| | |

(A B ) [B (B A)]B A U

A

B

A

na an( n) n a

a nbt ( )n a

b

a a

AB ABCACD

C B

A

A+B B A

sin sin sin sinA ...sin sin sin sincos cos cos cosB ...

cos cos cos cos

A A ( )

D

| | |

( ) ( )

x x

x x

( k)( k)

k

aAM m cm C BC cmABC

ABC C( , ) B( , ) A( , )

C(O,R) y=x+ y= x

C y x

(x x ) x x

( x ) ( x ) ( x)

y mx mxm

y kx (k )x k

xg(x) f ( )x fD [ , ]

nna

yx

y=f(x)

f (x) f (x)

B A y | x | dM( , ) AB

AB

x x

[ , ] f (x) x x

f (x) x x

g(x) | x | x | |

x f

f(x) (fof )(x) x (f (x) )

| | |

y=f(x)

g(x)f (x)

A AE AB

BE ˆCMB ACD

CB

AD

E

M

O R AT' AT A°

O ATOT'AT A

A AT

R P

PA ° AB

BA P

ABC OAO AD

CB

AD

A A

R

ABC

x

O

| | |

x R : x

x R : x

x R : x

n N : n n

[p (p q)] q

~ (p q) p ~ q

A B A C B CA B A C B C

A {a,b,c,d,e}A

[A (B C)] [A (A B)] A B

A (B C) (A B) (A C)

n B m n A

A n m

(A B) (C D) (A C) (B D)

g f

x

y

f

g

(gof)( )(fog)( )(fog)( )(gof)( )

g(x) x x f (x) | x |

fog( ) gof ( )

f(x)

y f (x)

y=f(x)

y f (x )

y f (x)

y f ( x)

[ , ] [ , ] ff of(x) fof (x)

y sin( x)

y sin x.cos x

y cos x sin x

| | |

cos x sin x cos x

tan xsin xtan x

tan xcos xtan x

sinsin cos

tan x cot xtan x cot x

sin x

sin x tan x

sin x cos x sin x

sin x cos x

y=f(x)f(x+|x|)

x

y

y y x

f(x)=x

g f R R g ffog g f

f (x) | x | | x |

p(x)=x +ax+b b a(x )(x+ )

xy sin

cos x sin x , [ , ]

sin x cos x , [ , ]

xf (x) tan( )

y=tanx xy tanxy tan

x

xlimx

xlim( )

x(x )

n m

x

xlimx mx n

m|n m n b>m n(b ) | (b )

( m+ ) ( m+ ) aa

| n n

| | |

aa+ a+ a

!A

A

A aba

m N Z x my

m

x y

An, ..., A , A A

A=A =A =...=An Aa b

Ac d

A

At( )=A Af e d c b a

aA b c

d e f

Rcos sin

Rsin cos

R R R

A A

A

A ×

A

A y xx

By

A=A A ×

a bA

c da

B b cd e f

a b cA d e

f

x y

m x y m

mmm

a

Am

AH BH.HC

C

B H

A

A A At

| | |

AB BH.BC AC CH.BC

C

B H

A

HM HOA AOH HP PM

HB

OB

M

B

PHO A

A xoyC oy B ox A d

AB AC

O O

x yx

AA

y

OKSA OBMN xoy oy ox

O

ˆ ˆo xoy

A

SB

xM

N

K

y

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DB ABDC AC

BA AD CBC AC AB K

DC DB

B D C

A

n nn

n nn

b a

c b a

a b( a.b)a b c a.b.c

d c b aa b cd a b c d a.b.c.d

| | |

n(A B) n(A) n(B) n(A B)n(A) n(U) n(A )n(A B) n(A) n(A B)

AB

U

UA B

x

n(A B) n( ) n((A B) ')n(A B) ,n(B A)

( ),( ),( )

n(A B) n(A) n(B) n(A B) ( )n(A B) n(A) n(A B) ( )n(B A) n(B) n(B A) ( )

n(A B) n(A B)n(A B) n(B A)

n(A B)

A B A B(A B) ' A ' B'(A B) ' A ' B'(A ') ' AA A 'A (B C) (A B)(A C)

((A ' B') [B (B A)]) '((A ' B') [B (B A ')]) '((A ' B') [B (B A ') ']) '((A ' B') [B (B' A)]) '

(A ' B') (B B') (B A) '

((A ' B') (B A)) '(A ' B') ' (B A) '(A B) (B' A ')

(A B B') (A B A ')

n(A B) n(A) n(B) n(A B)n(A ') n(U) n(A)

n(A) n(B)

n(A B) n(U)

n

n(A ' B') n(U)

n(U) n(A B) n(U) n(A B)

n(U)

n(A B) n(A) n(B) n(A B)

n(A) n(B)

n(B) n(B) n(U)

n( ) n(B) n(U)

n(B) n(U)

n(B) n(U)

n(A) n(U)

n(A ') n(U) n(A) n(U) n(U)

n(U)

A

d

t

n n

n

t t (n )d t (n )t n

t ' d '

n n

n

t ' t ' (n )d ' t ' (n )t ' n

n nt ' t ( n ) ( n ) n

n n n

n na an b

na an an n a ( a)nan a

n n

na a a na

a

nb bt ( )n a ( )n

a

bt b

b

td

nt t (n )d

| | |

a , a , a

a a+

a

t at t (( a ) ) d a

t ( a ) d a

a (a ) ( a )d a a( a )d

aa ( a )d da

( a )a

ACDCD AB

CD AB

C B

A

D

ACD

ABC : AC AB BC ( )

( ) BC

BC

BC ( )BC

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BD

CD BC BD

S AB CD

A ( ) ( )

( )

( ) ( ) ( )

A ( ( ))( ) ( )

(( ) ) (( ) )

m n mn m m m

(a b) a ab b ,

(a ) a ,a b (ab)

(a b)(a b) a b

( ) ( )

(( ) ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

(( )( ))

( )

( )

sin sinsin sinsin sinsin sin

sin sin sinA ...sin sin sin

sin

cos

cos coscos cos

Bcos coscos cos

sin

cos

cos cos ...cos cos

A B ( )

a b ab a,b

n na a a n

| | |

( )

( )( )

( ) ( )

b ( )Sa

( )

( )

( k)( k)

[ ( ) k][ ( ) k]

bSacPa

A

CBcm

M

cm BC CMC

AM= cm MC

CM AA B M

B

A

CB

cmM

AB AB M

AB

A

CBN

PM

AB M( , )

B AAB

B A

y ym

x x

mm

y (x ) y x

y xBC N

BCBC

BC N( , )

C BBC

C B

y ym

x xBC

BC

x

BC AB

xy

y x

O( , )

y xx x x

y xx , y

O( , )

y x

( )R OH

RO

Hx y =

(S R )

S ( )

( ) ( )k k

k k

k k

k k

( )k k

[ ( ) k][ ( ) k]

| | |

x x (x )(x )x , x

x x

a+b+c=a +b +c = abc

( x ) ( x ) ( x)( x )( x )( x)

x x

x x

x x

y=ax +bx+ca>

S>

a m

b ac ( m) (m)( )

m m m m mb ( m)S

a mcPa m

y=ax +bx+c

S>

k

a k

b ac (k ) k( )

(k ) kk k k

kcPa k

f (x) ax bx cc b a

(x y ) c

ba

b b a a ,ba

(x , y ) a b c

f (x) x x

f (x) f (x)

f (x) x x x

f (x) x x x

y=|x| AB

B=( b,b) A=(a,a)

M A

x

B

A BM

A BM

x x a bx a b

y y a by a b

a ,b

AB (a b) (a b)

x + =t

(t ) t(t ) tt t

tt t

t

t x x

[ , ] f

x xx x

x x xx xx | x | | x | x

x x

x ( x ) x x x

x x x x

g(x)

R ( , )

nn(n )

...

( ... )

nn

...

a , , ,...

a a ...a a (a q)(a q )...(a q )

a q a q

qa a ... a aq

( )

A

O( , ) A (a, )a

yx

OA a (a )aa

(a ) (a )a a

(a ) OAa

| | |

f (x) (x )

x x (x )

(x ) f (x)f [ , ]

f

f (x) (x )

f

y =

h(x) f (| x |) x | x |

h(x)

h(x)

g(x) | h(x) | | x | x | |

g(x)

T

C

T

OB A

ATOTOAT

OTA OAT A A

OT R Rtan A ATAT AT

S S ( AT OT)

R RR

AT TBT T A

RAT AT

TBT TCT R

R RAT TBT AT ( )

RR

( )R

O PC

ˆAOBAOB

AO=BO=AB=R

PA.PB PD.PC x.(x R) R( R)

x Rx R

R R R Rx

xx ( )RR

COR

RR

PB

D

A x

n f

x (f (x) ) n fof

n n n+

f (x) ax bx c n=x f

f (x) ax c

a(ax c) c x (ax c )

a x a cx ac c ax (c )x

a

c

a a a(a ) a c c ca c c

ac c a a

f (x) x

g

f (x) f (x)[ , ] Df=[ , ]

x=

gD [ , ) ( , )

( , )y= x ( , )

g x= f( )=

( ),( )

CD BE CD BE ( )

BC DEˆCMB BC DE ( )

BCE BE CD BEA

CD BE ( ) CD ,BE

AT' AT AT' T O

ˆ ˆT T

| | |

BH' AHˆ ˆH H

ˆˆAOH : O H A

ˆ ˆBHO : O H Bˆ ˆ ˆˆ ˆ ˆ ˆO H A O H B A B ( )

CB

AD

H

HO

DCB ( ) B

A

DCA ( )( )

ˆ ˆ ˆˆ( ), ( ) A B A A

AH' AODAOD

AO AD

ABC COC AB

COCB

OC OB BC ( ) OC

BC

O

A

O

C

B

A

D

E

ˆ ˆAB || DC,AD\ A Dˆ ˆˆ ˆA D A D

P ADP

P

M

ˆ ˆP M

C

BA

D

M

N

Q

P

°

ˆ ˆˆ ˆ ˆ ˆQ N M P Q N

°

PQMN

ˆˆQDC : D C DQ CQ( )BMC ADP

A BADP BMC AD BC DP CM( )

D C

TT d (R R )

d ( ) d dd R R

x

x

min

x d (R R ) ( )x / S ( / ) /

BE||CD CBEDACD

AE BE AE AEAD CD AE

AD AT

AT AE AD

AT AT

T

| | |

DQ DP CQ CM PQ QMPN NM

PQ NM QM PNPQMN

ABCD

AB+CD=AD+BC= ADAB+CD+BC+AD

= AD+ AD= AD

BA

CD

E

F

G

OR

DF DG AE AF

ˆ ˆAB || DC,AD\ A Dˆ ˆA D O

O AODOF AD F

AB DCOF AF FD R AE DG

R AB DC ( R) AB DC

ABC

R

rr

~ [ x Z : (x k) (x k )]x Z :~ [(x k) (x k )]x Z : (x k) (x k )

S { , }

x

S

x=

S R { } R

S=R

n N : n n n

[p^(p q)] qp^(p q)p qqp

p ~q~(p q)p q~qqp

x;x (A C) x A x Cx B x C (A B )x B C

x;[x (A C) x (B C)](A C) (B C)

x;x A C x A x Cx B x C (A B )x (B C)

x;[x (A B) x (B C)]A B B C

a

b

c

a b c

a b c

r

SrP a

( )r

r

r r r r

( )

R

r r r r ( )R

x x

D [ , )

x xx S { } D

pp(x)

~ [ p : p(x)] p :~ p(x)

| | |

{a},{b,c,d,e}

{a}, {b,c,d,e}{b}, {a,c,d,e}{c}, {a,b,d,e}{d}, {a,b,c,e}{e}, {a,b,c,d}

{a}, {b,c}, {d,e}{a}, {b,d}, {c,e}{a}, {b,e}, {c,d}

[A (B C)] [A (A B)][A (B C) ] [A (A B) ]A [(B C ) (A B )]A [((B C ) B ) A ]A [B A ](A B ) (A A )A B A B

(A B) (A C) (A B) (A C)(A B) (A C )[(A B) A ] [(A B) C ][(A A ) B] [A (B C )]

[A (B C)]A (B C)

n(A B) (m n) n(m n) nn m m n( n n) n

n n

n n nm

A m n

y f (x )y=f(x)

x

y

y f (x)y=f(x)

x

y

y f ( x)y=f(x)

x

y

f

f

fof (x) x ; x D

,

f of (x) x ; x D

fof (x)=xf of(x)=x [ , ]

[ , ]

(x, y);(x, y) [(A B) (C D)](x, y) A B

(x, y) (C D)

x A y B

x C y D(x A x C) (y B y D)(x A C) (y B D)(x, y) (A C) (B D)

g f

(gof )( ) g(f ( )) g( )

(fog)( ) f (g( )) f ( )

(fog)( ) f (g( )) f ( )

(gof )( ) g(f ( )) g( )

f (x) | x |

g(x) x x (x )

fog(x) (x ) (x )

gof (x) (| x | )

fog( ) ( ) ( )

gof ( ) ( ) ( )

( )

(fog)( ) gof ( )

f x x

x x

f fx x f (x ) f (x ) f (x ) f (x )

f (x ) f (x )

|f(x)|+

| | |

x

y

y=

fof

x

yfof

y=

T| a |

y cosax y sin ax

y sin( x) T

| |

y sin x.cos x sin x

T| |

y cos x sin xcos ( x) cos x

T| |

cos x sin x (cos x sin x)

(cos x sin x) cos x

sin x sin xtan x cos x cos xtan x sin x cos x sin x

cos x cos xsin x cos x sin x

cos x

cos x

cos x A

sin x cos x

f ( x) xf (x | x |)

f ( ) x

y x y x y (x )

y x

y x y ( x) y (x )

g

x x

g(x ) g(x ) f (g(x )) f (g(x ))(fog)(x ) (fog)(x )

x

x

x

x f (x) x x xx f (x) x x x

x f (x) x x

sin x cos x sin xtan x cos x cos x cos xtan x sin x cos x sin x

cos x cos x

(sin cos ) sin cos sin cos

sin cos sin cos ( sin )

sin( sin )

sin x sin cos x

sin x cos x

sin x cos xsin x cos x sin x.cos x

sin x cos xsin x.cos x sin x cos x

tan x cot xtan x cot xtan x cot x

(tan x cot x)(tan x cot x) (tan x cot x)

sin x cos x sin x sin x( cos x )cos x

sin x x k,k Z

cos x x k

sin xsin x tan x sin xcos x

sin x cos x sin x

sin x cos x cos x sin x

cos x(sin x ) (sin x )

(sin x )( cos x )

sin x x k,k Z

cos x x k

sin x cos x sin x cos x sin x

cos x( ) sin x cos x

cos x cos x A

| | |

f

[ , )

p( )= p( )=

p( ) a ba b

a ,bp( ) ( ) a b

a b

T| a |

y

x

xy=sinx

y=sinx

k

x

cos x= sinx=

x k

x k

n mq

m q n

m m q m q

q

n m m n

m | n n mq b b

[(b ) ] b

(b ) (b ) (b ) ...

(b ) (b ) q b | b

( ),( )

a m a ( m ) a m ( )

a m a ( m ) a m ( )

a ( m ) ( m ) a

a aa

n (n )

n k n n (n )

n k n k n n(n )

n n n

n k n n(n ) n n

n k n k n

(n ) [n(n )] n n

n k (n ) k k n

(n )(n ) n n n

aa= k a

a= k+ a= k+

a k a

a k a k k a

a k a k k a

n

a q a n q nn n

maxn

(a=bq+r)b an=b r

y=tanx

x

y y=

y= x

y= x )

-

x x

x

x x xlim limx x x

xlimx

x x

xlim( ) limx(x ) (x )

m( ) m( ) n n

x

m m

n n

| | |

r [ ]

! !

!

,

, ( ) ,

( )

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A aba a b a a b ( )

A aba a b a a b ( )

b ab= a= )

A=

Z x my

( ,m)

m { , , , , , , , } m

x y x

x ( , )

x x

x x k( k ) yy ky y y y

x ky y

y x

x y x y

x x x k( k ) y y ky k y k

k= k= k=y x

A

A

A A A A A A A

A A A A A A A

An=A

a b a b a bA A

c d c d c d

a bc a ( )ab bd b

a d ( )ac cd c

bc d d

a( d)+bc=a

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a+d=a b

Ac d

A ad=bc

ta b d a

A A c e b cf d e f

abcdef

cos sinR I

sin cos

cos sin cos sinR

sin cos sin cos

cos sin sin cos

sin cos cos sin

cos sinR

sin cos

| | |

A A

A I

A (A ) A ( I) A

I A

a b cA d e f

g h i| A | (aei bfg cdh) (ceg afh bdi)

a b cB d e f

g h i| B | ( aei bfg cdh) ( ceg afhbdi) | B | | A |

AK

K

x yA B

x y

x xB A

y y

x y xx x

x y y yy

a b d bA A

c d c a| A |a d

,| A |

A =I A=A

| A | (adf be( ) c( )( )) (cd( ) ae( )b( )f ) a.d.f

Aaij= i>j

A

| B | (acf ( )( )d ( )be) (( )cd a( )e

( )bf ) a.c.f

Aaij=

A

m mm

mmm

mmm

| A | ( ) ( )m

( )( )m

|A| (a )

m=m

ˆ ˆACH : A C ˆ ˆ ˆA B,HˆˆABC : B C

AH HCABH ACH AH BH.HCBH AH

B H

CAAAA

ˆˆ ˆ ˆB B,A H ABH ~ ABCAB BC AB BH.BCBH AB

ˆ ˆ ˆ ˆC C,A H ACH ~ ABCAC BC AC CH.BCCH AC

HPM : HP ,PM HM

HA

OA OBA BA

HO

BH OH HA BH

BH

OB OH OA OB

OB

B C

A

xha

x

a

a

aa

b b

ab c

c c

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h ( x)h bhh a hh ch , h /h a h

/

ox AAH´ H

ox H´ AHC oy

d CA

x

O

xy y

CA

B

dH

H

C

AA

Bd

H

H

O

AH ABABH ~ ACH AB ACAH AC

ˆ ˆ ˆ ˆ ˆ ˆAOS SOK,SOK NOM, NOM MOB

ˆ ˆO xOy

A

D

K

B C

ˆ ˆCK || DA, CA A C ˆK Cˆ ˆCK || DA, KA A K

AK AC ( )DB ABCK || DA ( )DC AK

DB AB( ), ( )DC AC

DB AB DB ABDC AC DC DB AC AB

DB DB DC

A A ...An An nAn A A

o An

O

A

A-

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H AH

AH

HA

H -

A-

H-

n

n n n

n n n

n n n

ˆo A oH oHˆo A oH oHˆo A oH oH

ˆo A oH oHˆo A oH oH

ˆoH oH o A

A A ...An An nA A A A

o An An

O

A-

A- A

AA

AA

A

A-

n n n n

n n n n

n n

o A A oA oAo A A oA oAo A A oA oA

o A A oA oAo A A oA oA

oA oA o A A

a b a.b

c d c.d

a b c da.b c.d a.b. c.d

a b c d a.b.c.d

a b cd

a b ca b c a b ca.b.c.

a b c a b ca.b.c.

a b c a b c( ) a.b.c.

a b c a b c( ) a.b.c a.b.c