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

~

+

1300 Math Formulas

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