NN += =~= OO += =~=~===
NO
N = =I=~I= N ON = K=
q= = M`_^ NNN =++ = ~= M`_^ OOO =++ = ~=~==
M__^^ ONON =+ K==

148
==
Figure 110. =
638. ^=_=q=i=
ON
NO
N
~
+
= I==
OO
OO
ON
ON
ONON
_^_^
__^^
++
+= K=
=

149
==
Figure 111. =
639. f==q=i=f=== M`_^ NNN =++ =~= M`_^ OOO =++ =-I====~=~=
NOON
NOONM _^_^
_`_`
+
= I=NOON
NOONM _^_^
`^`^
+
= K=
===
7.4 Circle =o~W=o=`==W= ( )I~ =m=~W=I=I= N I= N I==o~=W=^I=_I=`I=aI=bI=cI==

150
640. b~==~=`=`=~==l=Ep~~=cF=
OOO o =+ =
====== ==
Figure 112. =
641. b~==~=`=`=~=^=m= ( )I~ ( ) ( ) OOO o~ =+
Figure 113.

151
642. q=m=c
M
N
N
N
N
PPOP
OP
OOOO
OO
NNON
ON
OO
=
++++
=
==
Figure 114. =
643. m~~=c
==
o
oI= OM K
=644. d~=c
Mcba^^ OO =++++ =E^=I= ^cQba OO >+ FK==q=====~=~= ( )I~ I===
^O
a~ = I=
^O
b = K=
q=~====

152
Ô
^cQbao
OO += K
===
7.5 Ellipse =p~=~W=~=p=~W==cW= ( )MIcN I= ( )MIcO =a~===W=O= =bW===o~=W=Î=_I=Ì=aI=bI=cI==mW=i=^~W=p===
645. b~==~=b=Ep~~=cF
N
~
O
O
O
O
=+
==
Figure 115.

153
646. ~O ON =+ I= == N I== O ==~==~====~==== ( )Im ========K==
==
Figure 116. =
647. OOO ~ += =
648. b
N~

154
651. d~=c Mcba`_^ OO =+++++ I==
= M^`Q_O K =
653. ` ( )~bQi = I==
=====b========~=====K===
654. ^~=c~===` ( )( )~~RKNi += I==( )OO ~Oi += K=
=655. ~p = =
===
7.6 Hyperbola =q~=~W=~=`~=~W==cW= ( )MIcN I= ( )MIcO =a~===W=O= =bW===^W=I==o~=W=^I=_I=`I=aI=bI=cI=I=====

155
656. b~==~=e~=Ep~~=cF=
N
~
O
O
O
O
= =
=
==
Figure 117. =
657. ~O ON = I= = = N I= = O = =~= =~= ===~== = ( )Im ====~====K==

156
==
Figure 118. =
658. b~==^=
~
= =
=659. OOO ~ += =
=660. b
N~
>= =
=661. b~==a
~
~
O
== =
===

157
662. m~~=b~===o=_~==~=e~=
==
~I= OM K
=663. d~=c
Mcba`_^ OO =+++++ I=== M^`Q_O > K==
664. d~=c==^=m~~===`~=^ Mcba`^ OO =++++ I==
= M^` < K=
665. ^=c=
Q
O
= I==
==
= I==
Q
O
= K=
f= = ~= I= = ~= ~= ~= M = = ~=M = K==
=

158
==
Figure 119. ===
7.7 Parabola =c~=~~W==cW=c=sW= ( )MM Ij =o~=W=^I=_I=`I=aI=bI=cI=I=~I=I====
666. b~==~=m~~~=Ep~~=cF OO =
=

159
==
Figure 120. =b~===
O
= I=
`~====
MIO
c I=
`~====( )MIMj K=
=667. d~=c
Mcba`_^ OO =+++++ I=== M^`Q_O = K==
668. O~ = I=~O
N = K=
b~===

160
O
= I=
`~====
O
IMc I=
`~====( )MIMj K=
=
==
Figure 121. =
669. d~=cI=^=m~~===-~==Mcba^ O =+++ =E^I=b=FI==
~ O ++= I=~O
N = K==
b~===
O
M = I=
`~====

161
+
O
Ic MM I=
`~====
~O
M = I= ~Q
~Q~
O
MOMM
=++= K=
=
==
Figure 122. ===
7.8 Three-Dimensional Coordinate System =m=~W= M I= M I= M I= N I= N I= N I==
o~=W= ==a~===W==^~W=p=sW=s==

162
670. a~=_=q=m=( ) ( ) ( )ONOONOONO ^_ ++== =
=
=== ==
Figure 123. =
671. a=~=i=p===o~= =
++
=N
ONM I= +
+=
N
ONM I= +
+=
N
ONM I==
=
`_
^`= I= N K=
=

163
======== ==
Figure 124. =
=
Figure 125.

164
672. j==~=i=p=
O
ONM
+= I=
O
ONM
+= I=
O
ONM
+= I= N= K=
=673. ^~==~=q~=
q=~~==~=~=== ( )NNNN IIm I=( )OOOO IIm I=~= ( )PPPP IIm =====
O
PP
OO
NN
O
PP
OO
NN
O
PP
OO
NN
N
N
N
N
N
N
N
N
N
O
Np ++= K=
=674. s==~=q~=
q===~=~=== ( )NNNN IIm I=( )OOOO IIm I= ( )PPPP IIm I=~= ( )QQQQ IIm =====
N
N
N
N
S
Ns
QQQ
PPP
OOO
NNN
= I==
=
QPQPQP
QOQOQO
QNQNQN
S
Ns
= K=
kW=t====EHF==EF==~===~==~==K===

165
==== ==
Figure 126. ===
7.9 Plane =m=~W=I=I=I= M I= M I= M I= N I= N I= N I==
o~=W=^I=_I=`I=aI= N^ I= O^ I=~I=I=I= N~ I= O~ I= I=I=I===k~=W=
rI= Nr
I= Or
=a=W= I= I= =a~====~W====
675. d~=b~==~=m~=Ma`_^ =+++ =
==

166
676. k~=s==~=m~=q== ( )`I_I^r ==~===~=
Ma`_^ =+++ K==
=== ==
Figure 127. =
677. m~~=`~===b~==~=m~=Ma`_^ =+++ =
=f= M^ = I==~==~~===-~K=f= M_ = I==~==~~===-~K=f= M` = I==~==~~===-~K=f= Ma= I==~====K===f= M_^ == I==~==~~===-~K=f= M`_ == I==~==~~===-~K=f= M`^ == I==~==~~===-~K==

167
678. m=a=c=( ) ( ) ( ) M`_^ MMM =++ I==
=== ( )MMM IIm == ==~I=~==-= ( )`I_I^ ==~===~K====
==== ==
Figure 128. =
679. f=c=
N
~
=++ =
=

168
===== ==
Figure 129. =
680. q=m=c=
M
POPOPO
PNPNPN
PPP
=
I==
==
M
N
N
N
N
PPP
OOO
NNN = K=
=

169
===== ==
Figure 130. =
681. k~=c=M =++ I==
========~==~========~=I=~= I= I= =~=====~==~===~K===

170
====== ==
Figure 131. =
682. m~~=c=
++=++=++=
~~
ONN
ONN
ONN
I==
= ( )II =~==~==~======I=== ( )NNN IIm ====~I===( )NNN II~ =~= ( )OOO II~ =~=~~===~K==

171
===== ==
Figure 132. =
683. a~=^=_=q=m~=f==~=~====
Ma`_^ NNNN =+++ I==Ma`_^ OOOO =+++ I==
==~=~=====
OO
OO
OO
ON
ON
ON
ONONON
ON
ON
`_^`_^
``__^^
++++
++=
= rrrr
K=
=

172
====== ==
Figure 133. =
684. m~~=m~=q=~= Ma`_^ NNNN =+++ =~=
Ma`_^ OOOO =+++ =~=~~===
O
N
O
N
O
N
`
`
_
_
^
^== K=
=685. m~=m~=
q=~= Ma`_^ NNNN =+++ =~=Ma`_^ OOOO =+++ =~=~===
M``__^^ ONONON =++ K==
686. b~==~=m~=q= ( )NNN IIm =~=m~~=q==s= ( )NNN II~ =~= ( )OOO II~ =EcKNPOF=

173
M
~
~
OOO
NNN
NNN
=
=
=687. b~==~=m~=q= ( )NNNN IIm =~= ( )OOOO IIm I=
~=m~~=q==s= ( )II~ =
M
~
NONONO
NNN
=
=
=
=
Figure 134. =
688. a~=c=~=m=q=~=m~=q=~==== ( )NNNN IIm ===~=
Ma`_^ =+++ ===

174
OOO
NNN
`_^
a`_^
++
+++= K=
=
====== ==
Figure 135. =
689. f==q=m~=f==~= Ma`_^ NNNN =+++ =~=
Ma`_^ OOOO =+++ = I= = = ~=====
+=+=+=
~
N
N
N
I==
==
~
NNN =
=
I==
==

175
OO
NN
`_
`_~ = I=
OO
NN
^`
^` = I=
OO
NN
_^
_^ = I==
OOO
OO
NN
OO
NN
N ~
_a
_a
à
à
++
= I==
OOO
OO
NN
OO
NN
N ~
à
à~
â
â
++
= I==
OOO
OO
NN
OO
NN
N ~
â
â
_a
_a~
++
= K==
===
7.10 Straight Line in Space =m=~W=I=I=I= N I= N I= N I==a=W= I= I= =o~=W=Î=_I=Ì=aI=~I=I=I= N~ I= O~ I=I===a===~=W=
rI= Nr
I= Or
=k~===~=~W=
r=
^===W= ==
690. m=a=c===b~==~=i==
~
NNN =
=
I==
=== ( )NNNN IIm ====I=~= ( )II~ =======K===

176
===== ==
Figure 136. =
691. q=m=c=
NO
N
NO
N
NO
N
=
=
==
=
===== =Figure 137.

177
692. m~~=c==
+=+=+=
N
N
N
I==
=== ( )NNNN IIm ====~=I= I= I= =~=======
===I==~~===~=~=K===
===== ==
Figure 138. =
693. ^=_=q=p~=i=
OO
OO
OO
ON
ON
ON
ONONON
ON
ON
~~
~~
++++
++=
= rrrr
=
=

178
===== ==
Figure 139. =
694. m~~=i=q==~=~~===
ON rr
I====
O
N
O
N
O
N
~
~== K=
=695. m~=i=
q==~=~~===M ON =
rrI==
==M~~ ONONON =++ K=
=696. f==q=i=
q==N
N
N
N
N
N
~
=
=
=~=

179
O
O
O
O
O
O
~
=
=
====
M
~
~
OOO
NNN
NONONO
=
K=
=697. m~~=i=~=m~==
q=~==
~
NNN =
=
=~==~=
Ma`_^ =+++ =~=~~==M =
rrI==
==M`_^~ =++ K=
=
===== ==
Figure 140. ==

180
698. m~=i=~=m~==
q=~==
~
NNN =
=
=~==~=
Ma`_^ =+++ =~=~==rr
I====
`
_
~
^== K=
=
==== ==
Figure 141. ==
7.11 Quadric Surfaces =m=~===~=~W=I=I==o~=W=^I=_I=`I=~I=I=I= PON II I==

181
699. d~=n~~=b~=MaoOnOmOeOdOcO`_^ OOO =+++++++++
=700. `~~==n~=p~=
=`~= o~EF= o~EbF= = == q==p~=
N= P= Q= M< = p~= o~=b=O= P= Q= M> = p~= f~~=b=P= P= Q= M> = a= e==N=p=Q= P= Q= M< = a= e==O=p=R= P= P= = a= o~=n~=`=S= P= P= = p~= f~~=n~=`=T= O= Q= M< = p~= b=m~~=U= O= Q= M> = a= e=m~~=V= O= P= = p~= o~=b=`=
NM= O= P= = p~= f~~=b=`=NN= O= P= = a= e=`=NO= O= O= = a= o~=f=m~=NP= O= O= = p~= f~~=f=m~=NQ= N= P= = = m~~=`=NR= N= O= = = o~=m~~=m~=NS= N= O= = = f~~=m~~=m~=NT= N= N= = = `=m~=
=e==
=
`cd
c_e
de^
I=
=
aonm
o`cd
nc_e
mne^
b I= ( )b= I==
PON II =~=====~I==
M
`cd
c_e
de^
=
K=
==

182
701. o~=b=E`~=NF=
N
~
O
O
O
O
O
O
=++ =
=
===== ==
Figure 142. =
702. f~~=b=E`~=OF=
N
~
O
O
O
O
O
O
=++ =
=703. e==N=p=E`~=PF=
N
~
O
O
O
O
O
O
=+ =
=

183
===== ==
Figure 143. =
704. e==O=p=E`~=QF=
N
~
O
O
O
O
O
O
=+ =
=
= ==
Figure 144.

184
705. o~=n~=`=E`~=RF=
M
~
O
O
O
O
O
O
=+ =
=
== ==
Figure 145. =
706. f~~=n~=`=E`~=SF=
M
~
O
O
O
O
O
O
=++ =
=707. b=m~~=E`~=TF=
M
~
O
O
O
O
=+ =
=

185
====== =Figure 146.
708. e=m~~=E`~=UF=
M
~
O
O
O
O
= =
==== =Figure 147.

186
709. o~=b=`=E`~=VF=
N
~
O
O
O
O
=+ =
=
===== ==
Figure 148. =
710. f~~=b=`=E`~=NMF=
N
~
O
O
O
O
=+ =
=711. e=`=E`~=NNF=
N
~
O
O
O
O
= =
=

187
====== ==
Figure 149. =
712. o~=f=m~=E`~=NOF=
M
~
O
O
O
O
= =
=713. f~~=f=m~=E`~=NPF=
M
~
O
O
O
O
=+ =
=714. m~~=`=E`~=NQF=
M~
O
O
= =
=

188
===== ==
Figure 150. =
715. o~=m~~=m~=E`~=NRF=
N~
O
O
= =
=716. f~~=m~~=m~=E`~=NSF=
N~
O
O
= =
=717. `=m~=E`~=NTF=
MO = ======

189
7.12 Sphere =o~==~=W=o=m=~W=I=I=I= N I= N I= N I==`==~=W= ( )II~ =o~=W=^I=aI=bI=cI=j===
718. b~==~=p=`=~==l=Ep~~=cF=
OOOO o =++ ==
====== ==
Figure 151. =
719. b~==~=`=`=~=^=m= ( )II~ ( ) ( ) ( ) OOOO o~ =++
720. a~=c
( )( ) ( )( ) ( )( ) M ONONON =++ I==

190
==( )NNNN IIm I= ( )OOOO IIm =~====~=~K==
=721. c=m=c=
M
N
N
N
N
N
QQQOQ
OQ
OQ
PPPOP
OP
OP
OOOOO
OO
OO
NNNON
ON
ON
OOO
=
++++++++++
=
=722. d~=c
Mjcba^^^ OOO =++++++ =E^==FK==q=====~=~= ( )II~ I===
Ô
a~ = I=
Ô
b = I=
Ô
c = K=
q=~====
Ô
j^Qcbao
OOOO ++= K
==

191
Chapter 8
Differential Calculus ====cW=I=I=I=I==^=E=~~FW==o~=W=~I=I=I==k~~=W==^W= =f=W= N ===
8.1 Functions and Their Graphs =
723. b=c=( ) ( ) = =
=724. l=c=
( ) ( ) = ==
725. m=c=( ) ( )q =+ =
=726. f=c=
( ) = ==~=I= ( ) = == ( ) N= ====K===

CHAPTER 8. DIFFERENTIAL CALCULUS
192
===== ==
Figure 152. =
727. `=c=( ) = I= ( ) = I= ( )( ) = ==~==K=
=728. i~=c=
~ += I= = o I= = = ~~ =======I======-K==

193
====== ==
Figure 153. =
729. n~~=c==O = I= o K=
=
====== ==
Figure 154. =

194
730. ~ O ++= I= o K==
=== ==
Figure 155. =
731. `=c==P = I= o K=
=

195
= ==
Figure 156. =
732. ~ OP +++= I= o K==

196
== ==
Figure 157. =
733. m=c== = I= k K=
=

197
====== =Figure 158.
=
=Figure 159.
=

198
734. p~=o=c== = I= [ ) IM K=
=
======= =Figure 160.
=735. b~=c=
~ = I= M~ > I= N~ I= = == ~ = I= KSTNUOUNUOUQKO = =
=
= ==
Figure 161.

199
736. i~=c= ~= I= ( ) IM I= M~ > I= N~ I=
= == ~ = I= M > K==
==
Figure 162. =
737. e=p=c==
= I=O
= I= o K=
=

200
==== ==
Figure 163. =
738. e=`=c==
= I=O
+= I= o K=
====== ==
Figure 164.

201
739. e=q~=c==
~ = I=
~
+
=== I= o K=
=
====== ==
Figure 165. =
740. e=`~=c==
= I=
+
=== I= o I= M K=
=

202
====== ==
Figure 166. =
741. e=p~=c==
= I=
O
N +
=== I= o K=
=Figure 167.

203
742. e=`~=c==
= I=
O
N
=== I= o I= M K=
=
====== ==
Figure 168. =
743. f=e=p=c==~ = I= o K=
=

204
===== ==
Figure 169. =
744. f=e=`=c==~ = I= [ ) IN K=
=
===== ==
Figure 170. =
745. f=e=q~=c==~~ = I= ( )NIN K=
=

205
===== ==
Figure 171. =
746. f=e=`~=c==~ = I= ( ) ( ) INNI K==
=

206
===== ==
Figure 172. =
747. f=e=p~=c==~ = I= ( ]NIM K=
=

207
=Figure 173.
=748. f=e=`~=c==
~ = I= o I= M K==
= Figure 174.
=

208
8.2 Limits of Functions =cW= ( ) I= ( ) =^W==o~=~W=~I====
749. ( ) ( )[ ] ( ) ( )~~~
+=+ =
=750. ( ) ( )[ ] ( ) ( )
~~~ = =
=751. ( ) ( )[ ] ( ) ( )
~~~ = =
=
752. ( )( )( )( )
~
~
~
= I== ( ) M
~
K=
=753. ( )[ ] ( )
~~ = =
=754. ( )( ) ( )( )
~~ = =
=755. ( ) ( )~
~=
I==== ( ) ===~= ~ = K=
=
756. N
M=
=
=
757. N
~
M=
=
=
758. N
N
M=
=

209
759. N
~
N
M=
=
=
760. ( ) N
N
M=
+
=
=
761.
NN
=
+
=
=
762.
N =
+
=
=763. N~
M=
=
===
8.3 Definition and Properties of the Derivative =cW=I=I=I=I==f=~~W==o~=~W==^W= ===
764. ( ) ( ) ( )
MM=
=
+=
==
=

210
== ==
Figure 175. =
765. = ~
==
=
766. ( )
+=
+=
=
767. ( )
=
=
=
768. ( )
= =
=769. m=o=
( )
+=
==
==

211
770. n=o=
O
=
=
=771. `~=o=
( )( ) = I= ( ) = I==
= K=
=772. a~==f=c=
N
= I==
= ( ) ===== ( ) K===
773. o~=o=
O
N
=
=
=774. i~=a~=
( ) = I= ( ) = I==
( ) ( )[ ]
= K=
==
8.4 Table of Derivatives =f=~~W==o~=~W=`I=~I=I==k~~=W==

212
775. ( ) M`
= =
=
776. ( ) N
= =
=
777. ( ) ~~
=+ =
=
778. ( ) ~~
O +=++ =
=
779. ( ) N
= =
=
780. ( )N
+
= =
=
781. O
N
N
=
=
=
782. ( )O
N
= =
=
783. ( ) N
N
= =
=
784. ( )
N
= =
=
785. ( )~
N
~ = I= M~ > I= N~ K=
=

213
786. ( ) ~~~
= I= M~ > I= N~ K=
=
787. ( )
= =
=
788. ( )
= =
=
789. ( )
= =
=
790. ( )
N~
OO
== =
=
791. ( )
N
OO
== =
=
792. ( ) ~
= =
=
793. ( )
= =
=
794. ( )ON
N~
= =
=
795. ( )ON
N~
= =
=
796. ( )ON
N~~
+= =
=

214
797. ( )ON
N~
+= =
=
798. ( )N
N~
O
= =
=
799. ( )N
N~
O
= =
=
800. ( )
= =
=
801. ( )
= =
=
802. ( )
N~
OO
== =
=
803. ( )
N
OO
== =
=
804. ( ) ~
= =
=
805. ( )
= =
=
806. ( )N
N=~
O +
= =
=
807. ( )N
N=~
O
= =

215
808. ( )ON
N=~~
= I= N < K=
=
809. ( )N
N=~
O
= I= N > K=
=
810. ( )
N += =
===
8.5 Higher Order Derivatives =cW=I=I=I==f=~~W==k~~=W====
811. p=~=
( )O
O
=
=
== =
=812. e-l=~=
( ) ( ) ( )( ) === N
=
=
813. ( )( ) ( ) ( ) +=+ ==
814. ( )( ) ( ) ( ) = ==
815. i=c~=( ) O ++= =

216
( ) PP +++= =
( )( ) ( ) ( ) ( ) ( ) ( )ON ON
N ++
++= K =
=
816. ( )( ) ( )
>
>
= =
=
817. ( )( ) > = ==
818. ( )( ) ( ) ( )~
>NN
N
~
=
=
=
819. ( )( ) ( ) ( )
N
>NN
=
=
=
820. ( )( ) ~~~ = ==
821. ( )( ) = ==
822. ( )( ) ~~~ = ==
823. ( )( )
+=
O
=
=
824. ( )( )
+=
O
=
===

217
8.6 Applications of Derivative =cW=I=I==m==~=W===sW==^~W==f=~~W==qW==k~~=W====
825. s=~=^~=( ) = =====~==~==~==
~==~=~==I==( ) == ===~~====I=
( ) === ===~~=~~===K===
826. q~=i=( )( )MMM = =
=

218
==
Figure 176. =
827. k~=i=
( )( )MMM
N
= =Ec=NTSF=
=828. f~=~=a~=cK==
f= ( ) M M > I==EF==~=~= M K=Ec=NTTI= N < I=O < FI=
f= ( ) M M

219
==
Figure 177. =
829. i~=~=^==EF=~=~=~=~=~= N ==~======~=~= N ==~=( ) ( ) N ==~====~=EcKNTTFK==
=^==EF=~=~=~==~= O ==~======~=~= O ==~=( ) ( ) O ==~====~=EcKNTTFK=
=830. `~=m=
^=~===EF==~= M ==~====
( )M =====~==K==
831. c=a~=q==i~=b~K=f=EF===~==E ( ) M > F===~========~=( ]NI~ ==~==EF====~==E ( ) M

220
832. f=EF==~=E ( ) M F==~====~= [ )I O I==EF=~=~=~==~= O K==EcKNTTFK==
833. p=a~=q==i~=b~K=f= ( ) M N = =~= ( ) M N I==EF=~=~=~==~= O K=EcKNTTF==
834. `~K==EF= = = ~= ~= ~= = M = = = = ~= = = = = = ( ) = ============~=~= M =EcKNTTI= P < FK===EF= = =~= =~=~= = M = = =~== = = ( ) = = ===============~=~= M K=EcKNTTI= P < FK====
835. p=a~=q==`~K==f= ( ) M M > I==EF==~=~=~= M K==f= ( ) M M

221
8.7 Differential =cW=I=I==f=~~W==a~==~=W= ( ) I= ( ) =o~=~W=`=a~=== ( ) = W==a~==W==p~=~==W= =p~=~==W= ===
838. = ==
839. ( ) ( ) ( ) +=+ ==
==
Figure 178.

222
840. p~=`~===( ) ( ) += =
=841. ( ) +=+ =
=842. ( ) = =
=843. ( ) `` = =
=844. ( ) += =
=
845. O
=
=
===
8.8 Multivariable Functions =c===~~W= ( )I I= ( )I I= ( )I I= ( )I ==^W=I=I==p~=~==I=I=I=W= I= I= K===
846. c=l=m~~=a~=q=~~=~=====
=
=E~=
=
FI=
q=~~=~=====
=
=E~=
=
FK=
==

223
847. p=l=m~~=a~=
O
O
=
=
I==
O
O
=
=
I==
O
=
=
I==
O
=
=
K==
f==~=~=I===
OO
=
K==
=848. `~=o==
f= ( ) ( )( )II = =E==~====~~=FI===
( )( )
I
=
I= ( )( )
I
=
K==
=
f= ( ) ( ) ( )( )I = I== ( )
+
= K==
=f= ( ) ( )( )III = I===
+
=
I=
+
=
K==
=849. p~=`~=
+
=
==

224
850. i~=j~~=~=j~=( )I =~=~=~=~=~= ( )MM I == ( ) ( )MM II =
=~= ( )I ==== ( )MM I K===( )I =~=~=~==~= ( )MM I == ( ) ( )MM II =
=~= ( )I ==== ( )MM I K==
851. p~~=m=
M
=
=
K=
i~=~~=~=~=~==~=~~=K===
852. p~=m=^=~~= == == = = == =~= = ~= =~==~=~===
853. p=a~=q==p~~=m=
i= ( )MM I ==~=~~==E M
=
=
FK==
( ) ( )( ) ( )MMMM
MMMM
II
IIa = K==
=f= Ma> I= ( ) MI MM > I== ( )MM I ===~===~=~K=f= Ma> I= ( ) MI MM < I== ( )MM I ===~===~=~~K=f= Ma< I= ( )MM I ==~=~=K=f= Ma= I===~K==
854. q~=m~=q=~===~=~===~= ( )I = =~= ( )MMM II ===
( )( ) ( )( )MMMMMMM II += K==

225
855. k~==p~=q=~===~===~= ( )I = =~=( )MMM II ===
( ) ( ) N
I
I
M
MM
M
MM
M
=
=
K=
===
8.9 Differential Operators =
r==~==~=~W= r
I= r
I= r
=p~~==E~~=FW= ( )II I= ( )ON III K =d~==~=~~=W= ~ I= =
a~=~W=
=
s==E=FW= ( )oInImcr
=
a==~==W= cr
I= cr =
`==~==W= cr
I= cr
=i~~~=~W= O ===
856. d~==~=p~~=c=
==
I
I
~ I==
==ON
II
I
~ K K=
=857. a~=a~=
+
+
=
I==

226
========
( ) IIr
I= N OOO =++ K===
858. a==~=s=c=
o
n
mcc
+
+
==rr
=
=859. `==~=s=c=
onm
cc
==
rrr
rr=
m
n
o
m
n
o rrr
+
+
= =
=860. i~~~=l~=
O
O
O
O
O
OO
+
+
= =
=
861. ( ) ( ) Mcc = rr ==
862. ( ) ( ) M~ = ==
863. ( ) ( ) ~ O== ==
864. ( ) ( ) ( ) cccc~c OO rrrrr == ===

227
Chapter 9
Integral Calculus ====cW=I=I=I==f=~~W=I=I= =
f=~==~=W= ( ) I= ( ) I==a~==~=W= ( ) I= ( ) I= ( )c I==o~=~W=`I=~I=I=I=I==k~~=W=I=I=I====
9.1 Indefinite Integral =
865. ( ) ( ) `c += == ( ) ( )c = K==
866. ( )( ) ( ) = ==
867. ( ) ( ) = ==
868. ( ) ( )[ ] ( ) ( ) +=+ ==
869. ( ) ( )[ ] ( ) ( ) = ==
870. ( ) ( ) `~c~
N~ += =

CHAPTER 9. INTEGRAL CALCULUS
228
871. ( ) ( ) `~c~
N~ ++=+ =
=
872. ( ) ( ) ( ) `O
N O += =
=
873. ( )( ) ( ) `
+=
==
874. j==p=( ) ( )( ) ( ) = == ( ) = K=
=875. f~==m~=
= I=== ( ) I= ( ) =~=~=K=====
9.2 Integrals of Rational Functions =
876. `~~ += ==
877. `O
O
+= ==
878. `P
PO += =
=
879. `N
N +
+=
+
I= N K==

229
880. ( ) ( )( ) `N~~
~N
++
+=+
+
I= N K==
881. `
+= =
=
882. `~~
N
~
++=
+ ==
883. `
~
~
~O
++
+=++
==
884. ( )( ) `~
~
N
~
+
++
=
++ I= ~ K==
885. ( ) `~~~
N
~
O
+++=+ =
=
886. ( ) ( ) `~~~~O~O
N
N
~
OOP
O
+
++++=
+ ==
887. ( ) `~
~
N
~
+
+=
+ ==
888. ( ) `~
~
~
N
~
OO
++
+=+ =
=
889. ( )
`~
~~
N
~
OO
+
+++=
+ ==

230
890. ( )
`~
~~~O~
N
~
O
PO
O
+
+
++=+ =
=
891. ( ) ( )
`
~
~
N
~~
N
~
OO
++
++
=+ =
=
892. `N
N
O
N
N
O
++
= =
=
893. `N
N
O
N
N
O
++
= =
=
894. `~
~
~O
N
~
OO
++
= =
=
895. `~
~
~O
N
~
OO
++
= =
=
896. `~N
NO
+=+
==
897. `~
~
~
N
~
NOO
+=+
==
898. ( ) `~O
N
~
OOOO
++=+ =
=
899. `~
~~
~
N
~
O
+
=
+ I= M~ > K=
=

231
900. `
~
O
N
~
OO
++=+ =
=
901. ( ) `~
~O
N
~
O
O
O+
+=
+ ==
902. `~
~
~O
N
~
OOO
++
= =
=
903. `~Q~O
~Q~O
~Q
N
~
O
O
OO+
++
+
=
++ I=
M~QO > K==
904. `~Q
~O~~
~Q
O
~
OOO+
+
=
++ I=
M~QO

232
908. ( )( ) `~~NR
O~PO~ O
P
O++
=+ =
=
909. ( )
`~~
~~
~
N
~
+
+++
=
++ I==
M~ > K==
910. ( )
`~
~~~
~
N
~
+
+
=
++ I==M~ K=
=
912. ( )( ) ++=++
~N
~=
( )( ) `~
~~~
~
~+
++
I=E M~ < I= M > FK==
=
913. ( ) ( ) `~NMR
NR~NO~UO~ P
P
OOOO ++
+=+ =
=
914. ( ) `~NR
P~Q~UO
~
P
OOOO
+++
=+ =
=
915. `~~
~~
~
N
~
+
+++
=+ I= M~ > K=
=

233
916. `~
~~~
~
O
~
+

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