1300 Math Formulas
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fp_k= =VVQVNMTTQN=
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`çéóêáÖÜí=«=OMMQ=^Kpîá...
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qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK=

i
Preface
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qÜáë= Ü~åÇÄççâ= áë= ~= ÅçãéäÉíÉ= ÇÉëâíçé= êÉÑÉêÉåÅÉ= Ñçê= ëíìÇÉåíë= ~åÇ= ÉåÖáåÉÉêëK= fí= Ü~ë= ÉîÉêóíÜáåÖ= ...
Contents
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1 krj_bo=pbqp=
NKN= pÉí=fÇÉåíáíáÉë==1=
NKO= pÉíë=çÑ=kìãÄÉêë==5=
NKP= _~ëáÅ=fÇÉåíáíáÉë==7=
NKQ= `çãéäÉñ=kì...
PKNP= háíÉ==42=
PKNQ= `óÅäáÅ=nì~Çêáä~íÉê~ä==43=
PKNR= q~åÖÉåíá~ä=nì~Çêáä~íÉê~ä==45=
PKNS= dÉåÉê~ä=nì~Çêáä~íÉê~ä==46=
PKNT=...
QKS= oÉÇìÅíáçå=cçêãìä~ë==89=
QKT= mÉêáçÇáÅáíó=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90=
QKU= oÉä~íáçåë=ÄÉíïÉÉå=qêáÖçåçãÉíêáÅ=cìåÅíáç...
TKO= qïç=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==131=
TKP= píê~áÖÜí=iáåÉ=áå=mä~åÉ==139=
TKQ= `áêÅäÉ==149=
TKR= bääáéëÉ==152=
TKS= ...
VKNO= iáåÉ=fåíÉÖê~ä==275=
VKNP= pìêÑ~ÅÉ=fåíÉÖê~ä==285=
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10 afccbobkqf^i=bnr^qflkp=
NMKN= cáêëí=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåí...
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qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK=
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viii
Chapter 1

Number Sets
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1.1 Set Identities
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pÉíëW=^I=_I=`=
råáîÉêë~ä=ëÉíW=f=
`çãéäÉãÉåí=W= ^′ =
mêçéÉê=ëìÄëÉíW= ^...
CHAPTER 1. NUMBER SETS

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=====

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Figure 1.

6.

=
7.
=
8.

=
`çããìí~íáîáíó=
^∪_ = _∪^=
^ëëçÅá~íáîáíó=
^ ∪ (_ ∪ ` ) = (^...
CHAPTER 1. NUMBER SETS

11.

=
12.
=
13.
=
14.

aáëíêáÄìíáîáíó=
^ ∪ (_ ∩ ` ) = (^ ∪ _ ) ∩ (^ ∪ ` ) I=
^ ∩ (_ ∪ ` ) = (^ ∩ ...
CHAPTER 1. NUMBER SETS

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=====

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Figure 3.

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19.

_ y ^ = _ y (^ ∩ _ )
=

20.

_ y ^ = _ ∩ ^′

21.

^y^=∅

22.

^ y _ ...
CHAPTER 1. NUMBER SETS

1.2 Sets of Numbers
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26.

27.

=
28.

=
29.

=
30.

k~íìê~ä=åìãÄÉêëW=k=
tÜçäÉ=åìãÄÉêëW= kM =
fåí...
CHAPTER 1. NUMBER SETS

31.

oÉ~ä=kìãÄÉêë==
råáçå=çÑ=ê~íáçå~ä=~åÇ=áêê~íáçå~ä=åìãÄÉêëW=oK=

=
32.

`çãéäÉñ=kìãÄÉêë
` = {ñ +...
CHAPTER 1. NUMBER SETS

1.3 Basic Identities
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oÉ~ä=åìãÄÉêëW=~I=ÄI=Å=
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=
34.

^ÇÇáíáîÉ=fÇÉåíáíó=
~+M=~ =
=

35.

^ÇÇáíáîÉ...
CHAPTER 1. NUMBER SETS

43.

^ëëçÅá~íáîÉ=çÑ=jìäíáéäáÅ~íáçå=
(~ ⋅ Ä)⋅ Å = ~ ⋅ (Ä ⋅ Å )
=
aáëíêáÄìíáîÉ=i~ï=
~ (Ä + Å ) = ~Ä ...
CHAPTER 1. NUMBER SETS

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Figure 6.

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49.
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50.
=
51.
=

(~ + Äá ) − (Å + Çá ) = (~ − Å ) + (Ä − Ç)á =
(~ + Äá ...
CHAPTER 1. NUMBER SETS

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Figure 7.

55.

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56.

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mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë=
~ + Äá = ê(Åçë ϕ + á ëáå ϕ) =...
CHAPTER 1. NUMBER SETS

60.

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61.
=
62.
=
63.

=
64.

nìçíáÉåí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
ò N êN (Åçë ϕN + á ëáå ϕN ) êN
= ...
Chapter 2

Algebra
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2.1 Factoring Formulas
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oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==
k~íìê~ä=åìãÄÉêW=å=
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65.
=
66.
=
67.
=
68.
...
CHAPTER 2. ALGEBRA

~ å + Äå = (~ + Ä)(~ å−N − ~ å −O Ä + ~ å −P ÄO − K + ~Äå−O − Äå −N ) K=
=
=
=

2.2 Product Formulas

...
CHAPTER 2. ALGEBRA

2.3 Powers
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_~ëÉë=EéçëáíáîÉ=êÉ~ä=åìãÄÉêëFW=~I=Ä==
mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=
=
=
~ ã ~ å = ~ ã+...
CHAPTER 2. ALGEBRA

2.4 Roots
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91.
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_~ëÉëW=~I=Ä==
mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=
~ I Ä ≥ M =Ñçê=ÉîÉå=êççíë=E å = Oâ ...
CHAPTER 2. ALGEBRA

N å ~ å −N
=
I= ~ ≠ M K=
å
~
~

101.

=
~± Ä =

102.

~ + ~O − Ä
~ − ~O − Ä
±
=
O
O

=
N
~m Ä
=
=
~−Ä
...
CHAPTER 2. ALGEBRA

110. äçÖ ~ (ñ å ) = å äçÖ ~ ñ =
=
N
111. äçÖ ~ å ñ = äçÖ ~ ñ =
å
=
äçÖ Å ñ
112. äçÖ ~ ñ =
= äçÖ Å ñ ⋅ ...
CHAPTER 2. ALGEBRA

2.6 Equations
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oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=éI=èI=ìI=î=
pçäìíáçåëW= ñ N I= ñ O I= ó N I= ó O I= ó P =
=
=
1...
CHAPTER 2. ALGEBRA

ó N = ì + î I= ó OI P = −

N
(ì + î ) ± P (ì + î ) á I==
O
O

ïÜÉêÉ==
O

ì=P −

O

O

O

è
è
è  é
...
CHAPTER 2. ALGEBRA

127.
=
128.
=
129.
=
130.
=
131.
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132.
=
133.
=

fÑ= ~ > Ä I=íÜÉå= Ä < ~ K=
fÑ= ~ > Ä I=íÜÉå= ~ − Ä >...
CHAPTER 2. ALGEBRA

142.

å

~N~ O K~ å ≤

~N + ~ O + K + ~ å
I=ïÜÉêÉ= ~N I ~ O I KI ~ å > M K=
å

=
Ä
143. fÑ= ~ñ + Ä > M...
CHAPTER 2. ALGEBRA

~+Ä ≤ ~ + Ä =

146.
=
147.
=
148.
=
149.
=
150.
=

fÑ= ñ < ~ I=íÜÉå= − ~ < ñ < ~ I=ïÜÉêÉ= ~ > M K=
fÑ=...
CHAPTER 2. ALGEBRA

154. páãéäáÑáÉÇ=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=
fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=çåÅÉ=éÉê=óÉ~êI=íÜÉå=íÜÉ=éêÉîáçìë=...
Chapter 3

Geometry
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3.1 Right Triangle
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iÉÖë=çÑ=~=êáÖÜí=íêá~åÖäÉW=~I=Ä=
eóéçíÉåìëÉW=Å=
^äíáíìÇÉW=Ü=
jÉÇá~åëW= ã ...
CHAPTER 3. GEOMETRY

157. ëáå α =

~
= Åçë β =
Å

=
158. Åçë α =

Ä
= ëáå β =
Å
=

159. í~å α =

~
= Åçí β =
Ä
=

Ä
160. Å...
CHAPTER 3. GEOMETRY

165. Ü O = ÑÖ I===
ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK==
=
O
O
~
Ä
166. ã O = ÄO − I= ã O =...
CHAPTER 3. GEOMETRY

171. p =

~Ä ÅÜ
=
=
O
O

=
=
=

3.2 Isosceles Triangle
=
_~ëÉW=~=
iÉÖëW=Ä=
_~ëÉ=~åÖäÉW= β =
sÉêíÉñ=~å...
CHAPTER 3. GEOMETRY

174. i = ~ + OÄ =

=
175. p =

O

~Ü Ä
= ëáå α =
O
O

=
=
=

3.3 Equilateral Triangle
=
páÇÉ=çÑ=~=Éèì...
CHAPTER 3. GEOMETRY

O
~ P
=
177. o = Ü =
P
P
=
N
~ P o
= =
178. ê = Ü =
P
S
O
=

179. i = P~ =

=
180. p =

O

~Ü ~ P
=
=...
CHAPTER 3. GEOMETRY

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Figure 13.

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181. α + β + γ = NUM° =
182. ~ + Ä > Å I==
Ä + Å > ~ I==
~ + Å > Ä K=
=
183...
CHAPTER 3. GEOMETRY

185. i~ï=çÑ=`çëáåÉë=
~ O = ÄO + Å O − OÄÅ Åçë α I=
ÄO = ~ O + Å O − O~Å Åçë β I=
Å O = ~ O + ÄO − O~Ä...
CHAPTER 3. GEOMETRY

191. Ü ~ = Ä ëáå γ = Å ëáå β I=
Ü Ä = ~ ëáå γ = Å ëáå α I=
Ü Å = ~ ëáå β = Ä ëáå α K=

=
Ä +Å ~
− I==...
CHAPTER 3. GEOMETRY

~Ü ~ ÄÜ Ä ÅÜ Å
=
=
I==
O
O
O
~Ä ëáå γ ~Å ëáå β ÄÅ ëáå α
I==
p=
=
=
O
O
O
p = é(é − ~ )(é − Ä)(é − Å )...
CHAPTER 3. GEOMETRY

196. Ç = ~ O ==

=
197. o =

Ç ~ O
=
=
O
O
=

~
198. ê = =
O
199. i = Q~ =

=
=

200. p = ~ =
=
=
=
O...
CHAPTER 3. GEOMETRY

202. o =

Ç
=
O
=

203. i = O(~ + Ä) =

=

204. p = ~Ä =
=
=
=

3.7 Parallelogram
=
páÇÉë=çÑ=~=é~ê~ää...
CHAPTER 3. GEOMETRY

207. Ü = Ä ëáå α = Ä ëáå β =
208. i = O(~ + Ä) =
209. p = ~Ü = ~Ä ëáå α I==
N
p = ÇNÇ O ëáå ϕ K=
O
=
...
CHAPTER 3. GEOMETRY

210. α + β = NUM° =

=
211. Ç + Ç = Q~ =
O
N

O
O

O

=
212. Ü = ~ ëáå α =

ÇNÇ O
=
O~
=

Ü ÇÇ
~ ëáå ...
CHAPTER 3. GEOMETRY

=

=
Figure 20.

=
216. è =
217. p =

~+Ä
=
O
~+Ä
⋅ Ü = èÜ =
O

=

=
=
=

3.10 Isosceles Trapezoid
=
...
CHAPTER 3. GEOMETRY

=

=
Figure 21.

=
218. è =

~+Ä
=
O
=

219. Ç = ~Ä + Å =
=
N
O
220. Ü = Å O − (Ä − ~ ) =
Q
O

=
Å ~Ä...
CHAPTER 3. GEOMETRY

3.11 Isosceles Trapezoid with
Inscribed Circle
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
iÉÖW=Å=
jáÇäáåÉW=è=
^äíá...
CHAPTER 3. GEOMETRY

226. ê =

Ü
~Ä
=
=
O
O
=

Ä
ÅÇ ÅÇ Å
Å
Å
~+Ä ~
N+
ÜO + Å O =
=
=
=
+S+ =
OÜ Qê O
~Ä OÜ
U
Ä
~
=
228. i ...
CHAPTER 3. GEOMETRY

=

=
Figure 23.

=
230. ~ + Ä = Å + Ç =
~+Ä Å+Ç
=
=
231. è =
O
O
232. i = O(~ + Ä) = O(Å + Ç ) =

=

...
CHAPTER 3. GEOMETRY

=

=
Figure 24.

=

234. α + β + Oγ = PSM° =
235. i = O(~ + Ä) =

=
=

236. p =

ÇNÇ O
=
O

=
=
=

3....
CHAPTER 3. GEOMETRY

=

=
Figure 25.

=

237. α + γ = β + δ = NUM° =

=
238. míçäÉãó∞ë=qÜÉçêÉã=
~Å + ÄÇ = ÇNÇ O =
239. i =...
CHAPTER 3. GEOMETRY

3.15 Tangential Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= ÇN I Ç O =
^åÖäÉ=ÄÉ...
CHAPTER 3. GEOMETRY

N
245. p = éê = ÇNÇ O ëáå ϕ =
O
=
=
=

3.16 General Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=Å...
CHAPTER 3. GEOMETRY

N
248. p = ÇNÇ O ëáå ϕ =
O
=
=
=

3.17 Regular Hexagon
=
páÇÉW=~=
fåíÉêå~ä=~åÖäÉW= α =
pä~åí=ÜÉáÖÜíW=...
CHAPTER 3. GEOMETRY

251. o = ~ =

=

252. i = S~ =

=
O

~ P P
I==
O
i
ïÜÉêÉ= é = K=
O
=
=
=

253. p = éê =

3.18 Regular...
CHAPTER 3. GEOMETRY

=

=
Figure 29.

=
254. α =

255. α =

å−O
⋅ NUM° =
O
=

å−O
⋅ NUM° =
O

=
256. o =

~

π
O ëáå
å

=
...
CHAPTER 3. GEOMETRY

ïÜÉêÉ= é =

i
K==
O

=
=
=

3.19 Circle
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
`ÜçêÇW=~=
pÉÅ~åí=ëÉÖãÉåíëW=ÉI=Ñ=
q~...
CHAPTER 3. GEOMETRY

261. ~N~ O = ÄNÄO =
=

=

=
Figure 31.

=

262. ÉÉN = ÑÑN =
=

=

=

=====
Figure 32.

=
263. Ö O = Ñ...
CHAPTER 3. GEOMETRY

=

=====

=
Figure 33.

=
264. β =

α
=
O

=

=

=
Figure 34.

=
265. i = Oπo = πÇ =
=
266. p = πo O ...
CHAPTER 3. GEOMETRY

3.20 Sector of a Circle
=
o~Çáìë=çÑ=~=ÅáêÅäÉW=o=
^êÅ=äÉåÖíÜW=ë=
`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=
`Éåíê~...
CHAPTER 3. GEOMETRY

3.21 Segment of a Circle
=
o~Çáìë=çÑ=~=ÅáêÅäÉW=o=
^êÅ=äÉåÖíÜW=ë=
`ÜçêÇW=~=
`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëF...
CHAPTER 3. GEOMETRY

O
O
N
[ëo − ~(o − Ü )] = o  απ − ëáå α  = o (ñ − ëáå ñ ) I==


O
O  NUM°
 O
O
p ≈ Ü~ K=
P

274....
CHAPTER 3. GEOMETRY

277. o =

~ P
=
O
=

278. p = S~ =
O

=
279. s = ~ ==
=
=
=
P

3.23 Rectangular Parallelepiped
=
bÇÖÉ...
CHAPTER 3. GEOMETRY

3.24 Prism
=
i~íÉê~ä=ÉÇÖÉW=ä=
eÉáÖÜíW=Ü=
i~íÉê~ä=~êÉ~W= p i =
^êÉ~=çÑ=Ä~ëÉW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W...
CHAPTER 3. GEOMETRY

286. s = p_ Ü =

=
287. `~î~äáÉêáDë=mêáåÅáéäÉ==
dáîÉå=íïç=ëçäáÇë=áåÅäìÇÉÇ=ÄÉíïÉÉå=é~ê~ääÉä=éä~åÉëK=fÑ...
CHAPTER 3. GEOMETRY

289. p_ =

P~ O
=
Q
=

290. p = P~ =
=
N
~P
291. s = p_ Ü =
K==
P
S O
=
=
=
O

3.26 Regular Pyramid
=...
CHAPTER 3. GEOMETRY

=

=
Figure 41.

=
292. ã = ÄO −

~O
=
Q
=

293. Ü =

π O
−~
å
=
π
O ëáå
å

QÄO ëáå O

=
N
N
294. p i...
CHAPTER 3. GEOMETRY

3.27 Frustum of a Regular Pyramid
=
~N I ~ O I ~ P IKI ~ å
=
_~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW= 
ÄN I ÄO ...
CHAPTER 3. GEOMETRY

299.

pO
= âO =
pN
=

ã(mN + mO )
=
300. p i =
O

=

301. p = p i + pN + pO =

=
Ü
302. s = pN + pNpO...
CHAPTER 3. GEOMETRY

=

=
Figure 43.

=
N
(~ + Å ) QÜO + ÄO + Ä ÜO + (~ − Å )O =
O
=
305. p_ = ~Ä =
=
306. p = p_ + p i =
...
CHAPTER 3. GEOMETRY

308. cáîÉ=mä~íçåáÅ=pçäáÇë=
qÜÉ= éä~íçåáÅ= ëçäáÇë= ~êÉ= ÅçåîÉñ= éçäóÜÉÇê~= ïáíÜ= Éèìáî~äÉåí=
Ñ~ÅÉë=Åçã...
CHAPTER 3. GEOMETRY

311. p = O~ O P =
=
~P O
312. s =
=
P
=
=

Icosahedron
=

=

=
Figure 45.

=
313. ê =

(

=
314. o =
...
CHAPTER 3. GEOMETRY

Dodecahedron
=

=

=
Figure 46.

317. ê =

(

~ NM OR + NN R
=
O

=
318. o =

)

=

(

)

~ P N+ R
=
...
CHAPTER 3. GEOMETRY

eÉáÖÜíW=e=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i =
^êÉ~=çÑ=Ä~ëÉW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=

=
...
CHAPTER 3. GEOMETRY

3.31 Right Circular Cylinder with
an Oblique Plane Face
=
o~Çáìë=çÑ=Ä~ëÉW=o=
qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~...
CHAPTER 3. GEOMETRY

O

 ÜN − Ü O  
O
326. p = p i + p_ = πo ÜN + Ü O + o + o + 
 =
 O  



=
πo O
(ÜN + ÜO ...
CHAPTER 3. GEOMETRY

328. e = ã O − o O =
=
πãÇ
329. p i = πoã =
=
O
=
330. p_ = πo O =
=

N 
Ç
331. p = p i + p_ = πo (...
CHAPTER 3. GEOMETRY

=

=
Figure 50.

=
333. e = ã O − (o − ê ) =
=
o
334.
=â=
ê
=
p oO
335. O = O = â O =
pN ê
=
336. p i...
CHAPTER 3. GEOMETRY

3.34 Sphere
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=

=

=
Figure 51.

=
340. p = Qπo ...
CHAPTER 3. GEOMETRY

=

=
Figure 52.

=
342. o =

ê O + ÜO
=
OÜ

=
343. p_ = πê O =
=
344. p` = π(Ü O + ê O )=
=
345. p = ...
CHAPTER 3. GEOMETRY

======

=

===

=

Figure 53.

=
347. p = πo(OÜ + ê ) =
=
O
348. s = πo O Ü =
P
=
kçíÉW= qÜÉ= ÖáîÉå= ...
CHAPTER 3. GEOMETRY

=

=====

=
Figure 54.

=
349. pp = OπoÜ =
=
350. p = pp + pN + pO = π(OoÜ + êNO + êOO ) =
=
N
351. s...
CHAPTER 3. GEOMETRY

=

=
Figure 55.

=
352. p i =

πo O
α = Oo O ñ =
VM

=
353. p = πo O +

πo O
α = πo O + Oo O ñ =
VM

...
CHAPTER 3. GEOMETRY

=

=======

=
Figure 56.

=
Q
355. s = π~ÄÅ =
P
=
=
=

Prolate Spheroid
=
pÉãá-~ñÉëW=~I=ÄI=Ä=E ~ > Ä ...
CHAPTER 3. GEOMETRY

Oblate Spheroid
=
pÉãá-~ñÉëW=~I=ÄI=Ä=E ~ < Ä F=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=

 ÄÉ  
~ ~êÅëáåÜ ...
CHAPTER 3. GEOMETRY

==

Picture 57.

=
360. p = QπOoê =
=
361. s = OπOoê O =
=
=

79

=
Chapter 4

Trigonometry
=
=
=
=
^åÖäÉëW= α I= β =
oÉ~ä=åìãÄÉêë=EÅççêÇáå~íÉë=çÑ=~=éçáåíFW=ñI=ó==
tÜçäÉ=åìãÄÉêW=â=
=
=

4.1 ...
CHAPTER 4. TRIGONOMETRY

4.2 Definitions and Graphs of Trigonometric
Functions
=

=

=

=
Figure 58.

=
367. ëáå α =

ó
=
...
CHAPTER 4. TRIGONOMETRY

371. ëÉÅ α =

ê
=
ñ

=
372. ÅçëÉÅ α =

ê
=
ó
=

373. páåÉ=cìåÅíáçå=
ó = ëáå ñ I= − N ≤ ëáå ñ ≤ N ...
CHAPTER 4. TRIGONOMETRY

=

=
Figure 60.

=
375. q~åÖÉåí=cìåÅíáçå=

π
ó = í~å ñ I= ñ ≠ (Oâ + N) I= − ∞ ≤ í~å ñ ≤ ∞K =
O
=
...
CHAPTER 4. TRIGONOMETRY

376. `çí~åÖÉåí=cìåÅíáçå==
ó = Åçí ñ I= ñ ≠ âπ I== − ∞ ≤ Åçí ñ ≤ ∞ K=
=

=

=
Figure 62.

=
377. p...
CHAPTER 4. TRIGONOMETRY

=

=
Figure 63.

=
378. `çëÉÅ~åí=cìåÅíáçå==
ó = Åçë ÉÅ ñ I= ñ ≠ âπ K=

=
Figure 64.

85
CHAPTER 4. TRIGONOMETRY

4.3. Signs of Trigonometric Functions
379. =
=

=
=
380. =

nì~Çê~åí=

=

f=
ff=
fff=
fs=

páå
α=...
CHAPTER 4. TRIGONOMETRY

4.4 Trigonometric Functions of Common
Angles
381. =
α° = α ê~Ç =
M=
M=
π
=
PM=
S
π
=
QR=
Q
π
=
SM...
CHAPTER 4. TRIGONOMETRY

382. =
α° = α ê~Ç =
π
=
NR=
NO

ëáå α =

Åçë α =

í~å α =

Åçí α =

S− O
=
Q

S+ O
=
Q

O− P =

O...
CHAPTER 4. TRIGONOMETRY

387. Åçí α =

Åçë α
=
ëáå α

=
388. í~å α ⋅ Åçí α = N =
=
N
389. ëÉÅ α =
=
Åçë α
=
N
390. ÅçëÉÅ α...
CHAPTER 4. TRIGONOMETRY

4.7 Periodicity of Trigonometric Functions
=

392. ëáå(α ± Oπå ) = ëáå α I=éÉêáçÇ= Oπ =çê= PSM° K...
CHAPTER 4. TRIGONOMETRY

α
N − Åçë Oα
O =
=±
=
N + Åçë Oα
O α
N + í~å
O
O í~å

=
=

Åçë α
N + Åçë Oα
ëáå Oα
= ± ÅëÅ O α − ...
CHAPTER 4. TRIGONOMETRY

406. í~å(α + β ) =

=
407. í~å(α − β ) =

=
408. Åçí(α + β) =

=
409. Åçí(α − β) =

í~å α + í~å β...
CHAPTER 4. TRIGONOMETRY

4.11 Multiple Angle Formulas
=
414. ëáå Pα = P ëáå α − Q ëáå P α = P Åçë O α ⋅ ëáå α − ëáåP α =
=...
CHAPTER 4. TRIGONOMETRY

425. Åçí Rα =

N − NM í~å O α + R í~å Q α
=
í~å R α − NM í~å P α + R í~å α

=
=
=

4.12 Half Angl...
CHAPTER 4. TRIGONOMETRY

α
O=
431. Åçë α =
O α
N + í~å
O
=
α
O í~å
O =
432. í~å α =
α
N − í~å O
O
=
α
N − í~å O
O=
433. Åç...
CHAPTER 4. TRIGONOMETRY

438. í~å α + í~å β =

=
439. í~å α − í~å β =

=
440. Åçí α + Åçí β =

=
441. Åçí α − Åçí β =

ëáå...
CHAPTER 4. TRIGONOMETRY

π α
448. N + ëáå α = O Åçë O  −  =
Q O
=
π α
449. N − ëáå α = O ëáå O  −  =
Q O
=
=
=...
CHAPTER 4. TRIGONOMETRY

4.16 Powers of Trigonometric Functions
=
456. ëáå O α =

=
457. ëáå P α =

=
458. ëáå Q α =

=
45...
CHAPTER 4. TRIGONOMETRY

4.17 Graphs of Inverse Trigonometric
Functions
=
466. fåîÉêëÉ=páåÉ=cìåÅíáçå==
ó = ~êÅëáå ñ I= − N...
CHAPTER 4. TRIGONOMETRY

=

=
Figure 67.

=
468. fåîÉêëÉ=q~åÖÉåí=cìåÅíáçå==
ó = ~êÅí~å ñ I= − ∞ ≤ ñ ≤ ∞ I= −

π
π
< ~êÅí~å...
CHAPTER 4. TRIGONOMETRY

469. fåîÉêëÉ=`çí~åÖÉåí=cìåÅíáçå==
ó = ~êÅ Åçí ñ I= − ∞ ≤ ñ ≤ ∞ I= M < ~êÅ Åçí ñ < π K=

=====

=
...
CHAPTER 4. TRIGONOMETRY

471. fåîÉêëÉ=`çëÉÅ~åí=cìåÅíáçå==

 π   π
ó = ~êÅÅëÅ ñ I ñ ∈ (− ∞I − N] ∪ [NI ∞ )I ~êÅ ÅëÅ ñ ∈...
CHAPTER 4. TRIGONOMETRY

473.

ñ=

M=

P
P

N=

~êÅí~å ñ =

M° =

PM°

QR°

SM°

~êÅ Åçí ñ = VM°

SM°

QR°

PM°

P= −

P
P...
CHAPTER 4. TRIGONOMETRY

482. ~êÅÅçë ñ =

π
− ~êÅëáå ñ =
O

=
483. ~êÅÅçë ñ = ~êÅëáå N − ñ O I= M ≤ ñ ≤ N K=
=
484. ~êÅÅçë...
CHAPTER 4. TRIGONOMETRY

493. ~êÅí~å ñ =

π
N
− ~êÅí~å I= ñ > M K=
O
ñ

=

π
N
494. ~êÅí~å ñ = − − ~êÅí~å I= ñ < M K=
O
ñ
...
CHAPTER 4. TRIGONOMETRY

4.20 Trigonometric Equations

504.
505.
506.
507.

=
tÜçäÉ=åìãÄÉêW=å=
=
=
å
ëáå ñ = ~ I= ñ = (− N...
Chapter 5

Matrices and Determinants
=
=
=
=
j~íêáÅÉëW=^I=_I=`=
bäÉãÉåíë=çÑ=~=ã~íêáñW= ~ á I= Äá I= ~ áà I= Äáà I= Å áà =
...
CHAPTER 5. MATRICES AND DETERMINANTS

514. qÜáêÇ=lêÇÉê=aÉíÉêãáå~åí=
~NN ~NO ~NP
ÇÉí ^ = ~ ON ~ OO

~ OP = ~NN~ OO~ PP + ~N...
CHAPTER 5. MATRICES AND DETERMINANTS

518. `çÑ~Åíçê=
á +à
` áà = (− N) j áà =

=
519. i~éä~ÅÉ=bñé~åëáçå=çÑ=å-íÜ=lêÇÉê=aÉíÉ...
CHAPTER 5. MATRICES AND DETERMINANTS

523. fÑ==íÜÉ===ÉäÉãÉåíë==çÑ==~åó=êçï==Eçê=ÅçäìãåF=~êÉ=ãìäíáéäáÉÇ=Äó=====
~==Åçããçå==...
CHAPTER 5. MATRICES AND DETERMINANTS

529. aá~Öçå~ä=ã~íêáñ==áë==~=ëèì~êÉ==ã~íêáñ=ïáíÜ=~ää==ÉäÉãÉåíë==òÉêç=
ÉñÅÉéí=íÜçëÉ=çå...
CHAPTER 5. MATRICES AND DETERMINANTS

íÜÉå==

~NO + ÄNO K ~Nå + ÄNå 
 ~NN + ÄNN
~ +Ä
~ OO + ÄOO K ~ Oå + ÄOå 
 K=
 O...
CHAPTER 5. MATRICES AND DETERMINANTS

íÜÉå==
 ÅNN ÅNO K ÅNâ 
Å
Å OO K Å O â 
 I==
 ON
^_ = ` =
 M
M
M 


Ä ãN Å...
CHAPTER 5. MATRICES AND DETERMINANTS

539. ^Çàçáåí=çÑ=j~íêáñ=
fÑ=^=áë=~=ëèì~êÉ= å× å ã~íêáñI=áíë=~ÇàçáåíI=ÇÉåçíÉÇ=Äó= ~Çà ...
CHAPTER 5. MATRICES AND DETERMINANTS

aÉíÉêãáå~åíëW=aI= añ I= aó I= aò ==
j~íêáÅÉëW=^I=_I=u=
=
=

~ ñ + ÄNó = ÇN
I==
544....
CHAPTER 5. MATRICES AND DETERMINANTS

ïÜÉêÉ==
~N ÄN
a = ~ O ÄO
~ P ÄP

ÅN

ÇN

ÄN

ÅN

Å O I= añ = Ç O

ÄO

Å O I=

ÅP

ÄP...
CHAPTER 5. MATRICES AND DETERMINANTS

ïÜÉêÉ==
 ~ NN

~
^ =  ON
M

~
 åN

~ NO K ~ Nå 
 ñN 
 ÄN 

 
 
~ OO...
Chapter 6

Vectors
=
=
=
=
r r r r →
sÉÅíçêëW= ì I= î I= ï I= ê I= ^_ I=£=
r r
sÉÅíçê=äÉåÖíÜW= ì I= î I=£=
r r r
råáí=îÉÅí...
CHAPTER 6. VECTORS

=======
=

=
Figure 73.

=
→
r
ê = ^_ =

552.

(ñ N − ñ M )O + (óN − ó M )O + (òN − ò M )O =

=
→
→
r
...
CHAPTER 6. VECTORS

=

=====

=
Figure 75.

=
r
r
555. fÑ= ê (uI v I w ) = êN (uN I vN I wN ) I=íÜÉå==
u = uN I= v = vN I=...
CHAPTER 6. VECTORS

=

==

=
Figure 77.

=

r
r r r r
557. ï = ìN + ì O + ìP + K + ì å =
=

=

=

==
Figure 78.

=
558. `ç...
CHAPTER 6. VECTORS

6.3 Vector Subtraction
=

r r r r r r
561. ï = ì − î =áÑ= î + ï = ì K=
=

=

=
Figure 79.

=

=

==

=...
CHAPTER 6. VECTORS

=

=
Figure 81.

=
567.

r
r
ï = λ⋅ì=

=

r
568. λì = (λuI λv I λw ) =
=
r r
569. λì = ìλ =
=
r
r
r
57...
CHAPTER 6. VECTORS

=

=

=
Figure 82.

=
574. pÅ~ä~ê=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=
r
r
fÑ= ì = (uN I vN I wN ) I= î = (u O ...
CHAPTER 6. VECTORS

π
r r
581. ì ⋅ î < M =áÑ= < θ < π K=
O
=
r r r r
582. ì ⋅ î ≤ ì ⋅ î =
=
r r r r
r r
583. ì ⋅ î = ì ⋅ î...
CHAPTER 6. VECTORS

=

=======

=
Figure 83.

=

r
á
r r r
588. ï = ì × î = u N
uO

r
à
vN
vO

r
â
wN =
wO

=
uN wN uN vN ...
CHAPTER 6. VECTORS

594. aáëíêáÄìíáîÉ=mêçéÉêíó=
r r r r r r r
ì × (î + ï ) = ì × î + ì × ï =
=
r r r r
r
595. ì × î = M =á...
CHAPTER 6. VECTORS

=

============

=
Figure 84.

=
603. sçäìãÉ=çÑ=móê~ãáÇ=
Nr r r
s = ì ⋅ (î × ï ) =
S
=

=

=
Figure 85...
CHAPTER 6. VECTORS

606. sÉÅíçê=qêáéäÉ=mêçÇìÅí=
r r r
r r r r r r
ì × (î × ï ) = (ì ⋅ ï )î − (ì ⋅ î )ï ==
=
=
=
=
=
=
=
=
...
Chapter 7

Analytic Geometry
=
=
=
=

7.1 One-Dimensional Coordinate System
=
mçáåí=ÅççêÇáå~íÉëW= ñ M I= ñ N I= ñ O I= ó M...
CHAPTER 7. ANALYTIC GEOMETRY

609. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
ñ + ñO
ñM = N
I= λ = N K=
O
=
=
=

7.2 Two-Dimensional Coor...
CHAPTER 7. ANALYTIC GEOMETRY

611. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ =
ñ + λñ O
ó + λó O
ñM = N
I= ó M = N
I==
N+ λ
...
CHAPTER 7. ANALYTIC GEOMETRY

=======
=

=
Figure 90.

=
612. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
ñ + ñO
ó + óO
I= ó M = N
I= λ = ...
CHAPTER 7. ANALYTIC GEOMETRY

=========
=

=
Figure 91.

=
614. fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
...
CHAPTER 7. ANALYTIC GEOMETRY

615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê======================
_áëÉÅíçêëF=ç...
CHAPTER 7. ANALYTIC GEOMETRY

616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ=
O
O
óN ñ O ñ P + óN N
ñN + ó OóP...
CHAPTER 7. ANALYTIC GEOMETRY

618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä=
N
p = (± ) [(ñ N − ñ O )(ó N + ó O ) + (ñ O − ñ P )(ó O + ó P ...
CHAPTER 7. ANALYTIC GEOMETRY

=

=
Figure 96.

=
620. `çåîÉêíáåÖ=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=íç=mçä~ê=`ççêÇáå~íÉë=
ñ = ê Åçë ϕ...
CHAPTER 7. ANALYTIC GEOMETRY

7.3 Straight Line in Plane
=
mçáåí=ÅççêÇáå~íÉëW=uI=vI=ñI= ñ M I= ñ N I== ó M I= ó N I= ~N I=...
CHAPTER 7. ANALYTIC GEOMETRY

qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= â = í~å α K=
=

=

=
Figure 99.

=
625. dê~ÇáÉåí=çÑ=~=iáåÉ==
ó ...
CHAPTER 7. ANALYTIC GEOMETRY

626. bèì~íáçå=çÑ=~=iáåÉ=dáîÉå=~=mçáåí=~åÇ=íÜÉ=dê~ÇáÉåí=
ó = ó M + â (ñ − ñ M ) I==
ïÜÉêÉ=â=á...
CHAPTER 7. ANALYTIC GEOMETRY

=

=
Figure 102.

=
628. fåíÉêÅÉéí=cçêã=
ñ ó
+ =N=
~ Ä
=

=

=
Figure 103.

=
=

142
CHAPTER 7. ANALYTIC GEOMETRY

629. kçêã~ä=cçêã=
ñ Åçë β + ó ëáå β − é = M =
=

=

=
Figure 104.

=
630. mçáåí=aáêÉÅíáçå=cç...
CHAPTER 7. ANALYTIC GEOMETRY

=

=
Figure 105.

=
631. sÉêíáÅ~ä=iáåÉ=
ñ =~=
=
632. eçêáòçåí~ä=iáåÉ=
ó=Ä=
=
633. sÉÅíçê=bèì...
CHAPTER 7. ANALYTIC GEOMETRY

=

=
Figure 106.

=
634. píê~áÖÜí=iáåÉ=áå=m~ê~ãÉíêáÅ=cçêã=
ñ = ~N + íÄN
I==

ó = ~ O + íÄ...
CHAPTER 7. ANALYTIC GEOMETRY

=

Figure 107.

=
635. aáëí~åÅÉ=cêçã=~=mçáåí=qç=~=iáåÉ=
qÜÉ=Çáëí~åÅÉ=Ñêçã=íÜÉ=éçáåí= m(~ I Ä...
CHAPTER 7. ANALYTIC GEOMETRY

636. m~ê~ääÉä=iáåÉë=
qïç=äáåÉë= ó = â Nñ + ÄN =~åÇ= ó = â O ñ + ÄO =~êÉ=é~ê~ääÉä=áÑ==
â N = ...
CHAPTER 7. ANALYTIC GEOMETRY

=

=
Figure 110.

=
638. ^åÖäÉ=_ÉíïÉÉå=qïç=iáåÉë=
â − âN
í~å ϕ = O
I==
N + â Nâ O
^N^ O + _N...
CHAPTER 7. ANALYTIC GEOMETRY

=

=
Figure 111.

=
639. fåíÉêëÉÅíáçå=çÑ=qïç=iáåÉë=
fÑ=íïç=äáåÉë= ^Nñ + _Nó + `N = M =~åÇ= ^...
CHAPTER 7. ANALYTIC GEOMETRY

640. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ=
cçêãF=
ñ O + ó O = oO =

======
...
CHAPTER 7. ANALYTIC GEOMETRY

642. qÜêÉÉ=mçáåí=cçêã
ñO + óO ñ ó N
O
O
ñN + óN ñN óN N
=M
ñO + óO ñO óO N
O
O
O
O
ñP + óP ñ...
CHAPTER 7. ANALYTIC GEOMETRY

o=

aO + b O − Q ^c
K
O^

=
=
=

7.5 Ellipse
=
pÉãáã~àçê=~ñáëW=~=
pÉãáãáåçê=~ñáëW=Ä=
cçÅáW= ...
CHAPTER 7. ANALYTIC GEOMETRY

646. êN + êO = O~ I=
ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó==éçáåí== m(ñ I ó ) ==çå=
...
CHAPTER 7. ANALYTIC GEOMETRY

651. dÉåÉê~ä=cçêã
^ñ O + _ñó + `ó O + añ + bó + c = M I==
ïÜÉêÉ= _ O − Q ^` < M K=
=
652. dÉ...
CHAPTER 7. ANALYTIC GEOMETRY

656. bèì~íáçå=çÑ=~=eóéÉêÄçä~=Epí~åÇ~êÇ=cçêãF=
ñO óO
− = N=
~ O ÄO
=

=

=
Figure 117.

=

65...
CHAPTER 7. ANALYTIC GEOMETRY

=

=
Figure 118.

658.

659.
660.

661.

=
bèì~íáçåë=çÑ=^ëóãéíçíÉë=
Ä
ó=± ñ=
~
=
Å O = ~ O +...
CHAPTER 7. ANALYTIC GEOMETRY

662. m~ê~ãÉíêáÅ=bèì~íáçåë=çÑ=íÜÉ=oáÖÜí=_ê~åÅÜ=çÑ=~=eóéÉêÄçä~=
ñ = ~ ÅçëÜ í
I= M ≤ í ≤ Oπ K
...
CHAPTER 7. ANALYTIC GEOMETRY

=

=
Figure 119.

=
=
=

7.7 Parabola
=
cçÅ~ä=é~ê~ãÉíÉêW=é=
cçÅìëW=c=
sÉêíÉñW= j(ñ M I ó M )...
CHAPTER 7. ANALYTIC GEOMETRY

=

=
Figure 120.

=
bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ
é
ñ = − I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=
é 
c...
CHAPTER 7. ANALYTIC GEOMETRY

é
ó = − I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=
 é
c MI  I=
 O
`ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ=
j(MI ...
CHAPTER 7. ANALYTIC GEOMETRY

é

c ñ M I ó M +  I=
O

`ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ=
Ä
Q~Å − ÄO
K=
ñ M = − I= ó M = ~ñ O...
CHAPTER 7. ANALYTIC GEOMETRY

670. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=

Ç = ^_ =
=

=

(ñ O − ñ N )O + (ó O − óN )O + (ò O − òN )...
CHAPTER 7. ANALYTIC GEOMETRY

========
=

=
Figure 124.

=

=
Figure 125.

163
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1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
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1300 math formulas
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1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
1300 math formulas
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1300 math formulas

  1. 1. 1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë=oÉëÉêîÉÇK=
  2. 2. = qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK= i
  3. 3. Preface = = = = qÜáë= Ü~åÇÄççâ= áë= ~= ÅçãéäÉíÉ= ÇÉëâíçé= êÉÑÉêÉåÅÉ= Ñçê= ëíìÇÉåíë= ~åÇ= ÉåÖáåÉÉêëK= fí= Ü~ë= ÉîÉêóíÜáåÖ= Ñêçã= ÜáÖÜ= ëÅÜççä= ã~íÜ=íç=ã~íÜ=Ñçê=~Çî~åÅÉÇ=ìåÇÉêÖê~Çì~íÉë=áå=ÉåÖáåÉÉêáåÖI= ÉÅçåçãáÅëI=éÜóëáÅ~ä=ëÅáÉåÅÉëI=~åÇ=ã~íÜÉã~íáÅëK=qÜÉ=ÉÄççâ= Åçåí~áåë= ÜìåÇêÉÇë= çÑ= Ñçêãìä~ëI= í~ÄäÉëI= ~åÇ= ÑáÖìêÉë= Ñêçã= kìãÄÉê= pÉíëI= ^äÖÉÄê~I= dÉçãÉíêóI= qêáÖçåçãÉíêóI= j~íêáÅÉë= ~åÇ= aÉíÉêãáå~åíëI= sÉÅíçêëI= ^å~äóíáÅ= dÉçãÉíêóI= `~äÅìäìëI= aáÑÑÉêÉåíá~ä=bèì~íáçåëI=pÉêáÉëI=~åÇ=mêçÄ~Äáäáíó=qÜÉçêóK== qÜÉ= ëíêìÅíìêÉÇ= í~ÄäÉ= çÑ= ÅçåíÉåíëI= äáåâëI= ~åÇ= ä~óçìí= ã~âÉ= ÑáåÇáåÖ= íÜÉ= êÉäÉî~åí= áåÑçêã~íáçå= èìáÅâ= ~åÇ= é~áåäÉëëI= ëç= áí= Å~å=ÄÉ=ìëÉÇ=~ë=~å=ÉîÉêóÇ~ó=çåäáåÉ=êÉÑÉêÉåÅÉ=ÖìáÇÉK=== = = ii
  4. 4. Contents = = = = 1 krj_bo=pbqp= NKN= pÉí=fÇÉåíáíáÉë==1= NKO= pÉíë=çÑ=kìãÄÉêë==5= NKP= _~ëáÅ=fÇÉåíáíáÉë==7= NKQ= `çãéäÉñ=kìãÄÉêë==8= = 2 ^idb_o^= OKN= c~ÅíçêáåÖ=cçêãìä~ë==12= OKO= mêçÇìÅí=cçêãìä~ë==13= OKP= mçïÉêë==14= OKQ= oççíë==15= OKR= içÖ~êáíÜãë==16= OKS= bèì~íáçåë==18= OKT= fåÉèì~äáíáÉë==19= OKU= `çãéçìåÇ=fåíÉêÉëí=cçêãìä~ë==22= = 3 dbljbqov= PKN= oáÖÜí=qêá~åÖäÉ==24= PKO= fëçëÅÉäÉë=qêá~åÖäÉ==27= PKP= bèìáä~íÉê~ä=qêá~åÖäÉ==28= PKQ= pÅ~äÉåÉ=qêá~åÖäÉ==29= PKR= pèì~êÉ==33= PKS= oÉÅí~åÖäÉ==34= PKT= m~ê~ääÉäçÖê~ã==35= PKU= oÜçãÄìë==36= PKV= qê~éÉòçáÇ==37= PKNM= fëçëÅÉäÉë=qê~éÉòçáÇ==38= PKNN= fëçëÅÉäÉë=qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==40= PKNO= qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==41= iii
  5. 5. PKNP= háíÉ==42= PKNQ= `óÅäáÅ=nì~Çêáä~íÉê~ä==43= PKNR= q~åÖÉåíá~ä=nì~Çêáä~íÉê~ä==45= PKNS= dÉåÉê~ä=nì~Çêáä~íÉê~ä==46= PKNT= oÉÖìä~ê=eÉñ~Öçå==47= PKNU= oÉÖìä~ê=mçäóÖçå==48= PKNV= `áêÅäÉ==50= PKOM= pÉÅíçê=çÑ=~=`áêÅäÉ==53= PKON= pÉÖãÉåí=çÑ=~=`áêÅäÉ==54= PKOO= `ìÄÉ==55= PKOP= oÉÅí~åÖìä~ê=m~ê~ääÉäÉéáéÉÇ==56= PKOQ= mêáëã==57= PKOR= oÉÖìä~ê=qÉíê~ÜÉÇêçå==58= PKOS= oÉÖìä~ê=móê~ãáÇ==59= PKOT= cêìëíìã=çÑ=~=oÉÖìä~ê=móê~ãáÇ==61= PKOU= oÉÅí~åÖìä~ê=oáÖÜí=tÉÇÖÉ==62= PKOV= mä~íçåáÅ=pçäáÇë==63= PKPM= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê==66= PKPN= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê=ïáíÜ=~å=lÄäáèìÉ=mä~åÉ=c~ÅÉ==68= PKPO= oáÖÜí=`áêÅìä~ê=`çåÉ==69= PKPP= cêìëíìã=çÑ=~=oáÖÜí=`áêÅìä~ê=`çåÉ==70= PKPQ= péÜÉêÉ==72= PKPR= péÜÉêáÅ~ä=`~é==72= PKPS= péÜÉêáÅ~ä=pÉÅíçê==73= PKPT= péÜÉêáÅ~ä=pÉÖãÉåí==74= PKPU= péÜÉêáÅ~ä=tÉÇÖÉ==75= PKPV= bääáéëçáÇ==76= PKQM= `áêÅìä~ê=qçêìë==78= = = 4 qofdlkljbqov= QKN= o~Çá~å=~åÇ=aÉÖêÉÉ=jÉ~ëìêÉë=çÑ=^åÖäÉë==80= QKO= aÉÑáåáíáçåë=~åÇ=dê~éÜë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==81= QKP= páÖåë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==86= QKQ= qêáÖçåçãÉíêáÅ=cìåÅíáçåë=çÑ=`çããçå=^åÖäÉë==87= QKR= jçëí=fãéçêí~åí=cçêãìä~ë==88= iv
  6. 6. QKS= oÉÇìÅíáçå=cçêãìä~ë==89= QKT= mÉêáçÇáÅáíó=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90= QKU= oÉä~íáçåë=ÄÉíïÉÉå=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90= QKV= ^ÇÇáíáçå=~åÇ=pìÄíê~Åíáçå=cçêãìä~ë==91= QKNM= açìÄäÉ=^åÖäÉ=cçêãìä~ë==92= QKNN= jìäíáéäÉ=^åÖäÉ=cçêãìä~ë==93= QKNO= e~äÑ=^åÖäÉ=cçêãìä~ë==94= QKNP= e~äÑ=^åÖäÉ=q~åÖÉåí=fÇÉåíáíáÉë==94= QKNQ= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=mêçÇìÅí==95= QKNR= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=pìã==97=== QKNS= mçïÉêë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==98= QKNT= dê~éÜë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==99= QKNU= mêáåÅáé~ä=s~äìÉë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==102= QKNV= oÉä~íáçåë=ÄÉíïÉÉå=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==103= QKOM= qêáÖçåçãÉíêáÅ=bèì~íáçåë==106= QKON= oÉä~íáçåë=íç=eóéÉêÄçäáÅ=cìåÅíáçåë==106= = = 5 j^qof`bp=^ka=abqbojfk^kqp= RKN= aÉíÉêãáå~åíë==107= RKO= mêçéÉêíáÉë=çÑ=aÉíÉêãáå~åíë==109= RKP= j~íêáÅÉë==110= RKQ= léÉê~íáçåë=ïáíÜ=j~íêáÅÉë==111= RKR= póëíÉãë=çÑ=iáåÉ~ê=bèì~íáçåë==114= = = 6 sb`qlop= SKN= sÉÅíçê=`ççêÇáå~íÉë==118= SKO= sÉÅíçê=^ÇÇáíáçå==120= SKP= sÉÅíçê=pìÄíê~Åíáçå==122= SKQ= pÅ~äáåÖ=sÉÅíçêë==122= SKR= pÅ~ä~ê=mêçÇìÅí==123= SKS= sÉÅíçê=mêçÇìÅí==125= SKT= qêáéäÉ=mêçÇìÅí=127= = = 7 ^k^ivqf`=dbljbqov= TKN= låÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==130= v
  7. 7. TKO= qïç=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==131= TKP= píê~áÖÜí=iáåÉ=áå=mä~åÉ==139= TKQ= `áêÅäÉ==149= TKR= bääáéëÉ==152= TKS= eóéÉêÄçä~==154= TKT= m~ê~Äçä~==158= TKU= qÜêÉÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==161= TKV= mä~åÉ==165= TKNM= píê~áÖÜí=iáåÉ=áå=pé~ÅÉ==175= TKNN= nì~ÇêáÅ=pìêÑ~ÅÉë==180= TKNO= péÜÉêÉ==189= = = 8 afccbobkqf^i=`^i`rirp= UKN= cìåÅíáçåë=~åÇ=qÜÉáê=dê~éÜë==191= UKO= iáãáíë=çÑ=cìåÅíáçåë==208= UKP= aÉÑáåáíáçå=~åÇ=mêçéÉêíáÉë=çÑ=íÜÉ=aÉêáî~íáîÉ==209= UKQ= q~ÄäÉ=çÑ=aÉêáî~íáîÉë==211= UKR= eáÖÜÉê=lêÇÉê=aÉêáî~íáîÉë==215= UKS= ^ééäáÅ~íáçåë=çÑ=aÉêáî~íáîÉ==217= UKT= aáÑÑÉêÉåíá~ä==221= UKU= jìäíáî~êá~ÄäÉ=cìåÅíáçåë==222= UKV= aáÑÑÉêÉåíá~ä=léÉê~íçêë==225= = = 9 fkqbdo^i=`^i`rirp= VKN= fåÇÉÑáåáíÉ=fåíÉÖê~ä==227= VKO= fåíÉÖê~äë=çÑ=o~íáçå~ä=cìåÅíáçåë==228= VKP= fåíÉÖê~äë=çÑ=fêê~íáçå~ä=cìåÅíáçåë==231= VKQ= fåíÉÖê~äë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==237= VKR= fåíÉÖê~äë=çÑ=eóéÉêÄçäáÅ=cìåÅíáçåë==241= VKS= fåíÉÖê~äë=çÑ=bñéçåÉåíá~ä=~åÇ=içÖ~êáíÜãáÅ=cìåÅíáçåë==242= VKT= oÉÇìÅíáçå=cçêãìä~ë==243= VKU= aÉÑáåáíÉ=fåíÉÖê~ä==247= VKV= fãéêçéÉê=fåíÉÖê~ä==253= VKNM= açìÄäÉ=fåíÉÖê~ä==257= VKNN= qêáéäÉ=fåíÉÖê~ä==269= vi
  8. 8. VKNO= iáåÉ=fåíÉÖê~ä==275= VKNP= pìêÑ~ÅÉ=fåíÉÖê~ä==285= = = 10 afccbobkqf^i=bnr^qflkp= NMKN= cáêëí=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==295= NMKO= pÉÅçåÇ=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==298= NMKP= pçãÉ=m~êíá~ä=aáÑÑÉêÉåíá~ä=bèì~íáçåë==302= = = 11 pbofbp= NNKN= ^êáíÜãÉíáÅ=pÉêáÉë==304= NNKO= dÉçãÉíêáÅ=pÉêáÉë==305= NNKP= pçãÉ=cáåáíÉ=pÉêáÉë==305= NNKQ= fåÑáåáíÉ=pÉêáÉë==307= NNKR= mêçéÉêíáÉë=çÑ=`çåîÉêÖÉåí=pÉêáÉë==307= NNKS= `çåîÉêÖÉåÅÉ=qÉëíë==308= NNKT= ^äíÉêå~íáåÖ=pÉêáÉë==310= NNKU= mçïÉê=pÉêáÉë==311= NNKV= aáÑÑÉêÉåíá~íáçå=~åÇ=fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë==312= NNKNM= q~óäçê=~åÇ=j~Åä~ìêáå=pÉêáÉë==313= NNKNN= mçïÉê=pÉêáÉë=bñé~åëáçåë=Ñçê=pçãÉ=cìåÅíáçåë==314= NNKNO= _áåçãá~ä=pÉêáÉë==316= NNKNP= cçìêáÉê=pÉêáÉë==316= = = 12 mol_^_fifqv= NOKN= mÉêãìí~íáçåë=~åÇ=`çãÄáå~íáçåë==318= NOKO= mêçÄ~Äáäáíó=cçêãìä~ë==319= = = = = = vii
  9. 9. = qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK= = viii
  10. 10. Chapter 1 Number Sets = = = = 1.1 Set Identities = pÉíëW=^I=_I=`= råáîÉêë~ä=ëÉíW=f= `çãéäÉãÉåí=W= ^′ = mêçéÉê=ëìÄëÉíW= ^ ⊂ _ == bãéíó=ëÉíW= ∅ = råáçå=çÑ=ëÉíëW= ^ ∪ _ = fåíÉêëÉÅíáçå=çÑ=ëÉíëW= ^ ∩ _ = aáÑÑÉêÉåÅÉ=çÑ=ëÉíëW= ^ y _ = = = 1. = 2. = 3. 4. 5. ^ ⊂ f= ^ ⊂ ^= ^ = _ =áÑ= ^ ⊂ _ =~åÇ= _ ⊂ ^ .= = bãéíó=pÉí= ∅⊂^= = råáçå=çÑ=pÉíë== ` = ^ ∪ _ = {ñ ö ñ ∈ ^ çê ñ ∈ _}= = 1
  11. 11. CHAPTER 1. NUMBER SETS = ===== = Figure 1. 6. = 7. = 8. = `çããìí~íáîáíó= ^∪_ = _∪^= ^ëëçÅá~íáîáíó= ^ ∪ (_ ∪ ` ) = (^ ∪ _ ) ∪ ` = fåíÉêëÉÅíáçå=çÑ=pÉíë= ` = ^ ∪ _ = {ñ ö ñ ∈ ^ ~åÇ ñ ∈ _} = = = ===== = Figure 2. 9. = 10. = = `çããìí~íáîáíó= ^∩_ = _∩^= ^ëëçÅá~íáîáíó= ^ ∩ (_ ∩ ` ) = (^ ∩ _ ) ∩ ` = = 2
  12. 12. CHAPTER 1. NUMBER SETS 11. = 12. = 13. = 14. aáëíêáÄìíáîáíó= ^ ∪ (_ ∩ ` ) = (^ ∪ _ ) ∩ (^ ∪ ` ) I= ^ ∩ (_ ∪ ` ) = (^ ∩ _ ) ∪ (^ ∩ ` ) K= fÇÉãéçíÉåÅó= ^ ∩ ^ = ^ I== ^∪^ = ^= açãáå~íáçå= ^ ∩ ∅ = ∅ I= ^∪f= f= fÇÉåíáíó= ^ ∪ ∅ = ^ I== ^∩f= ^ = 15. 16. 17. 18. `çãéäÉãÉåí= ^′ = {ñ ∈ f ö ñ ∉ ^} = `çãéäÉãÉåí=çÑ=fåíÉêëÉÅíáçå=~åÇ=råáçå ^ ∪ ^′ = f I== ^ ∩ ^′ = ∅ = = aÉ=jçêÖ~å∞ë=i~ïë (^ ∪ _ )′ = ^′ ∩ _′ I== (^ ∩ _ )′ = ^′ ∪ _′ = = aáÑÑÉêÉåÅÉ=çÑ=pÉíë ` = _ y ^ = {ñ ö ñ ∈ _ ~åÇ ñ ∉ ^} = = 3
  13. 13. CHAPTER 1. NUMBER SETS = ===== = Figure 3. = 19. _ y ^ = _ y (^ ∩ _ ) = 20. _ y ^ = _ ∩ ^′ 21. ^y^=∅ 22. ^ y _ = ^ =áÑ= ^ ∩ _ = ∅ . = = = ===== = Figure 4. = 23. (^ y _) ∩ ` = (^ ∩ `) y (_ ∩ `) 24. ^′ = f y ^ 25. `~êíÉëá~å=mêçÇìÅí ` = ^ × _ = {(ñ I ó ) ö ñ ∈ ^ ~åÇ ó ∈ _} = = 4 =
  14. 14. CHAPTER 1. NUMBER SETS 1.2 Sets of Numbers = 26. 27. = 28. = 29. = 30. k~íìê~ä=åìãÄÉêëW=k= tÜçäÉ=åìãÄÉêëW= kM = fåíÉÖÉêëW=w= mçëáíáîÉ=áåíÉÖÉêëW= w + = kÉÖ~íáîÉ=áåíÉÖÉêëW= w − = o~íáçå~ä=åìãÄÉêëW=n= oÉ~ä=åìãÄÉêëW=o== `çãéäÉñ=åìãÄÉêëW=`== = = k~íìê~ä=kìãÄÉêë `çìåíáåÖ=åìãÄÉêëW k = {NI OI PI K} K= tÜçäÉ=kìãÄÉêë `çìåíáåÖ=åìãÄÉêë=~åÇ=òÉêçW= k M = {MI NI OI PI K} K= fåíÉÖÉêë tÜçäÉ=åìãÄÉêë=~åÇ=íÜÉáê=çééçëáíÉë=~åÇ=òÉêçW= w + = k = {NI OI PI K}I= w − = {KI − PI − OI − N} I= w = w − ∪ {M} ∪ w + = {KI − PI − OI − NI MI NI OI PI K} K= o~íáçå~ä=kìãÄÉêë oÉéÉ~íáåÖ=çê=íÉêãáå~íáåÖ=ÇÉÅáã~äëW== ~   n = ñ ö ñ = ~åÇ ~ ∈ w ~åÇ Ä ∈ w ~åÇ Ä ≠ M K= Ä   fêê~íáçå~ä=kìãÄÉêë kçåêÉéÉ~íáåÖ=~åÇ=åçåíÉêãáå~íáåÖ=ÇÉÅáã~äëK = 5
  15. 15. CHAPTER 1. NUMBER SETS 31. oÉ~ä=kìãÄÉêë== råáçå=çÑ=ê~íáçå~ä=~åÇ=áêê~íáçå~ä=åìãÄÉêëW=oK= = 32. `çãéäÉñ=kìãÄÉêë ` = {ñ + áó ö ñ ∈ o ~åÇ ó ∈ o}I== ïÜÉêÉ=á=áë=íÜÉ=áã~Öáå~êó=ìåáíK = 33. k⊂ w⊂n⊂ o ⊂ `= = === = = Figure 5. = = = = = = 6
  16. 16. CHAPTER 1. NUMBER SETS 1.3 Basic Identities = oÉ~ä=åìãÄÉêëW=~I=ÄI=Å= = = 34. ^ÇÇáíáîÉ=fÇÉåíáíó= ~+M=~ = = 35. ^ÇÇáíáîÉ=fåîÉêëÉ= ~ + (− ~ ) = M = = 36. `çããìí~íáîÉ=çÑ=^ÇÇáíáçå= ~ +Ä= Ä+~ = 37. ^ëëçÅá~íáîÉ=çÑ=^ÇÇáíáçå= (~ + Ä) + Å = ~ + (Ä + Å ) = = = 38. aÉÑáåáíáçå=çÑ=pìÄíê~Åíáçå= ~ − Ä = ~ + (− Ä) = = 39. = 40. 41. 42. jìäíáéäáÅ~íáîÉ=fÇÉåíáíó= ~ ⋅N = ~ = jìäíáéäáÅ~íáîÉ=fåîÉêëÉ= N ~ ⋅ = N I= ~ ≠ M ~ = jìäíáéäáÅ~íáçå=qáãÉë=M ~ ⋅M = M = `çããìí~íáîÉ=çÑ=jìäíáéäáÅ~íáçå= ~ ⋅Ä = Ä⋅~ = = 7
  17. 17. CHAPTER 1. NUMBER SETS 43. ^ëëçÅá~íáîÉ=çÑ=jìäíáéäáÅ~íáçå= (~ ⋅ Ä)⋅ Å = ~ ⋅ (Ä ⋅ Å ) = aáëíêáÄìíáîÉ=i~ï= ~ (Ä + Å ) = ~Ä + ~Å = 44. = 45. aÉÑáåáíáçå=çÑ=aáîáëáçå= ~ N = ~⋅ = Ä Ä = = = 1.4 Complex Numbers = k~íìê~ä=åìãÄÉêW=å= fã~Öáå~êó=ìåáíW=á= `çãéäÉñ=åìãÄÉêW=ò= oÉ~ä=é~êíW=~I=Å= fã~Öáå~êó=é~êíW=ÄáI=Çá= jçÇìäìë=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=êI= êN I= êO = ^êÖìãÉåí=çÑ=~=ÅçãéäÉñ=åìãÄÉêW= ϕ I= ϕN I= ϕO = = = 46. = 47. = 48. áN = á = á O = −N = á P = −á = áQ = N= áR = á = á S = −N = á T = −á = áU = N = á Q å +N = á = á Q å+ O = −N = á Q å + P = −á = á Qå = N = ò = ~ + Äá = `çãéäÉñ=mä~åÉ= = 8
  18. 18. CHAPTER 1. NUMBER SETS = ===== = Figure 6. = 49. = 50. = 51. = (~ + Äá ) − (Å + Çá ) = (~ − Å ) + (Ä − Ç)á = (~ + Äá )(Å + Çá ) = (~Å − ÄÇ ) + (~Ç + ÄÅ )á = ~ + Äá ~Å + ÄÇ ÄÅ − ~Ç = + ⋅á = Å + Çá Å O + Ç O Å O + Ç O 52. = 53. (~ + Äá ) + (Å + Çá ) = (~ + Å ) + (Ä + Ç )á = `çåàìÖ~íÉ=`çãéäÉñ=kìãÄÉêë= ||||||| ~ + Äá = ~ − Äá = = 54. ~ = ê Åçë ϕ I= Ä = ê ëáå ϕ == = 9
  19. 19. CHAPTER 1. NUMBER SETS = = Figure 7. 55. = 56. = mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë= ~ + Äá = ê(Åçë ϕ + á ëáå ϕ) = jçÇìäìë=~åÇ=^êÖìãÉåí=çÑ=~=`çãéäÉñ=kìãÄÉê= fÑ= ~ + Äá =áë=~=ÅçãéäÉñ=åìãÄÉêI=íÜÉå= ê = ~ O + ÄO =EãçÇìäìëFI== Ä ϕ = ~êÅí~å =E~êÖìãÉåíFK= ~ = 57. = 58. mêçÇìÅí=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ò N ⋅ ò O = êN (Åçë ϕN + á ëáå ϕN ) ⋅ êO (Åçë ϕO + á ëáå ϕO ) = = êNêO [Åçë(ϕN + ϕO ) + á ëáå(ϕN + ϕO )] = `çåàìÖ~íÉ=kìãÄÉêë=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ||||||||||||||||||||| ê(Åçë ϕ + á ëáå ϕ) = ê[Åçë(− ϕ) + á ëáå(− ϕ)] = = 59. fåîÉêëÉ=çÑ=~=`çãéäÉñ=kìãÄÉê=áå=mçä~ê=oÉéêÉëÉåí~íáçå= N N = [Åçë(− ϕ) + á ëáå(− ϕ)] = ê(Åçë ϕ + á ëáå ϕ) ê 10
  20. 20. CHAPTER 1. NUMBER SETS 60. = 61. = 62. = 63. = 64. nìçíáÉåí=áå=mçä~ê=oÉéêÉëÉåí~íáçå= ò N êN (Åçë ϕN + á ëáå ϕN ) êN = [Åçë(ϕN − ϕO ) + á ëáå(ϕN − ϕO )] = = ò O êO (Åçë ϕO + á ëáå ϕO ) êO mçïÉê=çÑ=~=`çãéäÉñ=kìãÄÉê= å ò å = [ê(Åçë ϕ + á ëáå ϕ)] = ê å [Åçë(åϕ) + á ëáå(åϕ)] = cçêãìä~=±aÉ=jçáîêÉ≤= (Åçë ϕ + á ëáå ϕ)å = Åçë(åϕ) + á ëáå(åϕ) = kíÜ=oççí=çÑ=~=`çãéäÉñ=kìãÄÉê= ϕ + Oπâ ϕ + Oπâ   å ò = å ê(Åçë ϕ + á ëáå ϕ) = å ê  Åçë + á ëáå  I== å å   ïÜÉêÉ== â = MI NI OI KI å − N K== bìäÉê∞ë=cçêãìä~= É áñ = Åçë ñ + á ëáå ñ = = = 11
  21. 21. Chapter 2 Algebra = = = = 2.1 Factoring Formulas = oÉ~ä=åìãÄÉêëW=~I=ÄI=Å== k~íìê~ä=åìãÄÉêW=å= = = 65. = 66. = 67. = 68. = 69. = 70. = 71. = 72. ~ O − ÄO = (~ + Ä)(~ − Ä) = ~ P − ÄP = (~ − Ä)(~ O + ~Ä + ÄO ) = ~ P + ÄP = (~ + Ä)(~ O − ~Ä + ÄO ) = ~ Q − ÄQ = (~ O − ÄO )(~ O + ÄO ) = (~ − Ä)(~ + Ä)(~ O + ÄO ) = ~ R − ÄR = (~ − Ä)(~ Q + ~ P Ä + ~ O ÄO + ~ÄP + ÄQ ) = ~ R + ÄR = (~ + Ä)(~ Q − ~ P Ä + ~ O ÄO − ~ÄP + ÄQ ) = fÑ=å=áë=çÇÇI=íÜÉå= ~ å + Äå = (~ + Ä)(~ å−N − ~ å −O Ä + ~ å −P ÄO − K − ~Äå −O + Äå −N ) K== fÑ=å=áë=ÉîÉåI=íÜÉå== ~ å − Äå = (~ − Ä)(~ å −N + ~ å −O Ä + ~ å −P ÄO + K + ~Äå−O + Äå −N ) I== 12
  22. 22. CHAPTER 2. ALGEBRA ~ å + Äå = (~ + Ä)(~ å−N − ~ å −O Ä + ~ å −P ÄO − K + ~Äå−O − Äå −N ) K= = = = 2.2 Product Formulas 73. = 74. = 75. = 76. = 77. = 78. = 79. = 80. = 81. oÉ~ä=åìãÄÉêëW=~I=ÄI=Å== tÜçäÉ=åìãÄÉêëW=åI=â= = = (~ − Ä)O = ~ O − O~Ä + ÄO = (~ + Ä)O = ~ O + O~Ä + ÄO = (~ − Ä)P = ~ P − P~ O Ä + P~ÄO − ÄP = (~ + Ä)P = ~ P + P~ OÄ + P~ÄO + ÄP = (~ − Ä)Q = ~ Q − Q~ P Ä + S~ O ÄO − Q~ÄP + ÄQ = (~ + Ä)Q = ~ Q + Q~ P Ä + S~ OÄO + Q~ÄP + ÄQ = _áåçãá~ä=cçêãìä~= (~ + Ä)å = å` M~ å + å`N~ å−NÄ + å` O~ å−OÄO + K + å` å−N~Äå−N + å` å Äå I å> ïÜÉêÉ= å ` â = =~êÉ=íÜÉ=Äáåçãá~ä=ÅçÉÑÑáÅáÉåíëK= â> (å − â )> (~ + Ä + Å )O = ~ O + ÄO + Å O + O~Ä + O~Å + OÄÅ = (~ + Ä + Å + K + ì + î )O = ~ O + ÄO + Å O + K + ì O + î O + = + O(~Ä + ~Å + K + ~ì + ~î + ÄÅ + K + Äì + Äî + K + ìî ) = 13
  23. 23. CHAPTER 2. ALGEBRA 2.3 Powers = _~ëÉë=EéçëáíáîÉ=êÉ~ä=åìãÄÉêëFW=~I=Ä== mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã= = = ~ ã ~ å = ~ ã+å = 82. = 83. ~ã = ~ ã −å = å ~ = 84. = (~Ä)ã = ~ ã Äã = 85. ~ã ~   = ã = Ä  Ä ã = 86. = 87. = 88. = (~ ) ã å = ~ ãå = ~ M = N I= ~ ≠ M = ~N = N = ~ −ã = 89. N = ~ã = ã å ~ = å ~ã = 90. = = = = = 14
  24. 24. CHAPTER 2. ALGEBRA 2.4 Roots = 91. = _~ëÉëW=~I=Ä== mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã= ~ I Ä ≥ M =Ñçê=ÉîÉå=êççíë=E å = Oâ I= â ∈ k F= = = å ~Ä = å ~ å Ä = 92. = å ~ ã Ä = åã ~ ã Äå = 93. å ~ å~ = I= Ä ≠ M = Ä åÄ = 94. = 95. = 96. = ~ åã ~ ã åã ~ ã I= Ä ≠ M K= = = ã Äå Ä åã Äå å (~ ) å ã ( ~) å å é = å ~ ãé = =~= åé 97. = å ~ã = 98. = å ~ =~ = 99. = ã å 100. = ã å ã ~ = ãå ~ = ( ~) å ~ ãé = ã = å ~ã = 15
  25. 25. CHAPTER 2. ALGEBRA N å ~ å −N = I= ~ ≠ M K= å ~ ~ 101. = ~± Ä = 102. ~ + ~O − Ä ~ − ~O − Ä ± = O O = N ~m Ä = = ~−Ä ~± Ä 103. = = = 2.5 Logarithms = 104. 105. 106. 107. 108. 109. mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=ñI=óI=~I=ÅI=â= k~íìê~ä=åìãÄÉêW=å== = = aÉÑáåáíáçå=çÑ=içÖ~êáíÜã= ó = äçÖ ~ ñ =áÑ=~åÇ=çåäó=áÑ= ñ = ~ ó I= ~ > M I= ~ ≠ N K= = äçÖ ~ N = M = = äçÖ ~ ~ = N = = − ∞ áÑ ~ > N äçÖ ~ M =  = + ∞ áÑ ~ < N = äçÖ ~ (ñó ) = äçÖ ~ ñ + äçÖ ~ ó = = ñ äçÖ ~ = äçÖ ~ ñ − äçÖ ~ ó = ó 16
  26. 26. CHAPTER 2. ALGEBRA 110. äçÖ ~ (ñ å ) = å äçÖ ~ ñ = = N 111. äçÖ ~ å ñ = äçÖ ~ ñ = å = äçÖ Å ñ 112. äçÖ ~ ñ = = äçÖ Å ñ ⋅ äçÖ ~ Å I= Å > M I= Å ≠ N K= äçÖ Å ~ = N 113. äçÖ ~ Å = = äçÖ Å ~ = 114. ñ = ~ äçÖ ~ ñ = = 115. içÖ~êáíÜã=íç=_~ëÉ=NM= äçÖ NM ñ = äçÖ ñ = = 116. k~íìê~ä=içÖ~êáíÜã= äçÖ É ñ = äå ñ I== â  N ïÜÉêÉ= É = äáã N +  = OKTNUOUNUOUK = â →∞  â = N 117. äçÖ ñ = äå ñ = MKQPQOVQ äå ñ = äå NM = N 118. äå ñ = äçÖ ñ = OKPMORUR äçÖ ñ = äçÖ É = = = = = 17
  27. 27. CHAPTER 2. ALGEBRA 2.6 Equations = oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=éI=èI=ìI=î= pçäìíáçåëW= ñ N I= ñ O I= ó N I= ó O I= ó P = = = 119. iáåÉ~ê=bèì~íáçå=áå=låÉ=s~êá~ÄäÉ= Ä ~ñ + Ä = M I= ñ = − K== ~ = 120. nì~Çê~íáÅ=bèì~íáçå= − Ä ± ÄO − Q~Å ~ñ + Äñ + Å = M I= ñ NI O = K= O~ = 121. aáëÅêáãáå~åí= a = ÄO − Q~Å = = 122. sáÉíÉ∞ë=cçêãìä~ë= fÑ= ñ O + éñ + è = M I=íÜÉå== ñ N + ñ O = −é K=  ñ Nñ O = è  = Ä 123. ~ñ O + Äñ = M I= ñ N = M I= ñ O = − K= ~ = Å 124. ~ñ O + Å = M I= ñ NI O = ± − K= ~ = 125. `ìÄáÅ=bèì~íáçåK=`~êÇ~åç∞ë=cçêãìä~K== ó P + éó + è = M I== O 18
  28. 28. CHAPTER 2. ALGEBRA ó N = ì + î I= ó OI P = − N (ì + î ) ± P (ì + î ) á I== O O ïÜÉêÉ== O ì=P − O O O è è è  é  è  é +   +   I= î = P − −   +   K== O O  O P  O P = = 2.7 Inequalities s~êá~ÄäÉëW=ñI=óI=ò= ~ I ÄI ÅI Ç oÉ~ä=åìãÄÉêëW=  I=ãI=å= ~N I ~ O I ~ P I KI ~ å aÉíÉêãáå~åíëW=aI= añ I= aó I= aò == = = 126. fåÉèì~äáíáÉëI=fåíÉêî~ä=kçí~íáçåë=~åÇ=dê~éÜë== = fåÉèì~äáíó= fåíÉêî~ä=kçí~íáçå= dê~éÜ= [~I Ä]= ~ ≤ ñ ≤ Ä= ~ < ñ ≤ Ä= (~I Ä] = = ~ ≤ ñ < Ä= [~I Ä) = = ~ < ñ < Ä= (~I Ä) = = − ∞ < ñ ≤ Ä I= ñ≤Ä= − ∞ < ñ < Ä I= ñ<Ä= ~ ≤ ñ < ∞ I= ñ≥~= ~ < ñ < ∞ I= ñ >~= (− ∞I Ä] = = = (− ∞I Ä) = = [~I ∞ ) = = (~I ∞ ) = = 19
  29. 29. CHAPTER 2. ALGEBRA 127. = 128. = 129. = 130. = 131. = 132. = 133. = fÑ= ~ > Ä I=íÜÉå= Ä < ~ K= fÑ= ~ > Ä I=íÜÉå= ~ − Ä > M =çê= Ä − ~ < M K= fÑ= ~ > Ä I=íÜÉå= ~ + Å > Ä + Å K= fÑ= ~ > Ä I=íÜÉå= ~ − Å > Ä − Å K= fÑ= ~ > Ä =~åÇ= Å > Ç I=íÜÉå= ~ + Å > Ä + Ç K= fÑ= ~ > Ä =~åÇ= Å > Ç I=íÜÉå= ~ − Ç > Ä − Å K= fÑ= ~ > Ä =~åÇ= ã > M I=íÜÉå= ã~ > ãÄ K= 134. fÑ= ~ > Ä =~åÇ= ã > M I=íÜÉå= ~ Ä > K= ã ã = 135. fÑ= ~ > Ä =~åÇ= ã < M I=íÜÉå= ã~ < ãÄ K= = ~ Ä 136. fÑ= ~ > Ä =~åÇ= ã < M I=íÜÉå= < K= ã ã = 137. fÑ= M < ~ < Ä =~åÇ= å > M I=íÜÉå= ~ å < Äå K= = 138. fÑ= M < ~ < Ä =~åÇ= å < M I=íÜÉå= ~ å > Äå K= = 139. fÑ= M < ~ < Ä I=íÜÉå= å ~ < å Ä K= = ~+Ä I== 140. ~Ä ≤ O ïÜÉêÉ= ~ > M =I= Ä > M X=~å=Éèì~äáíó=áë=î~äáÇ=çåäó=áÑ= ~ = Ä K== = N 141. ~ + ≥ O I=ïÜÉêÉ= ~ > M X=~å=Éèì~äáíó=í~âÉë=éä~ÅÉ=çåäó=~í= ~ = N K= ~ 20
  30. 30. CHAPTER 2. ALGEBRA 142. å ~N~ O K~ å ≤ ~N + ~ O + K + ~ å I=ïÜÉêÉ= ~N I ~ O I KI ~ å > M K= å = Ä 143. fÑ= ~ñ + Ä > M =~åÇ= ~ > M I=íÜÉå= ñ > − K= ~ = Ä 144. fÑ= ~ñ + Ä > M =~åÇ= ~ < M I=íÜÉå= ñ < − K== ~ = 145. ~ñ O + Äñ + Å > M = = = ~ > M= = = = = a>M= = = = a=M= = = = a<M= = ñ < ñ N I= ñ > ñ O = = ñ N < ñ I= ñ > ñ N = = = −∞< ñ <∞= = 21 ~ <M= = = ñN < ñ < ñ O = = ñ ∈∅ = = = ñ ∈∅ = = = =
  31. 31. CHAPTER 2. ALGEBRA ~+Ä ≤ ~ + Ä = 146. = 147. = 148. = 149. = 150. = fÑ= ñ < ~ I=íÜÉå= − ~ < ñ < ~ I=ïÜÉêÉ= ~ > M K= fÑ= ñ > ~ I=íÜÉå= ñ < −~ =~åÇ= ñ > ~ I=ïÜÉêÉ= ~ > M K= fÑ= ñ O < ~ I=íÜÉå= ñ < ~ I=ïÜÉêÉ= ~ > M K= fÑ= ñ O > ~ I=íÜÉå= ñ > ~ I=ïÜÉêÉ= ~ > M K= 151. fÑ= = Ñ (ñ ) ⋅ Ö (ñ ) > M Ñ (ñ ) > M I=íÜÉå=  K= Ö (ñ ) Ö (ñ ) ≠ M Ñ (ñ ) ⋅ Ö (ñ ) < M Ñ (ñ ) < M I=íÜÉå=  K= Ö (ñ ) Ö (ñ ) ≠ M 152. = = = 2.8 Compound Interest Formulas = cìíìêÉ=î~äìÉW=^= fåáíá~ä=ÇÉéçëáíW=`= ^ååì~ä=ê~íÉ=çÑ=áåíÉêÉëíW=ê= kìãÄÉê=çÑ=óÉ~êë=áåîÉëíÉÇW=í= kìãÄÉê=çÑ=íáãÉë=ÅçãéçìåÇÉÇ=éÉê=óÉ~êW=å= = = 153. dÉåÉê~ä=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~= åí  ê ^ = ` N +  =  å = 22
  32. 32. CHAPTER 2. ALGEBRA 154. páãéäáÑáÉÇ=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~= fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=çåÅÉ=éÉê=óÉ~êI=íÜÉå=íÜÉ=éêÉîáçìë= Ñçêãìä~=ëáãéäáÑáÉë=íçW= í ^ = `(N + ê ) K= = 155. `çåíáåìçìë=`çãéçìåÇ=fåíÉêÉëí= fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=Åçåíáåì~ääó=E å → ∞ FI=íÜÉå== ^ = `É êí K= = = 23
  33. 33. Chapter 3 Geometry = = = = 3.1 Right Triangle = iÉÖë=çÑ=~=êáÖÜí=íêá~åÖäÉW=~I=Ä= eóéçíÉåìëÉW=Å= ^äíáíìÇÉW=Ü= jÉÇá~åëW= ã ~ I= ã Ä I= ã Å = ^åÖäÉëW= α I β = o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= ^êÉ~W=p= = = = = Figure 8. = 156. α + β = VM° = = 24
  34. 34. CHAPTER 3. GEOMETRY 157. ëáå α = ~ = Åçë β = Å = 158. Åçë α = Ä = ëáå β = Å = 159. í~å α = ~ = Åçí β = Ä = Ä 160. Åçí α = = í~å β = ~ = Å 161. ëÉÅ α = = Åçë ÉÅ β = Ä = 162. Åçë ÉÅ α = Å = ëÉÅ β = ~ = 163. móíÜ~ÖçêÉ~å=qÜÉçêÉã= ~ O + ÄO = Å O = = 164. ~ = ÑÅ I= Ä = ÖÅ I== ïÜÉêÉ= Ñ= ~åÇ= Å= ~êÉ= éêçàÉÅíáçåë= çÑ= íÜÉ= äÉÖë= ~= ~åÇ= ÄI= êÉëéÉÅíáîÉäóI=çåíç=íÜÉ=ÜóéçíÉåìëÉ=ÅK= = O = O = ===== Figure 9. = 25
  35. 35. CHAPTER 3. GEOMETRY 165. Ü O = ÑÖ I=== ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK== = O O ~ Ä 166. ã O = ÄO − I= ã O = ~ O − I=== ~ Ä Q Q ïÜÉêÉ= ã ~ =~åÇ= ã Ä =~êÉ=íÜÉ=ãÉÇá~åë=íç=íÜÉ=äÉÖë=~=~åÇ=ÄK== = = = Figure 10. = Å 167. ã Å = I== O ïÜÉêÉ= ã Å =áë=íÜÉ=ãÉÇá~å=íç=íÜÉ=ÜóéçíÉåìëÉ=ÅK= = Å 168. o = = ã Å = O = ~ +Ä−Å ~Ä = = 169. ê = O ~ +Ä+Å = 170. ~Ä = ÅÜ = = = 26
  36. 36. CHAPTER 3. GEOMETRY 171. p = ~Ä ÅÜ = = O O = = = 3.2 Isosceles Triangle = _~ëÉW=~= iÉÖëW=Ä= _~ëÉ=~åÖäÉW= β = sÉêíÉñ=~åÖäÉW= α = ^äíáíìÇÉ=íç=íÜÉ=Ä~ëÉW=Ü= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 11. = 172. β = VM° − α = O = 173. Ü O = ÄO − O ~ = Q 27
  37. 37. CHAPTER 3. GEOMETRY 174. i = ~ + OÄ = = 175. p = O ~Ü Ä = ëáå α = O O = = = 3.3 Equilateral Triangle = páÇÉ=çÑ=~=Éèìáä~íÉê~ä=íêá~åÖäÉW=~= ^äíáíìÇÉW=Ü= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 12. = 176. Ü = ~ P = O = 28
  38. 38. CHAPTER 3. GEOMETRY O ~ P = 177. o = Ü = P P = N ~ P o = = 178. ê = Ü = P S O = 179. i = P~ = = 180. p = O ~Ü ~ P = = O Q = = = 3.4 Scalene Triangle E^=íêá~åÖäÉ=ïáíÜ=åç=íïç=ëáÇÉë=Éèì~äF= = = páÇÉë=çÑ=~=íêá~åÖäÉW=~I=ÄI=Å= ~ +Ä+Å == pÉãáéÉêáãÉíÉêW= é = O ^åÖäÉë=çÑ=~=íêá~åÖäÉW= αI βI γ = ^äíáíìÇÉë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= Ü ~ I Ü Ä I Ü Å = jÉÇá~åë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= ã ~ I ã Ä I ã Å = _áëÉÅíçêë=çÑ=íÜÉ=~åÖäÉë= αI βI γ W= í ~ I í Ä I í Å = o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= ^êÉ~W=p= = = 29
  39. 39. CHAPTER 3. GEOMETRY = ===== = Figure 13. = 181. α + β + γ = NUM° = 182. ~ + Ä > Å I== Ä + Å > ~ I== ~ + Å > Ä K= = 183. ~ − Ä < Å I== Ä − Å < ~ I== ~ − Å < Ä K= = = 184. jáÇäáåÉ= ~ è = I= è öö ~ K= O = = = ===== Figure 14. = 30
  40. 40. CHAPTER 3. GEOMETRY 185. i~ï=çÑ=`çëáåÉë= ~ O = ÄO + Å O − OÄÅ Åçë α I= ÄO = ~ O + Å O − O~Å Åçë β I= Å O = ~ O + ÄO − O~Ä Åçë γ K= = 186. i~ï=çÑ=páåÉë= ~ Ä Å = = = Oo I== ëáå α ëáå β ëáå γ ïÜÉêÉ=o=áë=íÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉK== = ~ Ä Å ÄÅ ~Å ~Ä ~ÄÅ = = = = = = 187. o = = O ëáå α O ëáå β O ëáå γ OÜ ~ OÜ Ä OÜ Å Qp = (é − ~ )(é − Ä)(é − Å ) I== 188. ê O = é N N N N = + + K= ê Ü~ ÜÄ ÜÅ = (é − Ä)(é − Å ) I= α 189. ëáå = O ÄÅ Åçë α é(é − ~ ) I= = O ÄÅ í~å α = O (é − Ä)(é − Å ) K= é(é − ~ ) = O 190. Ü ~ = é(é − ~ )(é − Ä)(é − Å ) I= ~ O é(é − ~ )(é − Ä)(é − Å ) I= ÜÄ = Ä O ÜÅ = é(é − ~ )(é − Ä)(é − Å ) K= Å 31
  41. 41. CHAPTER 3. GEOMETRY 191. Ü ~ = Ä ëáå γ = Å ëáå β I= Ü Ä = ~ ëáå γ = Å ëáå α I= Ü Å = ~ ëáå β = Ä ëáå α K= = Ä +Å ~ − I== O Q O O ~ + Å ÄO ãO = − I== Ä O Q O O ~ + Ä ÅO O ãÅ = − K= O Q 192. ã O = ~ O O O = = = ===== Figure 15. = O O O 193. ^j = ã ~ I= _j = ã Ä I= `j = ã Å =EcáÖKNRFK= P P P = QÄÅé(é − ~ ) 194. í O = I== ~ (Ä + Å )O Q~Åé(é − Ä) íO = I== Ä (~ + Å )O Q~Äé(é − Å ) íO = K= Å (~ + Ä)O = 32
  42. 42. CHAPTER 3. GEOMETRY ~Ü ~ ÄÜ Ä ÅÜ Å = = I== O O O ~Ä ëáå γ ~Å ëáå β ÄÅ ëáå α I== p= = = O O O p = é(é − ~ )(é − Ä)(é − Å ) =EeÉêçå∞ë=cçêãìä~FI= p = éê I== ~ÄÅ p= I= Qo p = Oo O ëáå α ëáå β ëáå γ I= α β γ p = éO í~å í~å í~å K= O O O 195. p = = = = 3.5 Square páÇÉ=çÑ=~=ëèì~êÉW=~= aá~Öçå~äW=Ç= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = Figure 16. 33
  43. 43. CHAPTER 3. GEOMETRY 196. Ç = ~ O == = 197. o = Ç ~ O = = O O = ~ 198. ê = = O 199. i = Q~ = = = 200. p = ~ = = = = O 3.6 Rectangle = páÇÉë=çÑ=~=êÉÅí~åÖäÉW=~I=Ä= aá~Öçå~äW=Ç= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 17. = 201. Ç = ~ O + ÄO == 34
  44. 44. CHAPTER 3. GEOMETRY 202. o = Ç = O = 203. i = O(~ + Ä) = = 204. p = ~Ä = = = = 3.7 Parallelogram = páÇÉë=çÑ=~=é~ê~ääÉäçÖê~ãW=~I=Ä= aá~Öçå~äëW= ÇN I Ç O = `çåëÉÅìíáîÉ=~åÖäÉëW= αI β = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ = ^äíáíìÇÉW=Ü== mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = ===== = Figure 18. = 205. α + β = NUM° = 206. Ç + Ç = O(~ + Ä ) = O N O O O = O = 35
  45. 45. CHAPTER 3. GEOMETRY 207. Ü = Ä ëáå α = Ä ëáå β = 208. i = O(~ + Ä) = 209. p = ~Ü = ~Ä ëáå α I== N p = ÇNÇ O ëáå ϕ K= O = = = = = 3.8 Rhombus = páÇÉ=çÑ=~=êÜçãÄìëW=~= aá~Öçå~äëW= ÇN I Ç O = `çåëÉÅìíáîÉ=~åÖäÉëW= αI β = ^äíáíìÇÉW=e= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = ===== Figure 19. = 36
  46. 46. CHAPTER 3. GEOMETRY 210. α + β = NUM° = = 211. Ç + Ç = Q~ = O N O O O = 212. Ü = ~ ëáå α = ÇNÇ O = O~ = Ü ÇÇ ~ ëáå α 213. ê = = N O = = O Q~ O = 214. i = Q~ = = 215. p = ~Ü = ~ ëáå α I== N p = ÇNÇ O K= O = = = O 3.9 Trapezoid = _~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä= jáÇäáåÉW=è= ^äíáíìÇÉW=Ü= ^êÉ~W=p= = = 37
  47. 47. CHAPTER 3. GEOMETRY = = Figure 20. = 216. è = 217. p = ~+Ä = O ~+Ä ⋅ Ü = èÜ = O = = = = 3.10 Isosceles Trapezoid = _~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä= iÉÖW=Å= jáÇäáåÉW=è= ^äíáíìÇÉW=Ü= aá~Öçå~äW=Ç= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= ^êÉ~W=p= = = 38
  48. 48. CHAPTER 3. GEOMETRY = = Figure 21. = 218. è = ~+Ä = O = 219. Ç = ~Ä + Å = = N O 220. Ü = Å O − (Ä − ~ ) = Q O = Å ~Ä + Å O = (OÅ − ~ + Ä)(OÅ + ~ − Ä) = ~+Ä 222. p = ⋅ Ü = èÜ = O = = = = = = 221. o = 39
  49. 49. CHAPTER 3. GEOMETRY 3.11 Isosceles Trapezoid with Inscribed Circle = _~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä= iÉÖW=Å= jáÇäáåÉW=è= ^äíáíìÇÉW=Ü= aá~Öçå~äW=Ç= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=o= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 22. = 223. ~ + Ä = OÅ = = ~+Ä 224. è = =Å= O = 225. Ç = Ü + Å = O O O = 40
  50. 50. CHAPTER 3. GEOMETRY 226. ê = Ü ~Ä = = O O = Ä ÅÇ ÅÇ Å Å Å ~+Ä ~ N+ ÜO + Å O = = = = +S+ = OÜ Qê O ~Ä OÜ U Ä ~ = 228. i = O(~ + Ä) = QÅ = = (~ + Ä) ~Ä = èÜ = ÅÜ = iê == ~+Ä ⋅Ü = 229. p = O O O = = = 227. o = O 3.12 Trapezoid with Inscribed Circle = _~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä= i~íÉê~ä=ëáÇÉëW=ÅI=Ç= jáÇäáåÉW=è= ^äíáíìÇÉW=Ü= aá~Öçå~äëW= ÇN I Ç O = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ = o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= ^êÉ~W=p= = 41
  51. 51. CHAPTER 3. GEOMETRY = = Figure 23. = 230. ~ + Ä = Å + Ç = ~+Ä Å+Ç = = 231. è = O O 232. i = O(~ + Ä) = O(Å + Ç ) = = = = ~+Ä Å+Ç ⋅Ü = ⋅ Ü = èÜ I== O O N p = ÇNÇ O ëáå ϕ K= O 233. p = = = = 3.13 Kite = páÇÉë=çÑ=~=âáíÉW=~I=Ä= aá~Öçå~äëW= ÇN I Ç O = ^åÖäÉëW= αI βI γ = mÉêáãÉíÉêW=i= ^êÉ~W=p= = = 42
  52. 52. CHAPTER 3. GEOMETRY = = Figure 24. = 234. α + β + Oγ = PSM° = 235. i = O(~ + Ä) = = = 236. p = ÇNÇ O = O = = = 3.14 Cyclic Quadrilateral páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç= aá~Öçå~äëW= ÇN I Ç O = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ = fåíÉêå~ä=~åÖäÉëW= αI βI γ I δ = o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= pÉãáéÉêáãÉíÉêW=é== ^êÉ~W=p= 43
  53. 53. CHAPTER 3. GEOMETRY = = Figure 25. = 237. α + γ = β + δ = NUM° = = 238. míçäÉãó∞ë=qÜÉçêÉã= ~Å + ÄÇ = ÇNÇ O = 239. i = ~ + Ä + Å + Ç = = = N (~Å + ÄÇ )(~Ç + ÄÅ )(~Ä + ÅÇ ) I== 240. o = Q (é − ~ )(é − Ä)(é − Å )(é − Ç ) i ïÜÉêÉ= é = K= O = N 241. p = ÇNÇ O ëáå ϕ I== O p = (é − ~ )(é − Ä)(é − Å )(é − Ç ) I== i ïÜÉêÉ= é = K= O = = = 44
  54. 54. CHAPTER 3. GEOMETRY 3.15 Tangential Quadrilateral = páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç= aá~Öçå~äëW= ÇN I Ç O = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ = o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= mÉêáãÉíÉêW=i= pÉãáéÉêáãÉíÉêW=é== ^êÉ~W=p= = = = = Figure 26. = 242. ~ + Å = Ä + Ç = = 243. i = ~ + Ä + Å + Ç = O(~ + Å ) = O(Ä + Ç ) = = O ÇN Ç O − (~ − Ä) (~ + Ä − é ) O I== Oé i ïÜÉêÉ= é = K== O = O O 244. ê = 45
  55. 55. CHAPTER 3. GEOMETRY N 245. p = éê = ÇNÇ O ëáå ϕ = O = = = 3.16 General Quadrilateral = páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç= aá~Öçå~äëW= ÇN I Ç O = ^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ = fåíÉêå~ä=~åÖäÉëW= αI βI γ I δ = mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = ======= Figure 27. = 246. α + β + γ + δ = PSM° = 247. i = ~ + Ä + Å + Ç = = = 46
  56. 56. CHAPTER 3. GEOMETRY N 248. p = ÇNÇ O ëáå ϕ = O = = = 3.17 Regular Hexagon = páÇÉW=~= fåíÉêå~ä=~åÖäÉW= α = pä~åí=ÜÉáÖÜíW=ã= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= pÉãáéÉêáãÉíÉêW=é== ^êÉ~W=p= = = = = Figure 28. = 249. α = NOM° = = 250. ê = ã = ~ P = O 47
  57. 57. CHAPTER 3. GEOMETRY 251. o = ~ = = 252. i = S~ = = O ~ P P I== O i ïÜÉêÉ= é = K= O = = = 253. p = éê = 3.18 Regular Polygon = páÇÉW=~= kìãÄÉê=çÑ=ëáÇÉëW=å= fåíÉêå~ä=~åÖäÉW= α = pä~åí=ÜÉáÖÜíW=ã= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= mÉêáãÉíÉêW=i= pÉãáéÉêáãÉíÉêW=é== ^êÉ~W=p= = = 48
  58. 58. CHAPTER 3. GEOMETRY = = Figure 29. = 254. α = 255. α = å−O ⋅ NUM° = O = å−O ⋅ NUM° = O = 256. o = ~ π O ëáå å = = 257. ê = ã = ~ O í~å π å = oO − ~O = Q = 258. i = å~ = = 259. p = åo Oπ ëáå I== O å O p = éê = é o O − ~O I== Q 49
  59. 59. CHAPTER 3. GEOMETRY ïÜÉêÉ= é = i K== O = = = 3.19 Circle = o~ÇáìëW=o= aá~ãÉíÉêW=Ç= `ÜçêÇW=~= pÉÅ~åí=ëÉÖãÉåíëW=ÉI=Ñ= q~åÖÉåí=ëÉÖãÉåíW=Ö= `Éåíê~ä=~åÖäÉW= α = fåëÅêáÄÉÇ=~åÖäÉW= β = mÉêáãÉíÉêW=i= ^êÉ~W=p= = = α 260. ~ = Oo ëáå = O = = = Figure 30. = 50
  60. 60. CHAPTER 3. GEOMETRY 261. ~N~ O = ÄNÄO = = = = Figure 31. = 262. ÉÉN = ÑÑN = = = = ===== Figure 32. = 263. Ö O = ÑÑN = = 51
  61. 61. CHAPTER 3. GEOMETRY = ===== = Figure 33. = 264. β = α = O = = = Figure 34. = 265. i = Oπo = πÇ = = 266. p = πo O = io πÇ = == Q O O = 52
  62. 62. CHAPTER 3. GEOMETRY 3.20 Sector of a Circle = o~Çáìë=çÑ=~=ÅáêÅäÉW=o= ^êÅ=äÉåÖíÜW=ë= `Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ= `Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW= α = mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 35. = 267. ë = oñ = 268. ë = = πoα = NUM° = 269. i = ë + Oo = = 270. p = oë o ñ πo α = = == O O PSM° O O = = 53
  63. 63. CHAPTER 3. GEOMETRY 3.21 Segment of a Circle = o~Çáìë=çÑ=~=ÅáêÅäÉW=o= ^êÅ=äÉåÖíÜW=ë= `ÜçêÇW=~= `Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ= `Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW= α = eÉáÖÜí=çÑ=íÜÉ=ëÉÖãÉåíW=Ü= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = = = Figure 36. = 271. ~ = O OÜo − Ü O = = N 272. Ü = o − Qo O − ~ O I= Ü < o = O = 273. i = ë + ~ = = 54
  64. 64. CHAPTER 3. GEOMETRY O O N [ëo − ~(o − Ü )] = o  απ − ëáå α  = o (ñ − ëáå ñ ) I==   O O  NUM°  O O p ≈ Ü~ K= P 274. p = = = = 3.22 Cube = bÇÖÉW=~== aá~Öçå~äW=Ç= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ëéÜÉêÉW=ê= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = === Figure 37. = 275. Ç = ~ P = = ~ 276. ê = = O = 55
  65. 65. CHAPTER 3. GEOMETRY 277. o = ~ P = O = 278. p = S~ = O = 279. s = ~ == = = = P 3.23 Rectangular Parallelepiped = bÇÖÉëW=~I=ÄI=Å== aá~Öçå~äW=Ç= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = ===== Figure 38. = 280. Ç = ~ O + ÄO + Å O = 281. p = O(~Ä + ~Å + ÄÅ ) = 282. s = ~ÄÅ == = = 56
  66. 66. CHAPTER 3. GEOMETRY 3.24 Prism = i~íÉê~ä=ÉÇÖÉW=ä= eÉáÖÜíW=Ü= i~íÉê~ä=~êÉ~W= p i = ^êÉ~=çÑ=Ä~ëÉW= p_ = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = ===== Figure 39. = 283. p = p i + Op_ K== = 284. i~íÉê~ä=^êÉ~=çÑ=~=oáÖÜí=mêáëã= p i = (~ N + ~ O + ~ P + K + ~ å )ä = = 285. i~íÉê~ä=^êÉ~=çÑ=~å=lÄäáèìÉ=mêáëã= p i = éä I== ïÜÉêÉ=é=áë=íÜÉ=éÉêáãÉíÉê=çÑ=íÜÉ=Åêçëë=ëÉÅíáçåK= = 57
  67. 67. CHAPTER 3. GEOMETRY 286. s = p_ Ü = = 287. `~î~äáÉêáDë=mêáåÅáéäÉ== dáîÉå=íïç=ëçäáÇë=áåÅäìÇÉÇ=ÄÉíïÉÉå=é~ê~ääÉä=éä~åÉëK=fÑ=ÉîÉêó= éä~åÉ=Åêçëë=ëÉÅíáçå=é~ê~ääÉä=íç=íÜÉ=ÖáîÉå=éä~åÉë=Ü~ë=íÜÉ=ë~ãÉ= ~êÉ~=áå=ÄçíÜ=ëçäáÇëI=íÜÉå=íÜÉ=îçäìãÉë=çÑ=íÜÉ=ëçäáÇë=~êÉ=Éèì~äK= = = = 3.25 Regular Tetrahedron = qêá~åÖäÉ=ëáÇÉ=äÉåÖíÜW=~= eÉáÖÜíW=Ü= ^êÉ~=çÑ=Ä~ëÉW= p_ = pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 40. = 288. Ü = O ~= P = 58
  68. 68. CHAPTER 3. GEOMETRY 289. p_ = P~ O = Q = 290. p = P~ = = N ~P 291. s = p_ Ü = K== P S O = = = O 3.26 Regular Pyramid = páÇÉ=çÑ=Ä~ëÉW=~= i~íÉê~ä=ÉÇÖÉW=Ä= eÉáÖÜíW=Ü= pä~åí=ÜÉáÖÜíW=ã== kìãÄÉê=çÑ=ëáÇÉëW=å== pÉãáéÉêáãÉíÉê=çÑ=Ä~ëÉW=é= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉ=çÑ=Ä~ëÉW=ê= ^êÉ~=çÑ=Ä~ëÉW= p_ = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= pi = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 59
  69. 69. CHAPTER 3. GEOMETRY = = Figure 41. = 292. ã = ÄO − ~O = Q = 293. Ü = π O −~ å = π O ëáå å QÄO ëáå O = N N 294. p i = å~ã = å~ QÄO − ~ O = éã = O Q = 295. p_ = éê = = 296. p = p_ + p i = = N N 297. s = p_ Ü = éêÜ == P P = = = 60
  70. 70. CHAPTER 3. GEOMETRY 3.27 Frustum of a Regular Pyramid = ~N I ~ O I ~ P IKI ~ å = _~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW=  ÄN I ÄO I ÄP IKI Äå eÉáÖÜíW=Ü= pä~åí=ÜÉáÖÜíW=ã== ^êÉ~=çÑ=Ä~ëÉëW= pN I= pO = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i = mÉêáãÉíÉê=çÑ=Ä~ëÉëW= mN I= mO = pÅ~äÉ=Ñ~ÅíçêW=â= qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 42. = 298. ÄN ÄO ÄP Ä Ä = = =K= å = = â = ~N ~ O ~ P ~å ~ = 61
  71. 71. CHAPTER 3. GEOMETRY 299. pO = âO = pN = ã(mN + mO ) = 300. p i = O = 301. p = p i + pN + pO = = Ü 302. s = pN + pNpO + pO = P = O Üp  Ä  Ä   Üp 303. s = N N + +    = N N + â + â O = P  ~ ~  P   = = = ( ) [ ] 3.28 Rectangular Right Wedge = páÇÉë=çÑ=Ä~ëÉW=~I=Ä= qçé=ÉÇÖÉW=Å= eÉáÖÜíW=Ü= i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i = ^êÉ~=çÑ=Ä~ëÉW= p_ = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 62
  72. 72. CHAPTER 3. GEOMETRY = = Figure 43. = N (~ + Å ) QÜO + ÄO + Ä ÜO + (~ − Å )O = O = 305. p_ = ~Ä = = 306. p = p_ + p i = = ÄÜ (O~ + Å ) = 307. s = S = = = 304. p i = 3.29 Platonic Solids = bÇÖÉW=~= o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 63
  73. 73. CHAPTER 3. GEOMETRY 308. cáîÉ=mä~íçåáÅ=pçäáÇë= qÜÉ= éä~íçåáÅ= ëçäáÇë= ~êÉ= ÅçåîÉñ= éçäóÜÉÇê~= ïáíÜ= Éèìáî~äÉåí= Ñ~ÅÉë=ÅçãéçëÉÇ=çÑ=ÅçåÖêìÉåí=ÅçåîÉñ=êÉÖìä~ê=éçäóÖçåëK== = kìãÄÉê= kìãÄÉê= pÉÅíáçå= pçäáÇ= kìãÄÉê= çÑ=sÉêíáÅÉë çÑ=bÇÖÉë= çÑ=c~ÅÉë= qÉíê~ÜÉÇêçå== Q= S= Q= PKOR= `ìÄÉ= U= NO= S= PKOO= lÅí~ÜÉÇêçå= S= NO= U= PKOT= fÅçë~ÜÉÇêçå= NO= PM= OM= PKOT= açÇÉÅ~ÜÉÇêçå= OM= PM= NO= PKOT= = = Octahedron = = = Figure 44. = 309. ê = ~ S = S = 310. o = ~ O = O = 64
  74. 74. CHAPTER 3. GEOMETRY 311. p = O~ O P = = ~P O 312. s = = P = = Icosahedron = = = Figure 45. = 313. ê = ( = 314. o = ) ~ P P+ R = NO ( ) ~ O R+ R = Q = 315. p = R~ O P = = R~ P P + R 316. s = = NO = = ( ) 65
  75. 75. CHAPTER 3. GEOMETRY Dodecahedron = = = Figure 46. 317. ê = ( ~ NM OR + NN R = O = 318. o = ) = ( ) ~ P N+ R = Q = ( ) 319. p = P~ O R R + O R = = ~ P NR + T R 320. s = = Q = = = ( ) 3.30 Right Circular Cylinder = o~Çáìë=çÑ=Ä~ëÉW=o= aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç= 66
  76. 76. CHAPTER 3. GEOMETRY eÉáÖÜíW=e= i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i = ^êÉ~=çÑ=Ä~ëÉW= p_ = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = ===== = Figure 47. = 321. p i = Oπoe = = Ç  322. p = p i + Op_ = Oπo(e + o ) = πÇ e +  = O  = 323. s = p_ e = πo O e = = = = 67
  77. 77. CHAPTER 3. GEOMETRY 3.31 Right Circular Cylinder with an Oblique Plane Face = o~Çáìë=çÑ=Ä~ëÉW=o= qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= ÜN = qÜÉ=ëÜçêíÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= Ü O = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i = ^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= p_ = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 48. = 324. p i = πo(ÜN + Ü O ) = = O  Ü − ÜO  325. p_ = πo + πo o +  N  =  O  = O O 68
  78. 78. CHAPTER 3. GEOMETRY O   ÜN − Ü O   O 326. p = p i + p_ = πo ÜN + Ü O + o + o +   =  O      = πo O (ÜN + ÜO ) = 327. s = O = = = 3.32 Right Circular Cone o~Çáìë=çÑ=Ä~ëÉW=o= aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç= eÉáÖÜíW=e= pä~åí=ÜÉáÖÜíW=ã= i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= pi = ^êÉ~=çÑ=Ä~ëÉW= p_ = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = = Figure 49. 69
  79. 79. CHAPTER 3. GEOMETRY 328. e = ã O − o O = = πãÇ 329. p i = πoã = = O = 330. p_ = πo O = = N  Ç 331. p = p i + p_ = πo (ã + o ) = πÇ ã +  = O  O = N N 332. s = p_ e = πo O e = P P = = = 3.33 Frustum of a Right Circular Cone = o~Çáìë=çÑ=Ä~ëÉëW=oI=ê= eÉáÖÜíW=e= pä~åí=ÜÉáÖÜíW=ã= pÅ~äÉ=Ñ~ÅíçêW=â= ^êÉ~=çÑ=Ä~ëÉëW= pN I= pO = i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 70
  80. 80. CHAPTER 3. GEOMETRY = = Figure 50. = 333. e = ã O − (o − ê ) = = o 334. =â= ê = p oO 335. O = O = â O = pN ê = 336. p i = πã(o + ê ) = = 337. p = pN + pO + p i = π o O + ê O + ã(o + ê ) = = Ü 338. s = pN + pNpO + pO = P = O ÜpN  o  o   ÜpN 339. s = N+ â + âO = N + +    = P  ê ê  P   = = = O [ ( ] ) [ 71 ]
  81. 81. CHAPTER 3. GEOMETRY 3.34 Sphere = o~ÇáìëW=o= aá~ãÉíÉêW=Ç= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = = Figure 51. = 340. p = Qπo O = = Q N N 341. s = πo P e = πÇ P = po = P S P = = = 3.35 Spherical Cap o~Çáìë=çÑ=ëéÜÉêÉW=o= o~Çáìë=çÑ=Ä~ëÉW=ê= eÉáÖÜíW=Ü= ^êÉ~=çÑ=éä~åÉ=Ñ~ÅÉW= p_ = ^êÉ~=çÑ=ëéÜÉêáÅ~ä=Å~éW= p` = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= 72
  82. 82. CHAPTER 3. GEOMETRY = = Figure 52. = 342. o = ê O + ÜO = OÜ = 343. p_ = πê O = = 344. p` = π(Ü O + ê O )= = 345. p = p_ + p` = π(Ü O + Oê O ) = π(OoÜ + ê O ) = = π π 346. s = Ü O (Po − Ü ) = Ü(Pê O + Ü O ) = S S = = = 3.36 Spherical Sector = o~Çáìë=çÑ=ëéÜÉêÉW=o= o~Çáìë=çÑ=Ä~ëÉ=çÑ=ëéÜÉêáÅ~ä=Å~éW=ê= eÉáÖÜíW=Ü= qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = 73
  83. 83. CHAPTER 3. GEOMETRY ====== = === = Figure 53. = 347. p = πo(OÜ + ê ) = = O 348. s = πo O Ü = P = kçíÉW= qÜÉ= ÖáîÉå= Ñçêãìä~ë= ~êÉ= ÅçêêÉÅí= ÄçíÜ= Ñçê= ±çéÉå≤= ~åÇ= ±ÅäçëÉÇ≤=ëéÜÉêáÅ~ä=ëÉÅíçêK= = = = 3.37 Spherical Segment = o~Çáìë=çÑ=ëéÜÉêÉW=o= o~Çáìë=çÑ=Ä~ëÉëW= êN I= êO = eÉáÖÜíW=Ü= ^êÉ~=çÑ=ëéÜÉêáÅ~ä=ëìêÑ~ÅÉW= pp = ^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= pN I= pO = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = 74
  84. 84. CHAPTER 3. GEOMETRY = ===== = Figure 54. = 349. pp = OπoÜ = = 350. p = pp + pN + pO = π(OoÜ + êNO + êOO ) = = N 351. s = πÜ(PêNO + PêOO + Ü O )= S = = = 3.38 Spherical Wedge = o~ÇáìëW=o= aáÜÉÇê~ä=~åÖäÉ=áå=ÇÉÖêÉÉëW=ñ= aáÜÉÇê~ä=~åÖäÉ=áå=ê~Çá~åëW= α = ^êÉ~=çÑ=ëéÜÉêáÅ~ä=äìåÉW= p i = qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = 75
  85. 85. CHAPTER 3. GEOMETRY = = Figure 55. = 352. p i = πo O α = Oo O ñ = VM = 353. p = πo O + πo O α = πo O + Oo O ñ = VM = 354. s = πoP O α = oP ñ = OTM P = = = 3.39 Ellipsoid = pÉãá-~ñÉëW=~I=ÄI=Å= sçäìãÉW=s= 76
  86. 86. CHAPTER 3. GEOMETRY = ======= = Figure 56. = Q 355. s = π~ÄÅ = P = = = Prolate Spheroid = pÉãá-~ñÉëW=~I=ÄI=Ä=E ~ > Ä F= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = = ~ ~êÅëáå É   356. p = OπÄ Ä +  I== É   ïÜÉêÉ= É = ~ O − ÄO K= ~ = Q 357. s = πÄO~ = P = 77
  87. 87. CHAPTER 3. GEOMETRY Oblate Spheroid = pÉãá-~ñÉëW=~I=ÄI=Ä=E ~ < Ä F= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = =   ÄÉ   ~ ~êÅëáåÜ      ~   I== 358. p = OπÄ Ä +   ÄÉ L ~     ïÜÉêÉ= É = ÄO − ~ O K= Ä = Q 359. s = πÄO~ = P = = = 3.40 Circular Torus = j~àçê=ê~ÇáìëW=o= jáåçê=ê~ÇáìëW=ê= pìêÑ~ÅÉ=~êÉ~W=p= sçäìãÉW=s= = 78
  88. 88. CHAPTER 3. GEOMETRY == Picture 57. = 360. p = QπOoê = = 361. s = OπOoê O = = = 79 =
  89. 89. Chapter 4 Trigonometry = = = = ^åÖäÉëW= α I= β = oÉ~ä=åìãÄÉêë=EÅççêÇáå~íÉë=çÑ=~=éçáåíFW=ñI=ó== tÜçäÉ=åìãÄÉêW=â= = = 4.1 Radian and Degree Measures of Angles = 362. N ê~Ç = = 363. N° = = 364. N D = = 365. N ? = = 366. = = = = = NUM° ≈ RT°NT DQR? = π π ê~Ç ≈ MKMNTQRP ê~Ç = NUM π ê~Ç ≈ MKMMMOVN ê~Ç = NUM ⋅ SM π ê~Ç ≈ MKMMMMMR ê~Ç = NUM ⋅ PSMM ^åÖäÉ= EÇÉÖêÉÉëF= ^åÖäÉ= Eê~Çá~åëF= M= PM= QR= SM= VM= NUM= OTM= PSM= M= π = S π = Q 80 π = P π = O π= Pπ = O Oπ =
  90. 90. CHAPTER 4. TRIGONOMETRY 4.2 Definitions and Graphs of Trigonometric Functions = = = = Figure 58. = 367. ëáå α = ó = ê = 368. Åçë α = ñ = ê = 369. í~å α = ó = ñ = 370. Åçí α = ñ = ó = 81
  91. 91. CHAPTER 4. TRIGONOMETRY 371. ëÉÅ α = ê = ñ = 372. ÅçëÉÅ α = ê = ó = 373. páåÉ=cìåÅíáçå= ó = ëáå ñ I= − N ≤ ëáå ñ ≤ N K= = = Figure 59. = 374. `çëáåÉ=cìåÅíáçå== ó = Åçë ñ I= − N ≤ Åçë ñ ≤ N K= 82
  92. 92. CHAPTER 4. TRIGONOMETRY = = Figure 60. = 375. q~åÖÉåí=cìåÅíáçå= π ó = í~å ñ I= ñ ≠ (Oâ + N) I= − ∞ ≤ í~å ñ ≤ ∞K = O = = = Figure 61. = 83
  93. 93. CHAPTER 4. TRIGONOMETRY 376. `çí~åÖÉåí=cìåÅíáçå== ó = Åçí ñ I= ñ ≠ âπ I== − ∞ ≤ Åçí ñ ≤ ∞ K= = = = Figure 62. = 377. pÉÅ~åí=cìåÅíáçå= π ó = ëÉÅ ñ I= ñ ≠ (Oâ + N) K= O == 84
  94. 94. CHAPTER 4. TRIGONOMETRY = = Figure 63. = 378. `çëÉÅ~åí=cìåÅíáçå== ó = Åçë ÉÅ ñ I= ñ ≠ âπ K= = Figure 64. 85
  95. 95. CHAPTER 4. TRIGONOMETRY 4.3. Signs of Trigonometric Functions 379. = = = = 380. = nì~Çê~åí= = f= ff= fff= fs= páå α= H= H= = = `çë α= H= = = H= q~å α= H= = H= = `çí α= H= = H= = pÉÅ α= H= = = H= `çëÉÅ= α= H= H= = = = = Figure 65. = = = = = = = = = = 86
  96. 96. CHAPTER 4. TRIGONOMETRY 4.4 Trigonometric Functions of Common Angles 381. = α° = α ê~Ç = M= M= π = PM= S π = QR= Q π = SM= P π = VM= O Oπ = NOM= P NUM= π= Pπ = OTM= O PSM= Oπ = = = = = = = = = = = = = = O = O P = O Åçë α = N= P = O O = O N = O N= M= P = O M= N − = O − N= − N= M= ëáå α = M= N = O í~å α = Åçí α M= ∞= N = P= P ëÉÅ α = N= O = P ÅçëÉÅ α = ∞= O= N= N= P= N = P O= O = P M= ∞= N= ∞= O= O= M= N P ∞= − N= O = P ∞= M= ∞= M= ∞= − N= N= M= ∞= N= ∞= − P= 87 − −O=
  97. 97. CHAPTER 4. TRIGONOMETRY 382. = α° = α ê~Ç = π = NR= NO ëáå α = Åçë α = í~å α = Åçí α = S− O = Q S+ O = Q O− P = O+ P = R−O R = R R+O R = NU= π = NM R −N = Q NM + O R Q PS= π = R NM − O R Q R +N = Q RQ= Pπ = NM R +N = Q NM − O R Q TO= Oπ = R NM + O R Q R −N = Q TR= Rπ = NO S+ O = Q S− O = Q = = = 4.5 Most Important Formulas = 383. ëáå O α + Åçë O α = N = = 384. ëÉÅ O α − í~å O α = N = = 385. ÅëÅ O α − Åçí O α = N = = ëáå α = 386. í~å α = Åçë α 88 NM − O R R +N R +N NM − O R R +N NM − O R = NM − O R R +N = R+O R = R−O R R = O+ P = O− P =
  98. 98. CHAPTER 4. TRIGONOMETRY 387. Åçí α = Åçë α = ëáå α = 388. í~å α ⋅ Åçí α = N = = N 389. ëÉÅ α = = Åçë α = N 390. ÅçëÉÅ α = = ëáå α = = = 4.6 Reduction Formulas = 391. = = = = = = = β= −α= VM° − α = VM° + α = NUM° − α NUM° + α OTM° − α OTM° + α PSM° − α = PSM° + α ëáå β = − ëáå α = + Åçë α = + Åçë α = + ëáå α = − ëáå α = − Åçë α = − Åçë α = − ëáå α = + ëáå α = 89 Åçë β = + Åçë α = + ëáå α = − ëáå α = − Åçë α = − Åçë α = − ëáå α = + ëáå α = + Åçë α = + Åçë α = í~å β = − í~å α = + Åçí α = − Åçí α = − í~å α = + í~å α = + Åçí α = − Åçí α = − í~å α = + í~å α = Åçí β = − Åçí α = + í~å α = − í~å α = − Åçí α = + Åçí α = + í~å α = − í~å α = − Åçí α = + Åçí α =
  99. 99. CHAPTER 4. TRIGONOMETRY 4.7 Periodicity of Trigonometric Functions = 392. ëáå(α ± Oπå ) = ëáå α I=éÉêáçÇ= Oπ =çê= PSM° K= = 393. Åçë(α ± Oπå ) = Åçë α I=éÉêáçÇ= Oπ =çê= PSM° K= = 394. í~å(α ± πå ) = í~å α I=éÉêáçÇ= π =çê= NUM° K= = 395. Åçí(α ± πå ) = Åçí α I=éÉêáçÇ= π =çê= NUM° K= = = = 4.8 Relations between Trigonometric Functions = 396. ëáå α = ± N − Åçë O α = ± α O = = α N + í~å O O N (N − Åçë Oα ) = O Åçë O  α − π  − N =   O  O Q O í~å = = 397. Åçë α = ± N − ëáå O α = ± α O= = α N + í~å O O N (N + Åçë Oα ) = O Åçë O α − N = O O N − í~å O = = 398. í~å α = ëáå α ëáå Oα N − Åçë Oα = ± ëÉÅ O α − N = = = Åçë α N + Åçë Oα ëáå Oα 90
  100. 100. CHAPTER 4. TRIGONOMETRY α N − Åçë Oα O = =± = N + Åçë Oα O α N + í~å O O í~å = = Åçë α N + Åçë Oα ëáå Oα = ± ÅëÅ O α − N = = = ëáå α ëáå Oα N − Åçë Oα α N − í~å O N + Åçë Oα O= = = =± α N − Åçë Oα O í~å O 399. Åçí α = = α N O= 400. ëÉÅ α = = ± N + í~å O α = α Åçë α N − í~å O O = α N + í~å O N O= 401. ÅëÅ α = = ± N + Åçí O α = α ëáå α O í~å O = = = N + í~å O 4.9 Addition and Subtraction Formulas = 402. ëáå(α + β) = ëáå α Åçë β + ëáå β Åçë α = = 403. ëáå(α − ó ) = ëáå α Åçë β − ëáå β Åçë α = = 404. Åçë(α + β ) = Åçë α Åçë β − ëáå α ëáå β = = 405. Åçë(α − β ) = Åçë α Åçë β + ëáå α ëáå β = 91
  101. 101. CHAPTER 4. TRIGONOMETRY 406. í~å(α + β ) = = 407. í~å(α − β ) = = 408. Åçí(α + β) = = 409. Åçí(α − β) = í~å α + í~å β = N − í~å α í~å β í~å α − í~å β = N + í~å α í~å β N − í~å α í~å β = í~å α + í~å β N + í~å α í~å β = í~å α − í~å β = = = 4.10 Double Angle Formulas = 410. ëáå Oα = O ëáå α ⋅ Åçë α = = 411. Åçë Oα = Åçë O α − ëáå O α = N − O ëáå O α = O Åçë O α − N = = O í~å α O 412. í~å Oα = = = O N − í~å α Åçí α − í~å α = Åçí O α − N Åçí α − í~å α = = 413. Åçí Oα = O Åçí α O = = = = = = 92
  102. 102. CHAPTER 4. TRIGONOMETRY 4.11 Multiple Angle Formulas = 414. ëáå Pα = P ëáå α − Q ëáå P α = P Åçë O α ⋅ ëáå α − ëáåP α = = 415. ëáå Qα = Q ëáå α ⋅ Åçë α − U ëáå P α ⋅ Åçë α = = 416. ëáå Rα = R ëáå α − OM ëáå P α + NS ëáå R α = = 417. Åçë Pα = Q ÅçëP α − P Åçë α = Åçë P α − P Åçë α ⋅ ëáå O α = = 418. Åçë Qα = U Åçë Q α − U Åçë O α + N = = 419. Åçë Rα = NS Åçë R α − OM Åçë P α + R Åçë α = = P í~å α − í~å P α 420. í~å Pα = = N − P í~å O α = Q í~å α − Q í~å P α = 421. í~å Qα = N − S í~å O α + í~å Q α = í~å R α − NM í~å P α + R í~å α = 422. í~å Rα = N − NM í~å O α + R í~å Q α = Åçí P α − P Åçí α 423. Åçí Pα = = P Åçí O α − N = N − S í~å O α + í~å Q α == 424. Åçí Qα = Q í~å α − Q í~å P α = 93
  103. 103. CHAPTER 4. TRIGONOMETRY 425. Åçí Rα = N − NM í~å O α + R í~å Q α = í~å R α − NM í~å P α + R í~å α = = = 4.12 Half Angle Formulas = 426. ëáå α N − Åçë α = =± O O = 427. Åçë α N + Åçë α = =± O O = 428. í~å α N − Åçë α ëáå α N − Åçë α =± = = = ÅëÅ α − Åçí α = O N + Åçë α N + Åçë α ëáå α = 429. Åçí α N + Åçë α ëáå α N + Åçë α =± = = = ÅëÅ α + Åçí α = O N − Åçë α N − Åçë α ëáå α = = = 4.13 Half Angle Tangent Identities = α O = 430. ëáå α = α N + í~å O O = O í~å 94
  104. 104. CHAPTER 4. TRIGONOMETRY α O= 431. Åçë α = O α N + í~å O = α O í~å O = 432. í~å α = α N − í~å O O = α N − í~å O O= 433. Åçí α = α O í~å O = = = N − í~å O 4.14 Transforming of Trigonometric Expressions to Product = 434. ëáå α + ëáå β = O ëáå = 435. ëáå α − ëáå β = O Åçë α+β α −β = Åçë O O α +β α −β = ëáå O O = 436. Åçë α + Åçë β = O Åçë α+β α −β = Åçë O O = 437. Åçë α − Åçë β = −O ëáå α +β α −β = ëáå O O = 95
  105. 105. CHAPTER 4. TRIGONOMETRY 438. í~å α + í~å β = = 439. í~å α − í~å β = = 440. Åçí α + Åçí β = = 441. Åçí α − Åçí β = ëáå(α + β ) = Åçë α ⋅ Åçë β ëáå(α − β ) = Åçë α ⋅ Åçë β ëáå(β + α ) = ëáå α ⋅ ëáå β ëáå(β − α ) = ëáå α ⋅ ëáå β = π  π  442. Åçë α + ëáå α = O Åçë − α  = O ëáå + α  = Q  Q  = π  π  443. Åçë α − ëáå α = O ëáå − α  = O Åçë + α  = Q  Q  = Åçë(α − β) = 444. í~å α + Åçí β = Åçë α ⋅ ëáå β = Åçë(α + β ) = 445. í~å α − Åçí β = − Åçë α ⋅ ëáå β = α 446. N + Åçë α = O Åçë O = O = α 447. N − Åçë α = O ëáå O = O = 96
  106. 106. CHAPTER 4. TRIGONOMETRY π α 448. N + ëáå α = O Åçë O  −  = Q O = π α 449. N − ëáå α = O ëáå O  −  = Q O = = = 4.15 Transforming of Trigonometric Expressions to Sum = 450. ëáå α ⋅ ëáå β = Åçë(α − β) − Åçë(α + β ) = O = 451. Åçë α ⋅ Åçë β = = 452. ëáå α ⋅ Åçë β = = 453. í~å α ⋅ í~å β = = 454. Åçí α ⋅ Åçí β = = 455. í~å α ⋅ Åçí β = Åçë(α − β ) + Åçë(α + β ) = O ëáå(α − β ) + ëáå(α + β ) = O í~å α + í~å β = Åçí α + Åçí β Åçí α + Åçí β = í~å α + í~å β í~å α + Åçí β = Åçí α + í~å β = = = 97
  107. 107. CHAPTER 4. TRIGONOMETRY 4.16 Powers of Trigonometric Functions = 456. ëáå O α = = 457. ëáå P α = = 458. ëáå Q α = = 459. ëáå R α = = 460. ëáå S α = = 461. Åçë O α = = 462. Åçë P α = = 463. Åçë Q α = = 464. Åçë R α = = 465. Åçë S α = N − Åçë Oα = O P ëáå α − ëáå Pα = Q Åçë Qα − Q Åçë Oα + P = U NM ëáå α − R ëáå Pα + ëáå Rα = NS NM − NR Åçë Oα + S Åçë Qα − Åçë Sα = PO N + Åçë Oα = O P Åçë α + Åçë Pα = Q Åçë Qα + Q Åçë Oα + P = U NM Åçë α + R ëáå Pα + Åçë Rα = NS NM + NR Åçë Oα + S Åçë Qα + Åçë Sα = PO = 98
  108. 108. CHAPTER 4. TRIGONOMETRY 4.17 Graphs of Inverse Trigonometric Functions = 466. fåîÉêëÉ=páåÉ=cìåÅíáçå== ó = ~êÅëáå ñ I= − N ≤ ñ ≤ N I= − π π ≤ ~êÅëáå ñ ≤ K= O O = = = Figure 66. = 467. fåîÉêëÉ=`çëáåÉ=cìåÅíáçå== ó = ~êÅÅçë ñ I= − N ≤ ñ ≤ N I= M ≤ ~êÅÅçë ñ ≤ π K= = 99
  109. 109. CHAPTER 4. TRIGONOMETRY = = Figure 67. = 468. fåîÉêëÉ=q~åÖÉåí=cìåÅíáçå== ó = ~êÅí~å ñ I= − ∞ ≤ ñ ≤ ∞ I= − π π < ~êÅí~å ñ < K= O O = = = ===== Figure 68. 100
  110. 110. CHAPTER 4. TRIGONOMETRY 469. fåîÉêëÉ=`çí~åÖÉåí=cìåÅíáçå== ó = ~êÅ Åçí ñ I= − ∞ ≤ ñ ≤ ∞ I= M < ~êÅ Åçí ñ < π K= ===== = Figure 69. = 470. fåîÉêëÉ=pÉÅ~åí=cìåÅíáçå==  π  π  ó = ~êÅëÉÅ=ñ I ñ ∈ (− ∞I − N] ∪ [NI ∞ )I ~êÅ ëÉÅ ñ ∈ MI  ∪  I πK  O  O  = Figure 70. 101
  111. 111. CHAPTER 4. TRIGONOMETRY 471. fåîÉêëÉ=`çëÉÅ~åí=cìåÅíáçå==  π   π ó = ~êÅÅëÅ ñ I ñ ∈ (− ∞I − N] ∪ [NI ∞ )I ~êÅ ÅëÅ ñ ∈ − I M  ∪  MI K  O   O = = Figure 71. = = 4.18 Principal Values of Inverse Trigonometric Functions 472. ñ= M= N = O PM° = SM° = O − O ~êÅëáå ñ = M° = ~êÅÅçë ñ = VM° N − ñ= O − PM° ~êÅëáå ñ = − QR° = NOM° ~êÅÅçë ñ = NPR° = = O = O QR° = QR° = P − O P O SM° PM° VM° M° = − N= = − VM° = NUM° NRM° = = − SM° 102 N= = =
  112. 112. CHAPTER 4. TRIGONOMETRY 473. ñ= M= P P N= ~êÅí~å ñ = M° = PM° QR° SM° ~êÅ Åçí ñ = VM° SM° QR° PM° P= − P P 4.19 Relations between Inverse Trigonometric Functions = 474. ~êÅëáå(− ñ ) = − ~êÅëáå ñ = = π 475. ~êÅëáå ñ = − ~êÅÅçë ñ = O = 476. ~êÅëáå ñ = ~êÅÅçë N − ñ O I= M ≤ ñ ≤ N K= = 477. ~êÅëáå ñ = − ~êÅÅçë N − ñ O I= − N ≤ ñ ≤ M K= = ñ O I= ñ < N K= 478. ~êÅëáå ñ = ~êÅí~å O N− ñ = N− ñO I= M < ñ ≤ N K= ñ = 480. ~êÅëáå ñ = ~êÅ Åçí N− ñO − π I= − N ≤ ñ < M K= ñ = 481. ~êÅÅçë(− ñ ) = π − ~êÅÅçë ñ = 103 − P= − QR° − SM° = = NPR° NOM° = NRM° = = − PM° = = = 479. ~êÅëáå ñ = ~êÅ Åçí − N=
  113. 113. CHAPTER 4. TRIGONOMETRY 482. ~êÅÅçë ñ = π − ~êÅëáå ñ = O = 483. ~êÅÅçë ñ = ~êÅëáå N − ñ O I= M ≤ ñ ≤ N K= = 484. ~êÅÅçë ñ = π − ~êÅëáå N − ñ O I= − N ≤ ñ ≤ M K= = 485. ~êÅÅçë ñ = ~êÅí~å N− ñO I= M < ñ ≤ N K= ñ = N− ñO I= − N ≤ ñ < M K= ñ 486. ~êÅÅçë ñ = π + ~êÅí~å = 487. ~êÅÅçë ñ = ~êÅ Åçí ñ N− ñO I= − N ≤ ñ ≤ N K= = 488. ~êÅí~å(− ñ ) = − ~êÅí~å ñ = = π 489. ~êÅí~å ñ = − ~êÅ Åçí ñ = O = ñ = 490. ~êÅí~å ñ = ~êÅëáå N+ ñO = N I= ñ ≥ M K= 491. ~êÅí~å ñ = ~êÅÅçë N+ ñO = N I= ñ ≤ M K= 492. ~êÅí~å ñ = − ~êÅÅçë N+ ñO = 104
  114. 114. CHAPTER 4. TRIGONOMETRY 493. ~êÅí~å ñ = π N − ~êÅí~å I= ñ > M K= O ñ = π N 494. ~êÅí~å ñ = − − ~êÅí~å I= ñ < M K= O ñ = N 495. ~êÅí~å ñ = ~êÅ Åçí I= ñ > M K= ñ = N 496. ~êÅí~å ñ = ~êÅ Åçí − π I= ñ < M K= ñ = 497. ~êÅ Åçí(− ñ ) = π − ~êÅ Åçí ñ = = π 498. ~êÅ Åçí ñ = − ~êÅí~å ñ = O = N I= ñ > M K= 499. ~êÅ Åçí ñ = ~êÅëáå N+ ñO = N I= ñ < M K= 500. ~êÅ Åçí ñ = π − ~êÅëáå N+ ñO = ñ = 501. ~êÅ Åçí ñ = ~êÅÅçë N+ ñO = N 502. ~êÅ Åçí ñ = ~êÅí~å I= ñ > M K= ñ = N 503. ~êÅ Åçí ñ = π + ~êÅí~å I= ñ < M K= ñ = = 105
  115. 115. CHAPTER 4. TRIGONOMETRY 4.20 Trigonometric Equations 504. 505. 506. 507. = tÜçäÉ=åìãÄÉêW=å= = = å ëáå ñ = ~ I= ñ = (− N) ~êÅëáå ~ + πå = = Åçë ñ = ~ I= ñ = ± ~êÅÅçë ~ + Oπå = = í~å ñ = ~ I= ñ = ~êÅí~å ~ + πå = = Åçí ñ = ~ I= ñ = ~êÅ Åçí ~ + πå = = = = 4.21 Relations to Hyperbolic Functions 508. 509. 510. 511. 512. = fã~Öáå~êó=ìåáíW=á= = = ëáå(áñ ) = á ëáåÜ ñ = = í~å(áñ ) = á í~åÜ ñ = = Åçí(áñ ) = −á ÅçíÜ ñ = = ëÉÅ(áñ ) = ëÉÅÜ ñ = = ÅëÅ(áñ ) = −á ÅëÅÜ ñ = = = = 106
  116. 116. Chapter 5 Matrices and Determinants = = = = j~íêáÅÉëW=^I=_I=`= bäÉãÉåíë=çÑ=~=ã~íêáñW= ~ á I= Äá I= ~ áà I= Äáà I= Å áà = aÉíÉêãáå~åí=çÑ=~=ã~íêáñW= ÇÉí ^ = jáåçê=çÑ=~å=ÉäÉãÉåí= ~ áà W= j áà = `çÑ~Åíçê=çÑ=~å=ÉäÉãÉåí= ~ áà W= ` áà = ú qê~åëéçëÉ=çÑ=~=ã~íêáñW= ^ q I= ^ = ^Çàçáåí=çÑ=~=ã~íêáñW= ~Çà ^ = qê~ÅÉ=çÑ=~=ã~íêáñW= íê ^ = fåîÉêëÉ=çÑ=~=ã~íêáñW= ^ −N = oÉ~ä=åìãÄÉêW=â= oÉ~ä=î~êá~ÄäÉëW= ñ á = k~íìê~ä=åìãÄÉêëW=ãI=å=== = = 5.1 Determinants = 513. pÉÅçåÇ=lêÇÉê=aÉíÉêãáå~åí= ~ ÄN ÇÉí ^ = N = ~ N Ä O − ~ O ÄN = ~ O ÄO = = = = = 107
  117. 117. CHAPTER 5. MATRICES AND DETERMINANTS 514. qÜáêÇ=lêÇÉê=aÉíÉêãáå~åí= ~NN ~NO ~NP ÇÉí ^ = ~ ON ~ OO ~ OP = ~NN~ OO~ PP + ~NO~ OP~ PN + ~ NP~ ON~ PO − = ~ PN ~ PO ~ PP − ~NN~ OP~ PO − ~NO~ ON~ PP − ~ NP~ OO~ PN = = 515. p~êêìë=oìäÉ=E^êêçï=oìäÉF= = = Figure 72. = 516. k-íÜ=lêÇÉê=aÉíÉêãáå~åí= ~NN ~NO K ~Nà ~ ON ~ OO K ~ O à K K K K ÇÉí ^ = ~ áN ~ á O K ~ áà K K K K ~ åN ~ å O K ~ åà K ~Nå K ~ Oå K K K ~ áå = K K K ~ åå = 517. jáåçê= qÜÉ=ãáåçê= j áà =~ëëçÅá~íÉÇ=ïáíÜ=íÜÉ=ÉäÉãÉåí= ~ áà =çÑ=å-íÜ=çêÇÉê= ã~íêáñ= ^= áë= íÜÉ= (å − N) -íÜ= çêÇÉê= ÇÉíÉêãáå~åí= ÇÉêáîÉÇ= Ñêçã= íÜÉ=ã~íêáñ=^=Äó=ÇÉäÉíáçå=çÑ=áíë=á-íÜ=êçï=~åÇ=à-íÜ=ÅçäìãåK=== = 108
  118. 118. CHAPTER 5. MATRICES AND DETERMINANTS 518. `çÑ~Åíçê= á +à ` áà = (− N) j áà = = 519. i~éä~ÅÉ=bñé~åëáçå=çÑ=å-íÜ=lêÇÉê=aÉíÉêãáå~åí= i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=á-íÜ=êçï= å ÇÉí ^ = ∑ ~ áà` áà I= á = NI OI KI å K= à=N i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=à-íÜ=Åçäìãå= å ÇÉí ^ = ∑ ~ áà` áà I= à = NI OI KI å K== á =N = = = 5.2 Properties of Determinants = 520. qÜÉ==î~äìÉ==çÑ=~=ÇÉíÉêãáå~åí=êÉã~áåë==ìåÅÜ~åÖÉÇ=áÑ=êçïë=~êÉ= ÅÜ~åÖÉÇ=íç=Åçäìãåë=~åÇ=Åçäìãåë=íç=êçïëK= ~ ~ O ~N ÄN = == = N ÄN ÄO ~ O ÄO = 521. fÑ=íïç==êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áåíÉêÅÜ~åÖÉÇI=íÜÉ=ëáÖå=çÑ= íÜÉ=ÇÉíÉêãáå~åí=áë=ÅÜ~åÖÉÇK= ~N ÄN ~ ÄO =− O = ~ O ÄO ~N ÄN = 522. fÑ=íïç=êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áÇÉåíáÅ~äI=íÜÉ=î~äìÉ=çÑ=íÜÉ= ÇÉíÉêãáå~åí=áë=òÉêçK= ~N ~N = M= ~O ~O = 109
  119. 119. CHAPTER 5. MATRICES AND DETERMINANTS 523. fÑ==íÜÉ===ÉäÉãÉåíë==çÑ==~åó=êçï==Eçê=ÅçäìãåF=~êÉ=ãìäíáéäáÉÇ=Äó===== ~==Åçããçå==Ñ~ÅíçêI==íÜÉ==ÇÉíÉêãáå~åí==áë==ãìäíáéäáÉÇ==Äó==íÜ~í= Ñ~ÅíçêK= â~ N âÄN ~ ÄN =â N = ~ O ÄO ~ O ÄO = 524. fÑ==íÜÉ==ÉäÉãÉåíë==çÑ==~åó==êçï==Eçê==ÅçäìãåF=~êÉ=áåÅêÉ~ëÉÇ=Eçê= ÇÉÅêÉ~ëÉÇFÄó=Éèì~ä=ãìäíáéäÉë=çÑ=íÜÉ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë= çÑ=~åó=çíÜÉê=êçï==Eçê=ÅçäìãåFI==íÜÉ=î~äìÉ=çÑ=íÜÉ=ÇÉíÉêãáå~åí= áë=ìåÅÜ~åÖÉÇK= ~N + âÄN ÄN ~N ÄN = = ~ O + âÄO ÄO ~ O ÄO = = = 5.3 Matrices = 525. aÉÑáåáíáçå= ^å= ã × å =ã~íêáñ=^=áë=~=êÉÅí~åÖìä~ê=~êê~ó=çÑ=ÉäÉãÉåíë=EåìãÄÉêë=çê=ÑìåÅíáçåëF=ïáíÜ=ã=êçïë=~åÇ=å=ÅçäìãåëK==  ~ NN ~ NO K ~ Nå  ~ ~ OO K ~ Oå   ==  ON ^ = ~ áà =  M M M    ~ ãN ~ ã O K ~ ãå  = 526. pèì~êÉ=ã~íêáñ=áë=~=ã~íêáñ=çÑ=çêÇÉê= å× å K== = 527. ^=ëèì~êÉ=ã~íêáñ== ~ áà ==áë==ëóããÉíêáÅ==áÑ== ~ áà = ~ àá I==áKÉK==áí==áë= [ ] [ ] ëóããÉíêáÅ=~Äçìí=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK== = 528. ^=ëèì~êÉ=ã~íêáñ= ~ áà =áë=ëâÉï-ëóããÉíêáÅ=áÑ= ~ áà = −~ àá K== = [ ] 110
  120. 120. CHAPTER 5. MATRICES AND DETERMINANTS 529. aá~Öçå~ä=ã~íêáñ==áë==~=ëèì~êÉ==ã~íêáñ=ïáíÜ=~ää==ÉäÉãÉåíë==òÉêç= ÉñÅÉéí=íÜçëÉ=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK== = 530. råáí=ã~íêáñ==áë==~=Çá~Öçå~ä==ã~íêáñ==áå=ïÜáÅÜ=íÜÉ=ÉäÉãÉåíë=çå= íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~ä=~êÉ=~ää=ìåáíóK=qÜÉ=ìåáí=ã~íêáñ=áë=========== ÇÉåçíÉÇ=Äó=fK== = 531. ^=åìää=ã~íêáñ=áë=çåÉ=ïÜçëÉ=ÉäÉãÉåíë=~êÉ=~ää=òÉêçK= = = = 5.4 Operations with Matrices = 532. qïç=ã~íêáÅÉë=^=~åÇ=_=~êÉ=Éèì~ä=áÑI=~åÇ=çåäó=áÑI=íÜÉó=~êÉ=ÄçíÜ= çÑ==íÜÉ==ë~ãÉ==ëÜ~éÉ== ã × å ==~åÇ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=~êÉ= Éèì~äK= = 533. qïç=ã~íêáÅÉë==^=~åÇ=_==Å~å=ÄÉ=~ÇÇÉÇ=Eçê=ëìÄíê~ÅíÉÇF=çÑI=~åÇ= çåäó=áÑI=íÜÉó=Ü~îÉ=íÜÉ=ë~ãÉ=ëÜ~éÉ= ã × å K=fÑ==  ~NN ~NO K ~Nå  ~ ~ OO K ~ Oå   I== ^ = ~ áà =  ON  M M M    ~ ãN ~ ã O K ~ ãå   ÄNN ÄNO K ÄNå  Ä ÄOO K ÄOå   I== _ = Äáà =  ON  M M M    ÄãN Äã O K Äãå  = = = = = [ ] [ ] 111
  121. 121. CHAPTER 5. MATRICES AND DETERMINANTS íÜÉå== ~NO + ÄNO K ~Nå + ÄNå   ~NN + ÄNN ~ +Ä ~ OO + ÄOO K ~ Oå + ÄOå   K=  ON ON ^+_=   M M M   ~ ãN + ÄãN ~ ã O + Äã O K ~ ãå + Äãå  = 534. fÑ=â=áë=~=ëÅ~ä~êI=~åÇ= ^ = ~ áà =áë=~=ã~íêáñI=íÜÉå= [ ]  â~NN â~NO K â~Nå   â~ â~ OO K â~ Oå   K=  ON â^ = â~ áà =  M M M    â~ ãN â~ ã O K â~ ãå  = 535. jìäíáéäáÅ~íáçå=çÑ=qïç=j~íêáÅÉë= qïç= ã~íêáÅÉë= Å~å= ÄÉ= ãìäíáéäáÉÇ= íçÖÉíÜÉê= çåäó= ïÜÉå= íÜÉ= åìãÄÉê= çÑ= Åçäìãåë= áå= íÜÉ= Ñáêëí= áë= Éèì~ä= íç= íÜÉ= åìãÄÉê= çÑ= êçïë=áå=íÜÉ=ëÉÅçåÇK== = fÑ=  ~NN ~NO K ~Nå  ~ ~ OO K ~ Oå   I== ^ = ~ áà =  ON  M M M    ~ ãN ~ ã O K ~ ãå   ÄNN ÄNO K ÄNâ  Ä ÄOO K ÄO â   I= _ = Äáà =  ON  M M M    ÄåN Äå O K Äåâ  = = = = = [ ] [ ] [ ] 112
  122. 122. CHAPTER 5. MATRICES AND DETERMINANTS íÜÉå==  ÅNN ÅNO K ÅNâ  Å Å OO K Å O â   I==  ON ^_ = ` =  M M M    Ä ãN Å ã O K Å ãâ  ïÜÉêÉ== å Å áà = ~ áNÄNà + ~ á O ÄO à + K + ~ áå Äåà = ∑ ~ á λ Äλ à = E á = NI OI KI ã X à = NI OI KI â FK== = qÜìë=áÑ= [ ] ~ NN ^ = ~ áà =  ~ ON ~ NO ~ OO λ =N  ÄN  ~ NP     I= _ = [Ä á ] = Ä O  I== ~ OP   ÄP    íÜÉå== ~ NN ~ NO ^_ =  ~ ON ~ OO Ä  ~ NP   N  ~ NNÄN ⋅ Ä = ~ OP   O  ~ ONÄN  Ä    P ~ NO Ä O ~ OO Ä O ~ NP ÄP  K== ~ OP ÄP   = 536. qê~åëéçëÉ=çÑ=~=j~íêáñ= fÑ=íÜÉ=êçïë=~åÇ=Åçäìãåë=çÑ=~=ã~íêáñ=~êÉ=áåíÉêÅÜ~åÖÉÇI=íÜÉå= íÜÉ=åÉï=ã~íêáñ=áë=Å~ääÉÇ=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=çêáÖáå~ä=ã~íêáñK=== fÑ= ^= áë= íÜÉ= çêáÖáå~ä= ã~íêáñI= áíë= íê~åëéçëÉ= áë= ÇÉåçíÉÇ= ^ q = çê= ú ^ K== = 537. qÜÉ=ã~íêáñ=^=áë=çêíÜçÖçå~ä=áÑ= ^^ q = f K== = 538. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå== (^_ )q = _ q ^ q K= = = 113
  123. 123. CHAPTER 5. MATRICES AND DETERMINANTS 539. ^Çàçáåí=çÑ=j~íêáñ= fÑ=^=áë=~=ëèì~êÉ= å× å ã~íêáñI=áíë=~ÇàçáåíI=ÇÉåçíÉÇ=Äó= ~Çà ^ I= áë=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=ã~íêáñ=çÑ=ÅçÑ~Åíçêë= ` áà =çÑ=^W= [ ] ~Çà ^ = ` áà K== = 540. qê~ÅÉ=çÑ=~=j~íêáñ= fÑ= ^= áë= ~= ëèì~êÉ= å × å ã~íêáñI= áíë= íê~ÅÉI= ÇÉåçíÉÇ= Äó= íê ^ I= áë= ÇÉÑáåÉÇ=íç=ÄÉ==íÜÉ=ëìã=çÑ==íÜÉ=íÉêãë=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äW= íê ^ = ~NN + ~ OO + K + ~ åå K= = 541. fåîÉêëÉ=çÑ=~=j~íêáñ= fÑ=^=áë=~=ëèì~êÉ= å× å ã~íêáñ=ïáíÜ=~=åçåëáåÖìä~ê=ÇÉíÉêãáå~åí= ÇÉí ^ I=íÜÉå=áíë=áåîÉêëÉ= ^ −N =áë=ÖáîÉå=Äó= ~Çà ^ ^ −N = K= ÇÉí ^ = 542. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå== (^_)−N = _ −N^ −N K= = 543. fÑ==^==áë=~=ëèì~êÉ=== å × å ==ã~íêáñI==íÜÉ==ÉáÖÉåîÉÅíçêë==u===ë~íáëÑó= íÜÉ=Éèì~íáçå= ^u = λu I== ïÜáäÉ=íÜÉ=ÉáÖÉåî~äìÉë= λ =ë~íáëÑó=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå= ^ − λf = M K=== = = = q 5.5 Systems of Linear Equations = = s~êá~ÄäÉëW=ñI=óI=òI= ñ N I= ñ O I K = oÉ~ä=åìãÄÉêëW= ~ N I ~ O I ~ P I ÄN I ~ NN I ~ NO I K = 114
  124. 124. CHAPTER 5. MATRICES AND DETERMINANTS aÉíÉêãáå~åíëW=aI= añ I= aó I= aò == j~íêáÅÉëW=^I=_I=u= = = ~ ñ + ÄNó = ÇN I== 544.  N ~ O ñ + ÄO ó = Ç O aó a =E`ê~ãÉê∞ë=êìäÉFI== ñ = ñ I= ó = a a ïÜÉêÉ== ~ ÄN a= N = ~NÄO − ~ O ÄN I== ~ O ÄO Ç ÄN añ = N = ÇNÄO − Ç O ÄN I== Ç O ÄO ~ ÇN aó = N = ~NÇ O − ~ OÇN K== ~ O ÇO = 545. fÑ= a ≠ M I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW== aó a K= ñ = ñ I= ó = a a fÑ= a = M = ~åÇ= añ ≠ M Eçê= aó ≠ M FI= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = åç== ëçäìíáçåK= fÑ= a = añ = aó = M I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = áåÑáåáíÉäó= = ã~åó== ëçäìíáçåëK= = ~Nñ + ÄNó + ÅNò = ÇN=  546. ~ O ñ + ÄO ó + Å Oò = Ç O I== ~ ñ + Ä ó + Å ò = Ç P P P  P ñ= aó añ a I= ó = I= ò = ò =E`ê~ãÉê∞ë=êìäÉFI== a a a = 115
  125. 125. CHAPTER 5. MATRICES AND DETERMINANTS ïÜÉêÉ== ~N ÄN a = ~ O ÄO ~ P ÄP ÅN ÇN ÄN ÅN Å O I= añ = Ç O ÄO Å O I= ÅP ÄP ÅP ÇP ~N ÇN ÅN ~N ÄN ÇN aó = ~ O ~P ÇO ÇP Å O I= aò = ~ O ÅP ~P ÄO ÄP Ç O K== ÇP = 547. fÑ= a ≠ M I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW== aó a a I= ò = ò K= ñ = ñ I= ó = a a a fÑ= a = M =~åÇ= añ ≠ M Eçê= aó ≠ M =çê= aò ≠ M FI=íÜÉå=íÜÉ=ëóëíÉã= Ü~ë=åç=ëçäìíáçåK= fÑ= a = añ = aó = aò = M I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= áåÑáåáíÉäó= ã~åó=ëçäìíáçåëK= = 548. j~íêáñ=cçêã=çÑ=~=póëíÉã=çÑ=å=iáåÉ~ê=bèì~íáçåë=áå================= å=råâåçïåë= qÜÉ=ëÉí=çÑ=äáåÉ~ê=Éèì~íáçåë== ~NNñ N + ~ NO ñ O + K + ~ Nå ñ å = ÄN ~ ñ + ~ ñ + K + ~ ñ = Ä  ON N OO O Oå å O =  KKKKKKKKKKKK  ~ åNñ N + ~ å O ñ O + K + ~ åå ñ å = Äå  Å~å=ÄÉ=ïêáííÉå=áå=ã~íêáñ=Ñçêã=  ~ NN ~ NO K ~ Nå   ñ N   ÄN         ~ ON ~ OO K ~ Oå   ñ O   Ä O  I== = ⋅  M M M   M   M             ~  åN ~ å O K ~ åå   ñ å   Ä å  áKÉK== ^ ⋅ u = _ I== 116
  126. 126. CHAPTER 5. MATRICES AND DETERMINANTS ïÜÉêÉ==  ~ NN  ~ ^ =  ON M  ~  åN ~ NO K ~ Nå   ñN   ÄN       ~ OO K ~ Oå   ñO  Ä  I= u =   I= _ =  O  K== M M  M M       ñ  Ä  ~ å O K ~ åå   å  å = 549. pçäìíáçå=çÑ=~=pÉí=çÑ=iáåÉ~ê=bèì~íáçåë= å × å = u = ^ −N ⋅ _ I== ïÜÉêÉ= ^ −N =áë=íÜÉ=áåîÉêëÉ=çÑ=^K= = = 117
  127. 127. Chapter 6 Vectors = = = = r r r r → sÉÅíçêëW= ì I= î I= ï I= ê I= ^_ I=£= r r sÉÅíçê=äÉåÖíÜW= ì I= î I=£= r r r råáí=îÉÅíçêëW= á I= à I= â = r kìää=îÉÅíçêW= M = r `ççêÇáå~íÉë=çÑ=îÉÅíçê= ì W= uN I vN I wN = r `ççêÇáå~íÉë=çÑ=îÉÅíçê= î W= u O I vO I wO = pÅ~ä~êëW= λ I µ = aáêÉÅíáçå=ÅçëáåÉëW= Åçë α I= Åçë β I= Åçë γ = ^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW= θ = = = 6.1 Vector Coordinates = 550. råáí=sÉÅíçêë= r á = (NI MI M) I= r à = (MI NI M) I= r â = (MI MI N) I= r r r á = à = â = N K= = r r r r → 551. ê = ^_ = (ñ N − ñ M ) á + (ó N − ó M ) à + (ò N − ò M ) â = = 118
  128. 128. CHAPTER 6. VECTORS ======= = = Figure 73. = → r ê = ^_ = 552. (ñ N − ñ M )O + (óN − ó M )O + (òN − ò M )O = = → → r r 553. fÑ= ^_ = ê I=íÜÉå= _^ = − ê K= = = = Figure 74. r 554. u = ê Åçë α I= r v = ê Åçë β I= r w = ê Åçë γ K= = 119
  129. 129. CHAPTER 6. VECTORS = ===== = Figure 75. = r r 555. fÑ= ê (uI v I w ) = êN (uN I vN I wN ) I=íÜÉå== u = uN I= v = vN I= w = wN K== == = 6.2 Vector Addition = r r r 556. ï = ì + î = = = == = Figure 76. 120
  130. 130. CHAPTER 6. VECTORS = == = Figure 77. = r r r r r 557. ï = ìN + ì O + ìP + K + ì å = = = = == Figure 78. = 558. `çããìí~íáîÉ=i~ï= r r r r ì+ î =î+ì= = 559. ^ëëçÅá~íáîÉ=i~ï= r r r r r r (ì + î ) + ï = ì + (î + ï ) = = r r 560. ì + î = (uN + u O I vN + vO I wN + wO ) = = = = = = = 121
  131. 131. CHAPTER 6. VECTORS 6.3 Vector Subtraction = r r r r r r 561. ï = ì − î =áÑ= î + ï = ì K= = = = Figure 79. = = == = Figure 80. = r r r r 562. ì − î = ì + (− î ) = = r r r 563. ì − ì = M = (MI MI M ) = = r 564. M = M = = r r 565. ì − î = (uN − u O I vN − vO I wN − w O ) I== = = = 6.4 Scaling Vectors = r r 566. ï = λì = 122
  132. 132. CHAPTER 6. VECTORS = = Figure 81. = 567. r r ï = λ⋅ì= = r 568. λì = (λuI λv I λw ) = = r r 569. λì = ìλ = = r r r 570. (λ + µ ) ì = λì + µì = = r r r 571. λ(µì ) = µ(λì ) = (λµ )ì = = r r r r 572. λ(ì + î ) = λì + λî = = = = 6.5 Scalar Product = r r 573. pÅ~ä~ê=mêçÇìÅí=çÑ=sÉÅíçêë= ì =~åÇ î = r r r r ì ⋅ î = ì ⋅ î ⋅ Åçë θ I== r r ïÜÉêÉ= θ =áë=íÜÉ=~åÖäÉ=ÄÉíïÉÉå=îÉÅíçêë= ì =~åÇ î K==== = 123
  133. 133. CHAPTER 6. VECTORS = = = Figure 82. = 574. pÅ~ä~ê=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã= r r fÑ= ì = (uN I vN I wN ) I= î = (u O I vO I w O ) I=íÜÉå== r r ì ⋅ î = uNu O + vNvO + wNwO K= = 575. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë== r r fÑ= ì = (uN I vN I wN ) I= î = (u O I vO I w O ) I=íÜÉå== uNu O + vNvO + wNw O K= Åçë θ = O O O O O O uN + vN + wN u O + vO + w O = 576. `çããìí~íáîÉ=mêçéÉêíó= r r r r ì⋅î = î ⋅ì= = 577. ^ëëçÅá~íáîÉ=mêçéÉêíó= r r r r (λì ) ⋅ (µî ) = λµì ⋅ î = = 578. aáëíêáÄìíáîÉ=mêçéÉêíó= r r r r r r r ì ⋅ (î + ï ) = ì ⋅ î + ì ⋅ ï = = π r r r r 579. ì ⋅ î = M =áÑ= ì I î =~êÉ=çêíÜçÖçå~ä=E θ = FK= O = π r r 580. ì ⋅ î > M =áÑ= M < θ < K= O = 124
  134. 134. CHAPTER 6. VECTORS π r r 581. ì ⋅ î < M =áÑ= < θ < π K= O = r r r r 582. ì ⋅ î ≤ ì ⋅ î = = r r r r r r 583. ì ⋅ î = ì ⋅ î =áÑ= ì I î =~êÉ=é~ê~ääÉä=E θ = M FK= = r 584. fÑ= ì = (uN I vN I wN ) I=íÜÉå== r r r rO O O O ì ⋅ ì = ì O = ì = uN + vN + wN K= = r r r r r r 585. á ⋅ á = à ⋅ à = â ⋅ â = N = = r r r r r r 586. á ⋅ à = à ⋅ â = â ⋅ á = M = = = = 6.6 Vector Product = r r 587. sÉÅíçê=mêçÇìÅí=çÑ=sÉÅíçêë= ì =~åÇ î = r r r ì × î = ï I=ïÜÉêÉ== π r r r • ï = ì ⋅ î ⋅ ëáå θ I=ïÜÉêÉ= M ≤ θ ≤ X= O r r r r • ï ⊥ì= ~åÇ= ï ⊥ î X= r r r • =sÉÅíçêë= ì I= î I= ï =Ñçêã=~=êáÖÜí-Ü~åÇÉÇ=ëÅêÉïK= = 125
  135. 135. CHAPTER 6. VECTORS = ======= = Figure 83. = r á r r r 588. ï = ì × î = u N uO r à vN vO r â wN = wO = uN wN uN vN  r r r  v wN = 589. ï = ì × î =  N I− I v w u O w O u O vO  O O   = r r r r 590. p = ì × î = ì ⋅ î ⋅ ëáå θ =EcáÖKUPF= = 591. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë=EcáÖKUPF= r r ì× î ëáå θ = r r = ì⋅î = 592. kçåÅçããìí~íáîÉ=mêçéÉêíó= r r r r ì × î = −(î × ì ) == = 593. ^ëëçÅá~íáîÉ=mêçéÉêíó= r r r r (λì )× (µî ) = λµì × î = = = 126
  136. 136. CHAPTER 6. VECTORS 594. aáëíêáÄìíáîÉ=mêçéÉêíó= r r r r r r r ì × (î + ï ) = ì × î + ì × ï = = r r r r r 595. ì × î = M =áÑ= ì =~åÇ= î =~êÉ=é~ê~ääÉä=E θ = M FK= = r r r r r r r 596. á × á = à × à = â × â = M = = r r r r r r r r r 597. á × à = â I= à × â = á I= â × á = à = = = = 6.7 Triple Product 598. 599. 600. 601. = pÅ~ä~ê=qêáéäÉ=mêçÇìÅí= rr r r r r r r r r r r [ìîï ] = ì ⋅ (î × ï ) = î ⋅ (ï × ì ) = ï ⋅ (ì × î ) = = rr r r rr rr r rr r r rr rrr [ìîï ] = [ïìî ] = [îïì] = −[îìï ] = −[ïîì] = −[ìïî ] = = r r r rr r âì ⋅ (î × ï ) = â[ìîï ] = = pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã= uN vN wN r r r ì ⋅ (î × ï ) = u O vO w O I== uP vP wP ïÜÉêÉ== r r r ì = (uN I vN I wN ) I= î = (u O I vO I w O ) I= ï = (uP I vP I wP ) K== = 602. sçäìãÉ=çÑ=m~ê~ääÉäÉéáéÉÇ= r r r s = ì ⋅ (î × ï ) = = 127
  137. 137. CHAPTER 6. VECTORS = ============ = Figure 84. = 603. sçäìãÉ=çÑ=móê~ãáÇ= Nr r r s = ì ⋅ (î × ï ) = S = = = Figure 85. = r r r r r r 604. fÑ== ì ⋅ (î × ï ) = M I=íÜÉå=íÜÉ=îÉÅíçêë== ì I= î I=~åÇ= ï =~êÉ=äáåÉ~êäó= r r r ÇÉéÉåÇÉåí=I=ëç= ï = λì + µî =Ñçê=ëçãÉ=ëÅ~ä~êë= λ =~åÇ= µ K== = r r r r r r 605. fÑ== ì ⋅ (î × ï ) ≠ M I=íÜÉå=íÜÉ=îÉÅíçêë== ì I= î I=~åÇ= ï =~êÉ=äáåÉ~êäó= áåÇÉéÉåÇÉåíK= = 128
  138. 138. CHAPTER 6. VECTORS 606. sÉÅíçê=qêáéäÉ=mêçÇìÅí= r r r r r r r r r ì × (î × ï ) = (ì ⋅ ï )î − (ì ⋅ î )ï == = = = = = = = = 129
  139. 139. Chapter 7 Analytic Geometry = = = = 7.1 One-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= ñ M I= ñ N I= ñ O I= ó M I= ó N I= ó O = oÉ~ä=åìãÄÉêW= λ == aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= = = 607. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë= Ç = ^_ = ñ O − ñ N = ñ N − ñ O = = = = Figure 86. = 608. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ = ñ + λñ O ^` I= λ = ñM = N I= λ ≠ −N K= N+ λ `_ = = ======== Figure 87. 130 =
  140. 140. CHAPTER 7. ANALYTIC GEOMETRY 609. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí= ñ + ñO ñM = N I= λ = N K= O = = = 7.2 Two-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= ñ M I= ñ N I= ñ O I= ó M I= ó N I= ó O = mçä~ê=ÅççêÇáå~íÉëW= êI ϕ = oÉ~ä=åìãÄÉêW= λ == mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI== aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= ^êÉ~W=p= = = 610. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë= Ç = ^_ = = (ñ O − ñ N )O + (ó O − óN )O = = = Figure 88. 131
  141. 141. CHAPTER 7. ANALYTIC GEOMETRY 611. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ = ñ + λñ O ó + λó O ñM = N I= ó M = N I== N+ λ N+ λ ^` λ= I= λ ≠ −N K= `_ = ======= = = Figure 89. = = 132
  142. 142. CHAPTER 7. ANALYTIC GEOMETRY ======= = = Figure 90. = 612. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí= ñ + ñO ó + óO I= ó M = N I= λ = N K= ñM = N O O = 613. `ÉåíêçáÇ=EfåíÉêëÉÅíáçå=çÑ=jÉÇá~åëF=çÑ=~=qêá~åÖäÉ= ñ + ñ O + ñP ó + óO + óP I= ó M = N ñM = N I== P P ïÜÉêÉ== ^(ñ N I ó N ) I== _(ñ O I ó O ) I==~åÇ== `(ñ P I ó P ) ==~êÉ=îÉêíáÅÉë=çÑ= íÜÉ=íêá~åÖäÉ= ^_` K= = = 133
  143. 143. CHAPTER 7. ANALYTIC GEOMETRY ========= = = Figure 91. = 614. fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ= ~ñ + Äñ O + Åñ P ~ó + Äó O + Åó P I= ó M = N ñM = N I== ~ +Ä+Å ~ +Ä+Å ïÜÉêÉ= ~ = _` I= Ä = `^ I= Å = ^_ K== = ======== = = Figure 92. 134
  144. 144. CHAPTER 7. ANALYTIC GEOMETRY 615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê====================== _áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ= O O O O ñN + óN óN N ñN ñN + óN N ñO + óO óO N ñO ñO + óO N O O O O O O O O ñP + óP óP N ñP ñP + óP N ñM = I= ó M = = ñN óN N ñN óN N O ñO ñP óO N óP N O ñO ñP óO N óP N = = ======== == Figure 93. = = = = = = = 135
  145. 145. CHAPTER 7. ANALYTIC GEOMETRY 616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ= O O óN ñ O ñ P + óN N ñN + ó OóP ñN N ó O ñPñN + ó O N ñ O + ó P óN ñ O N O O O O ó P ñ Nñ O + ó P N ñ P + ó Nó O ñ P N I= ó M = = ñM = ñN óN N ñN óN N ñO óO N ñO óO N ñP óP N ñP óP N = = ====== = Figure 94. = 617. ^êÉ~=çÑ=~=qêá~åÖäÉ= ñ N óN N N N ñ O − ñN p = (± ) ñ O ó O N = (± ) O O ñ P − ñN ñP óP N = = = 136 ó O − óN ó P − óN =
  146. 146. CHAPTER 7. ANALYTIC GEOMETRY 618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä= N p = (± ) [(ñ N − ñ O )(ó N + ó O ) + (ñ O − ñ P )(ó O + ó P ) + = O + (ñ P − ñ Q )(ó P + ó Q ) + (ñ Q − ñ N )(ó Q + ó N )] = = === = = Figure 95. = kçíÉW=få=Ñçêãìä~ë=SNTI=SNU=ïÉ=ÅÜççëÉ=íÜÉ=ëáÖå=EHF=çê=E¥F=ëç= íÜ~í=íç=ÖÉí=~=éçëáíáîÉ=~åëïÉê=Ñçê=~êÉ~K== = 619. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=áå=mçä~ê=`ççêÇáå~íÉë= Ç = ^_ = êNO + êOO − OêNêO Åçë(ϕ O − ϕN ) = = 137
  147. 147. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 96. = 620. `çåîÉêíáåÖ=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=íç=mçä~ê=`ççêÇáå~íÉë= ñ = ê Åçë ϕ I= ó = ê ëáå ϕ K= = = = Figure 97. = 621. `çåîÉêíáåÖ=mçä~ê=`ççêÇáå~íÉë=íç=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë= ó ê = ñ O + ó O I= í~å ϕ = K= ñ 138
  148. 148. CHAPTER 7. ANALYTIC GEOMETRY 7.3 Straight Line in Plane = mçáåí=ÅççêÇáå~íÉëW=uI=vI=ñI= ñ M I= ñ N I== ó M I= ó N I= ~N I= ~ O I=£== oÉ~ä=åìãÄÉêëW=âI=~I=ÄI=éI=íI=^I=_I=`I= ^N I= ^ O I=£= ^åÖäÉëW= α I= β = ^åÖäÉ=ÄÉíïÉÉå=íïç=äáåÉëW= ϕ = r kçêã~ä=îÉÅíçêW= å = r r r mçëáíáçå=îÉÅíçêëW= ê I= ~ I= Ä = = = 622. dÉåÉê~ä=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ= ^ñ + _ó + ` = M = = 623. kçêã~ä=sÉÅíçê=íç=~=píê~áÖÜí=iáåÉ= r qÜÉ=îÉÅíçê= å(^I _ ) =áë=åçêã~ä=íç=íÜÉ=äáåÉ= ^ñ + _ó + ` = M K= = = = Figure 98. = 624. bñéäáÅáí=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=EpäçéÉ-fåíÉêÅÉéí=cçêãF= ó = âñ + Ä K== 139
  149. 149. CHAPTER 7. ANALYTIC GEOMETRY qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= â = í~å α K= = = = Figure 99. = 625. dê~ÇáÉåí=çÑ=~=iáåÉ== ó − óN â = í~å α = O = ñ O − ñN = = = Figure 100. 140
  150. 150. CHAPTER 7. ANALYTIC GEOMETRY 626. bèì~íáçå=çÑ=~=iáåÉ=dáîÉå=~=mçáåí=~åÇ=íÜÉ=dê~ÇáÉåí= ó = ó M + â (ñ − ñ M ) I== ïÜÉêÉ=â=áë=íÜÉ=Öê~ÇáÉåíI= m(ñ M I ó M ) =áë=~=éçáåí=çå=íÜÉ=äáåÉK= = = = Figure 101. = 627. bèì~íáçå=çÑ=~=iáåÉ=qÜ~í=m~ëëÉë=qÜêçìÖÜ=qïç=mçáåíë= ó − óN ñ − ñN = == ó O − óN ñ O − ñN çê= ñ ó N ñ N ó N N = M K= ñO óO N = 141
  151. 151. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 102. = 628. fåíÉêÅÉéí=cçêã= ñ ó + =N= ~ Ä = = = Figure 103. = = 142
  152. 152. CHAPTER 7. ANALYTIC GEOMETRY 629. kçêã~ä=cçêã= ñ Åçë β + ó ëáå β − é = M = = = = Figure 104. = 630. mçáåí=aáêÉÅíáçå=cçêã= ñ − ñ N ó − óN = I== u v ïÜÉêÉ= (uI v ) = áë= íÜÉ= ÇáêÉÅíáçå= çÑ= íÜÉ= äáåÉ= ~åÇ= mN (ñ N I ó N ) = äáÉë= çå=íÜÉ=äáåÉK= = 143
  153. 153. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 105. = 631. sÉêíáÅ~ä=iáåÉ= ñ =~= = 632. eçêáòçåí~ä=iáåÉ= ó=Ä= = 633. sÉÅíçê=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ= r r r ê = ~ + íÄ I== ïÜÉêÉ== l=áë=íÜÉ=çêáÖáå=çÑ=íÜÉ=ÅççêÇáå~íÉëI= u=áë=~åó=î~êá~ÄäÉ=éçáåí=çå=íÜÉ=äáåÉI== r ~ =áë=íÜÉ=éçëáíáçå=îÉÅíçê=çÑ=~=âåçïå=éçáåí=^=çå=íÜÉ=äáåÉ=I= r Ä =áë=~=âåçïå=îÉÅíçê=çÑ=ÇáêÉÅíáçåI=é~ê~ääÉä=íç=íÜÉ=äáåÉI== í=áë=~=é~ê~ãÉíÉêI== r → ê = lu =áë=íÜÉ=éçëáíáçå=îÉÅíçê=çÑ=~åó=éçáåí=u=çå=íÜÉ=äáåÉK== = 144
  154. 154. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 106. = 634. píê~áÖÜí=iáåÉ=áå=m~ê~ãÉíêáÅ=cçêã= ñ = ~N + íÄN I==  ó = ~ O + íÄO ïÜÉêÉ== (ñ I ó ) ~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~åó=ìåâåçïå=éçáåí=çå=íÜÉ=äáåÉI== (~N I ~ O ) =~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~=âåçïå=éçáåí=çå=íÜÉ=äáåÉI== (ÄN I ÄO ) =~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~=îÉÅíçê=é~ê~ääÉä=íç=íÜÉ=äáåÉI== í=áë=~=é~ê~ãÉíÉêK= = 145
  155. 155. CHAPTER 7. ANALYTIC GEOMETRY = Figure 107. = 635. aáëí~åÅÉ=cêçã=~=mçáåí=qç=~=iáåÉ= qÜÉ=Çáëí~åÅÉ=Ñêçã=íÜÉ=éçáåí= m(~ I Ä) =íç=íÜÉ=äáåÉ= ^ñ + _ó + ` = M =áë== ^~ + _Ä + ` K= Ç= ^ O + _O = = = Figure 108. 146
  156. 156. CHAPTER 7. ANALYTIC GEOMETRY 636. m~ê~ääÉä=iáåÉë= qïç=äáåÉë= ó = â Nñ + ÄN =~åÇ= ó = â O ñ + ÄO =~êÉ=é~ê~ääÉä=áÑ== â N = â O K= qïç= äáåÉë= ^Nñ + _Nó + `N = M = ~åÇ= ^ O ñ + _O ó + ` O = M = ~êÉ= é~ê~ääÉä=áÑ= ^N _N = K= ^ O _O = = = Figure 109. = 637. mÉêéÉåÇáÅìä~ê=iáåÉë= qïç=äáåÉë= ó = â Nñ + ÄN =~åÇ= ó = â O ñ + ÄO =~êÉ=éÉêéÉåÇáÅìä~ê=áÑ== N â O = − =çêI=Éèìáî~äÉåíäóI= â Nâ O = −N K= âN qïç= äáåÉë= ^Nñ + _Nó + `N = M = ~åÇ= ^ O ñ + _ O ó + ` O = M = ~êÉ= éÉêéÉåÇáÅìä~ê=áÑ= ^N^ O + _N_ O = M K= = 147
  157. 157. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 110. = 638. ^åÖäÉ=_ÉíïÉÉå=qïç=iáåÉë= â − âN í~å ϕ = O I== N + â Nâ O ^N^ O + _N_ O Åçë ϕ = K= O O ^N + _N ⋅ ^ O + _ O O O = 148
  158. 158. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 111. = 639. fåíÉêëÉÅíáçå=çÑ=qïç=iáåÉë= fÑ=íïç=äáåÉë= ^Nñ + _Nó + `N = M =~åÇ= ^ O ñ + _ O ó + ` O = M =áåíÉêëÉÅíI=íÜÉ=áåíÉêëÉÅíáçå=éçáåí=Ü~ë=ÅççêÇáå~íÉë= − `N_ O + ` O_N − ^N` O + ^ O`N ñM = I= ó M = K= ^N_ O − ^ O_N ^N_ O − ^ O_N = = = 7.4 Circle = o~ÇáìëW=o= `ÉåíÉê=çÑ=ÅáêÅäÉW= (~ I Ä) = mçáåí=ÅççêÇáå~íÉëW=ñI=óI= ñ N I= ó N I=£= oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í= 149
  159. 159. CHAPTER 7. ANALYTIC GEOMETRY 640. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ= cçêãF= ñ O + ó O = oO = ====== = = Figure 112. = 641. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=^åó=mçáåí= (~I Ä) (ñ − ~ )O + (ó − Ä)O = o O Figure 113. 150
  160. 160. CHAPTER 7. ANALYTIC GEOMETRY 642. qÜêÉÉ=mçáåí=cçêã ñO + óO ñ ó N O O ñN + óN ñN óN N =M ñO + óO ñO óO N O O O O ñP + óP ñP óP N = = = Figure 114. = 643. m~ê~ãÉíêáÅ=cçêã ñ = o Åçë í I= M ≤ í ≤ Oπ K  ó = o ëáå í = 644. dÉåÉê~ä=cçêã ^ñ O + ^ó O + añ + bó + c = M =E^=åçåòÉêçI= aO + b O > Q ^c FK== qÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ÅáêÅäÉ=Ü~ë=ÅççêÇáå~íÉë= (~ I Ä) I=ïÜÉêÉ== a b ~=− I= Ä = − K= O^ O^ qÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅäÉ=áë 151
  161. 161. CHAPTER 7. ANALYTIC GEOMETRY o= aO + b O − Q ^c K O^ = = = 7.5 Ellipse = pÉãáã~àçê=~ñáëW=~= pÉãáãáåçê=~ñáëW=Ä= cçÅáW= cN (− ÅI M) I= cO (ÅI M) = aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= = bÅÅÉåíêáÅáíóW=É== oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = 645. bèì~íáçå=çÑ=~å=bääáéëÉ=Epí~åÇ~êÇ=cçêãF ñO óO + =N ~ O ÄO = = Figure 115. 152
  162. 162. CHAPTER 7. ANALYTIC GEOMETRY 646. êN + êO = O~ I= ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó==éçáåí== m(ñ I ó ) ==çå= íÜÉ=ÉääáéëÉ=íç=íÜÉ=íïç=ÑçÅáK= = = = Figure 116. = 647. ~ O = ÄO + Å O = 648. bÅÅÉåíêáÅáíó Å É = <N= ~ = 649. bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë ~ ~O ñ=± =± = É Å = 650. m~ê~ãÉíêáÅ=cçêã ñ = ~ Åçë í I= M ≤ í ≤ Oπ K  ó = Ä ëáå í = = 153
  163. 163. CHAPTER 7. ANALYTIC GEOMETRY 651. dÉåÉê~ä=cçêã ^ñ O + _ñó + `ó O + añ + bó + c = M I== ïÜÉêÉ= _ O − Q ^` < M K= = 652. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë ^ñ O + `ó O + añ + bó + c = M I== ïÜÉêÉ= ^` > M K = 653. `áêÅìãÑÉêÉåÅÉ i = Q~b(É ) I== ïÜÉêÉ==íÜÉ==ÑìåÅíáçå=b==áë==íÜÉ=ÅçãéäÉíÉ==ÉääáéíáÅ=áåíÉÖê~ä==çÑ= íÜÉ=ëÉÅçåÇ=âáåÇK== = 654. ^ééêçñáã~íÉ=cçêãìä~ë=çÑ=íÜÉ=`áêÅìãÑÉêÉåÅÉ i = π NKR(~ + Ä) − ~Ä I== ( i = π O(~ O + ÄO ) K= = 655. p = π~Ä = = = = ) 7.6 Hyperbola = qê~åëîÉêëÉ=~ñáëW=~= `çåàìÖ~íÉ=~ñáëW=Ä= cçÅáW= cN (− ÅI M) I= cO (ÅI M) = aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= = bÅÅÉåíêáÅáíóW=É== ^ëóãéíçíÉëW=ëI=í= oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=íI=â= = = = 154
  164. 164. CHAPTER 7. ANALYTIC GEOMETRY 656. bèì~íáçå=çÑ=~=eóéÉêÄçä~=Epí~åÇ~êÇ=cçêãF= ñO óO − = N= ~ O ÄO = = = Figure 117. = 657. êN − êO = O~ I= ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó=éçáåí== m(ñ I ó ) ==çå= íÜÉ=ÜóéÉêÄçä~=íç=íÜÉ=íïç=ÑçÅáK= = 155
  165. 165. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 118. 658. 659. 660. 661. = bèì~íáçåë=çÑ=^ëóãéíçíÉë= Ä ó=± ñ= ~ = Å O = ~ O + ÄO = = bÅÅÉåíêáÅáíó Å É = > N= ~ = bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë ~ ~O ñ=± =± = É Å = = = 156
  166. 166. CHAPTER 7. ANALYTIC GEOMETRY 662. m~ê~ãÉíêáÅ=bèì~íáçåë=çÑ=íÜÉ=oáÖÜí=_ê~åÅÜ=çÑ=~=eóéÉêÄçä~= ñ = ~ ÅçëÜ í I= M ≤ í ≤ Oπ K  ó = Ä ëáåÜ í = 663. dÉåÉê~ä=cçêã ^ñ O + _ñó + `ó O + añ + bó + c = M I== ïÜÉêÉ= _ O − Q ^` > M K= = 664. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë ^ñ O + `ó O + añ + bó + c = M I== ïÜÉêÉ= ^` < M K= 665. ^ëóãéíçíáÅ=cçêã= ÉO ñó = I== Q çê== ÉO â ó = I=ïÜÉêÉ= â = K= ñ Q få= íÜáë= Å~ëÉ= I= íÜÉ= ~ëóãéíçíÉë= Ü~îÉ= Éèì~íáçåë= ñ = M = ~åÇ= ó = M K== = 157
  167. 167. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 119. = = = 7.7 Parabola = cçÅ~ä=é~ê~ãÉíÉêW=é= cçÅìëW=c= sÉêíÉñW= j(ñ M I ó M ) = oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=éI=~I=ÄI=Å= = = 666. bèì~íáçå=çÑ=~=m~ê~Äçä~=Epí~åÇ~êÇ=cçêãF ó O = Oéñ = 158
  168. 168. CHAPTER 7. ANALYTIC GEOMETRY = = Figure 120. = bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ é ñ = − I= O `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë= é  c I M  I= O  `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= j(MI M) K= = 667. dÉåÉê~ä=cçêã ^ñ O + _ñó + `ó O + añ + bó + c = M I== ïÜÉêÉ= _ O − Q ^` = M K= = N 668. ó = ~ñ O I= é = K= O~ bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ 159
  169. 169. CHAPTER 7. ANALYTIC GEOMETRY é ó = − I= O `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=  é c MI  I=  O `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= j(MI M) K= = = = Figure 121. = 669. dÉåÉê~ä=cçêãI=^ñáë=m~ê~ääÉä=íç=íÜÉ=ó-~ñáë== ^ñ O + añ + bó + c = M =E^I=b=åçåòÉêçFI== N ó = ~ñ O + Äñ + Å I= é = K== O~ bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ é ó = ó M − I= O `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë= 160
  170. 170. CHAPTER 7. ANALYTIC GEOMETRY é  c ñ M I ó M +  I= O  `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= Ä Q~Å − ÄO K= ñ M = − I= ó M = ~ñ O + Äñ M + Å = M O~ Q~ = = = Figure 122. = = = 7.8 Three-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= ñ M I= ó M I= ò M I= ñ N I= ó N I= ò N I=£= oÉ~ä=åìãÄÉêW= λ == aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= ^êÉ~W=p= sçäìãÉW=s= = 161
  171. 171. CHAPTER 7. ANALYTIC GEOMETRY 670. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë= Ç = ^_ = = = (ñ O − ñ N )O + (ó O − óN )O + (ò O − òN )O = = === Figure 123. = 671. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ = ñ + λñ O ó + λó O ò + λò O ñM = N I= ó M = N I= ò M = N I== N+ λ N+ λ N+ λ ïÜÉêÉ= ^` λ= I= λ ≠ −N K= `_ = 162
  172. 172. CHAPTER 7. ANALYTIC GEOMETRY ======== = = Figure 124. = = Figure 125. 163

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