الرياضيات باك2 ملخص الدروس .pdf


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=0
x − x0

f (x ) − f (x 0 )
= a
x − x0
x →x 0
(a ≠ 0)

  lim +

f (x ) − f (x 0 )
=0
x − x0
x →x 0

  lim +

f (x ) − f (x 0 )
= −∞
x − x0
x →x 0

  lim +

f (x ) − f (x 0 )
= +∞
x − x0
x →x 0

  lim +

f (x ) − f (x 0 )
= a
x →x 0
x − x0
(a ≠ 0)

  lim −

f (x ) − f (x 0 )
=0
x →x 0
x − x0

  lim −

f (x ) − f (x 0 )
= −∞
x →x 0
x − x0

  lim −

f (x ) − f (x 0 )
= +∞
x →x 0
x − x0

  lim −

/-0‫ – 
\ א‬/-0‫  א‬5
4‫( א‬05 K2

]9J‫ א‬97
W/-0‫  א‬5↖

  (C f ) ^n,+) x7 x = a j( vZ‫א‬o/‫א‬O
 WO  ‫א‬O h!p‫א‬e/y‫א‬S&
∀x ∈ D f

  ∀x ∈ D f

(2a − x ) ∈ Df



f (2a − x ) = f (x )



W/-0‫↖
\ א‬
  (C f ) ^n,+) x8! I (a,b )0/‫א‬O WO  ‫א‬O h!p‫א‬e/y‫א‬S&
∀x ∈ D f

  ∀x ∈ D f

(2a − x ) ∈ D f



f (2a − x ) + f (x ) = 2b



W]9J‫ א‬97 - ^(%‫ – א‬7‫↖א‬
 #O 8‫א‬S&3 2^‫!א‬/‫^(א‬nO
 3 d‫אא‬Z.^j- @ JHzcy
  ∀x ∈ I
 

f ′′ (x ) ≤ 0 WO 8‫א‬S&

  I 3 d‫!א^א‬/O (C f ) ^n‫א‬WOTU
 #O 8‫א‬S&3 2^ #7‫^(א‬nO
 3 d‫אא‬Z.^j- @ JHz6U
 ∀x ∈ I

f ′′ (x ) ≥ 0 WO 8‫א‬S&

  I 3 d‫ ^א‬#7O (C f ) ^n‫א‬WOTU
  .#r‫^א‬n‫=א‬0/D.‫^(א‬n9 0‫א‬0/
 ^n‫אא‬Z.!/-KV
   %{‫א‬KV-H x 0 a#- f ′′ c 8‫א‬S&
  x 0  C4UP9 0‫א‬0/+</ (C f ) ^n‫א‬OTU
   %{‫א‬KV-O ( x 0 a#- f ′ c 8‫א‬S&
  x 0  C4UP9 0‫א‬0/+</ (C f ) ^n‫א‬OTU
12

lim f (x ) = ∞

lim f (x ) = a

lim [ f (x ) − (ax + b )] = 0

4‫( א‬05 K2

M=.‫א&! א‬

x →∞

f (x )
= a
(a ≠0)
x →∞ x
lim

lim [ f (x ) − ax ] = b

x →∞

 

 W+</ (C f )

 

 W+</ (C f )
*   /

 /UP  /
 Wj( 
 

y =a

  ∞ ‫א‬u

lim f (x ) = ∞

x →∞

x →∞

 Wj( 
y = ax + b
  ∞ ‫א‬u
 

x →a

f (x )
=∞
x →∞ x
lim

f (x )
=0
x →∞ x
lim

lim [ f (x ) − ax ] = ∞

x →∞

 W+</ (C f )
 ,|% !U
o/‫א‬j. }‫א‬
 j( vZ‫א‬
y = ax
  ∞ ‫א‬u
 

13

 

 

 W+</ (C f )

 W+</ (C f )

 ,|% !U

 ,|% !U

 j. }‫א‬

 j. }‫א‬

 t-‫אא‬7
  ∞ ‫א‬u

+> U‫א‬7
  ∞ ‫א‬u

 

 W+</ (C f )
 (,  /
 Wj( 
 x

=a

4‫( א‬05 K2



‫א(א א‬

  I 3 2^ 0g<- 4‫ (א‬f c 8‫א‬S&

WNH ↖

  I 3 d‫א‬w f (I ) 3 d‫=א‬U!‫(א‬+</- f OTU



f −1 W!  C!
 f −1 (y ) = x

f (x ) = y





⇔



x

I
y ∈ f (I )





∀x ∈ I
( f −1ο f )(x ) = x
∀y ∈ f (I )

WLM



( f ο f −1 )(y ) = y



W‫ א(א א‬YN ((_ ↖
  I 3 2^ 0g<- 4‫ (א‬f =
  I 3 d‫!א=א‬4 y   f (I ) 3 d‫!א=א‬4 x =
  f −1 (x ) = y ⇔ f (y ) = x WD ‫~א‬U   @5
  f (I ) = x !4+ f −1 (x ) V> x 5#  y ##n 
W‫ א(א א‬41‫↖א‬
  I 3 2^ 0g<- 4‫ (א‬f c 8‫א‬S&
  f (I ) 3 d‫^א‬4 f −1 ‫אא‬#‫א‬OTU
W‫ א(א א‬S7‫↖א‬
  I 3 2^ 0g<- 4‫ (א‬f =
  y0 = f (x 0 )   f (I ) 3 d‫!א=א‬4 x 0 =
  f ' (x 0 ) ≠ 0   x 0 a6 /%* g f c 8‫א‬S&
  y 0 a6 /%* g f −1 ‫אא‬#‫א‬OTU
'

  ( f −1 ) (y 0 ) =

1
 
f ' (x 0 )

W #

  I 3 2^ 0g<- 4‫ (א‬f =
  I 3 d‫^א‬#-5 f ′ /p‫  (א" א‬I 3 d‫א‬+6 /%* g f c 8‫א‬S&
  f (I ) 3 d‫^א‬6 /%* g f −1 ‫אא‬#‫א‬OTU
  ∀x ∈ f (I )

'

( f −1 ) (x ) = f '  f −11 (x )


14



W #

W‫ א(א א‬81↖
  I 3 2^ 0g<- 4‫ (א‬f =
  f ‫א‬#‫א‬KV-^nG C f −1 ‫אא‬#‫א‬
W‫ (א א‬Q #‫ א‬/0‫↖א‬
  I 3 2^ 0g<- 4‫ (א‬f =
o€J# oa f −1   f L‫א‬#O  <‫א‬O*E,‫א‬
 o,3 ‫א‬14,< O*) ,

W6 I).↖

(

)

  C f −1 ^n‫א‬

(

  A ' (b, a ) ∈ C f −1

)

 

  /UP  /+</
  y = a Wj( 
  (,  /+</
  x = b Wj( 
1
a

b
a

( )

  C f ^n‫א‬
 
 
 

  y = x + Wj( *   /+</
Wg*‫ =א‬g*0‫א (א‬##yo
  x = ay + b
 Eq J14 PF @ J+</
  /UP
 Eq J14 PF @ J+</
  (,

 

 
 

15

  A (a,b ) ∈ (C f )
  (,  /+</
  x = a Wj( 
  /UP  /+</
  y = b Wj( 
 *   /+</
  y = ax + b Wj( 
 Eq J14 PF @ J+</
  (,
 Eq J14 PF @ J+</
  /UP

(n ∈  *)n  1‫א א@?  א‬+
4‫( א‬05 K2



?@‫ ` א‬7‫א‬

W:1 NH↖
  n <-!‫=א‬Z;‫^(אא‬,-‫(א‬+</-  + ^U!‫ א‬x  x n W‫א‬#‫א‬
 

n

:  + → +
n

x  x

  ∀ (x ; y ) ∈ 2+

n



W!  C!

n

x = y ⇔ x = yn

WNH J)
x = 2x

x 3t‫א‬Z;‫^א‬, 3 x W(#‫א‬




( )

∀ (x ; y ) ∈  2+
n

∀ (m; n ) ∈ *

2

∀ (x ; y ) ∈  2+

x × n y = n x ×y

n

(n x )m = n x m
x
x
=n
ny
y
=

∀n ∈  *

xn = x

(n x )n = x

n

nmx

WNH↖

(y ≠ 0)

n×m x

n

x =ny ⇔x =y

n

x >ny ⇔x >y

W6 I).
3

x −y
3
x ² + 3 x 3 y + 3 y²

x −3y =

x− y =

x −y
x+ y

W:‫  א‬0;↖
 WD. f ‫א‬#‫א‬1!-,2

 WD ,8U! f ‫א‬#‫א‬

D f = [0; +∞[

f (x ) = n x

D f = {x ∈  / x ∈ Du  u (x ) ≥ 0}

f (x ) = n u (x )

W=‫↖א‬
lim
x →x0

lim u (x )

n u (x )

x →x0

l≥0
+∞

n

l
+∞

  −∞ # P +∞ # P ‫ ^א‬x 0 # PL,‫ ^א‬x 0 # >^/<-  "‫]א‬Z.
16



W41J‫↖א‬
   + ^4 x  n x ‫א‬#‫א‬


  I 3 2^U!‫ (א‬u =
  I 3 d‫^א‬4 x  n u (x ) ‫א‬#‫א‬OTU I 3 2^4 <‫ (א‬u c 8‫א‬S&

WS7J‫↖א‬
  I 3 2^U!‫ (א‬u =

  ]0;+∞[ 3 d‫^א‬6 /%* g x  n x ‫א‬#‫א‬

  I 3 2^6 /%* g  0g<‫ (א‬u c 8‫א‬S&
  I 3 d‫^א‬6 /%* g x  n u (x ) ‫א‬#‫א‬OTU
∀x ∈ I

(

n

u (x )

)′ =

u ′ (x )
n n [u (x )]n −1

 W #



(n x )′ =

∀x ∈ ]0; +∞[

 W #

1
n

n x

(a ∈  ) x ∈ 
 D X(# n

n −1

x n = a W+#‫ א‬/)↖

 v(!U(# n

S = {−n a ; n a }
S = {0}

S = {n a }
S = {0}

S =∅

S = {−n a }

a >0
a=0
a <0

W9T B  Q77) +( ?@‫ ` א‬7‫↖א‬
  q ∈ *  p ∈ * WM$#K? Z‫(א‬# r =

p
=
q

p

  ∀x ∈ ]0, +∞[

q

r

x = xq = xp

WI).
1
xn

 (r ∈  *)

nx =
  ∀x ∈ ]0; +∞[
WD ,8U! x D//$KV f (#‫(א‬1!-,2

f (x ) = [u (x )]r




  Df = {x ∈  / x ∈ Du  u (x ) > 0} WD.


1 ′





(n u (x ) )′ = (u (x ))n 

=

1
1
× u ' (x ) × [u (x )]n −1 •
n

   * = r ′  r =!4+ *+ = y x =!4+
  x r × x r ' = x r +r '
  (x × y ) = x × y
r


   x

r

r 

r

 = x r −r ′
 x r ′ 

r'



  (x r ) = x r×r '



   x  = x r
y
 y 

 r

 



17

1
x

r'



r

= x −r '






4‫( א‬05 K2

+(‫ א‬#‫א‬



W(Z‫ א‬#‫ – א‬8,‫ א‬#‫↖א‬
 @#. 

  $ 

  un +1 = q × un

  un +1 = un + r

 q @‫א‬. q
  un = u p × q n −p

 

 q @‫א‬. r
  un = u p + (n − p )r

  (p ≤ n )

 1!  ‫א‬# ‫א‬

  (p ≤ n )

q n −p +1 − 1

u p + ... + un = u p × 
 n −p +1 − 1 ( #$:,2
 q − 1    u + ... + u = u × q

p
n
p
 q − 1    

  (q

≠ 1)

 c  b  a
( #$)*)

  2b = a + c

 b ² = a × c

  

:* Y#‫ א‬#‫ * – א‬#‫ א‬#‫↖א‬
 (#  (un )n ∈I =
M (#  < (un )n ∈I ⇔ ∀n ∈ I

un ≤ M



m (#  V4 (un )n ∈I ⇔ ∀n ∈ I

un ≥ m



  ( #7 (un )n ∈I ⇔  V4  < (un )n ∈I



W+(  81↖
 (#  (un )n ∈I =
4g - (un )n ∈I ⇔ ∀n ∈ I

un +1 ≤ un



#‫א‬- (un )n ∈I ⇔ ∀n ∈ I

un +1 ≥ un



  ) (un )n ∈I ⇔ ∀n ∈ I

un +1



18

= un

W =↖
W α ∈  * Wa) (n α ) #‫= א‬

α<0

α>0

lim n α = 0

lim n α = +∞

n →+∞

n →+∞

Wq ∈  Wa) (q n ) (Z‫ א‬#‫= א‬
q ≤ −1

  (q n )  ‫א‬
  " CG

−1 < q < 1

  lim q n = 0
n →+∞

q =1

q >1

  lim q n = 1   lim q n = +∞
n →+∞

n →+∞

W^7‫ א‬b+↖
  / D. < #‫א‬- +8



  / D. V4 4g - +8




vn ≤ un ≤ wn 

lim vn = l  ⇒ lim un = l
 n →∞
n →+∞
lim vn = l 
n →+∞


un − l ≤ vn 
 ⇒ lim u = l
lim vn = 0 n →∞ n
n →+∞


un ≥ vn

un ≤ vn



⇒ lim u = +∞
lim vn = +∞ n →+∞ n

n →+∞



⇒ lim u = −∞
lim vn = −∞ n →+∞ n

n →+∞

W un +1 = f (un ) ! ‫↖  א‬
 WD ,8U!‫( א‬un )  ‫א‬Y

u0 = a
 
un +1 = f (un )

  I =‫!א‬4 a  f (I ) ⊂ I Mf I 3 2^4‫ (א‬f M$

  f (x ) = x W( ,+$ l  " " OTU  / (un ) c 8‫א‬S&
 

19

4‫( א‬05 K2

Nc‫ א‬4‫א(א‬



W4; D  ‫ (א‬Nc‫ א‬4‫↖א(א‬
W:1

  I 3 2^U!(#‫ (א‬f =
  I 3 d‫ ^א‬f ‫א‬#>P‫(א‬D. F OP3/
 WO  ‫א‬O h!p‫א‬e/y‫א‬S&
I 3 d‫^א‬6 /%* g F

∀x ∈ I



F ' (x ) = f (x ) •

WNH

 3 d‫אא‬Z.^>P‫(א‬+</-3 2^4‫(א‬+8
  I 3 2^U!(#‫ (א‬f =
 WOTU I 3 d‫ ^א‬f ‫א‬#>P‫ (א‬F c 8‫א‬S&
 WD W I ^U! f ‫א‬#>‫א‬3‫ א‬#‫א‬Hz
x  F (x ) + k

(k ∈  )

  I 3 2^>P‫(א‬+</-(#‫ (א‬f =
  » =‫!א‬4 y0  I =‫!א‬4 x 0 =
I 3 d‫ ^א‬f ‫א‬# F  #$ >P‫(א‬#-

  F (x 0 ) = y0 WD#<‫!א‬p‫א‬e/y
WQ77) +(  ‫א‬+ F ‫ @(א‬-E‫א‬+ ! 0d WNc‫ א‬4‫↖א(א‬
WNH

  //$‫(א‬# k  I 3 2^LU!L(#L‫ (א‬g   f =
 WOTUD‫ ^אא‬I 3 d‫ ^א‬g   f L‫א‬#L>PL‫(א‬G   F c 8‫א‬S&
I 3 d‫ ^א‬f + g ‫א‬#>P‫ (א‬F + G



I 3 d‫ ^א‬kf ‫א‬#>P‫ (א‬kF



20

W+J‫ א‬4‫  < א(א‬Nc‫ א‬4‫ א(א‬4( ↖
f (x )

F (x )

a∈

ax + k
1
x² + k
2
−1
+k
x

x
1

1
x

(r ∈ » * − {-1})

x

2 x +k
x r +1
+k
r +1

r

sin x

− cos x + k

cos x

sin x + k

1 + tan ²x =

1
cos ²x

tan x + k

1
x

ln x + k

ex

ex + k

(k ∈  )

WNc‫ א‬4‫< א(א‬8 ((% S7J‫ א‬eN 40‫↖א‬

(r ∈ » * − {-1})

f (x )

F (x )

u ' (x )
u (x )

2 u (x ) + k

−v ' (x )
[v (x )] ²

1
+k
v (x )

u ' (x ) ×[u (x )]

[u (x )] r +1

r

r +1

u ' (x )
u (x )

+k

ln u (x ) + k
( )

u ' (x ) ×eu x

eu x + k

(a ≠ 0)

cos (ax + b )

(a ≠ 0)

sin (ax + b )

1
sin (ax + b ) + k
a
1
− cos (ax + b ) + k
a

( )

21

(k ∈  )

4‫( א‬05 K2 Q‫^ א‬,‫א‬


W9T D  ‫א‬+ /1↖
I 3 d‫ ^א‬f ‫א‬#>P‫ (א‬F  I 3 2^4‫ (א‬f = W:1

  I 3 d‫!==א‬4b a



WD// ‫(א‬#‫א‬.b k& a = f ‫א‬#‫א‬+ b
∫ f (x ) dx = [F (x )]ba = F (b ) − F (a )
a

WNH↖
 W9f‫א‬
a

a

b

  ∫ f (x ) dx = 0

  ∫ f (x ) dx = −∫ f (x ) dx

a
b
b
b
b
b
  ∫ [ f (x ) + g (x )]dx = ∫ f (x )dx + ∫ g (x )dx  (k ∈  ) ∫ kf (x ) dx = k ∫ f (x ) dx
a
a
a
a
a
W4 T.
b
c
b
(x ) dx =
(x ) dx +
f
f
∫a
∫a
∫c f (x ) dx
b

a

WB1C‫  א‬/‫↖א‬
  ∀x ∈ [a,b ]

f (x ) ≤ g (x ) WO 8‫א‬S&
b
b
  ∫ f (x ) dx ≤ ∫ g (x ) dx WOTU
a
a

  ∀x ∈ [a,b ]

f (x ) ≥ 0 WO 8‫א‬S&

b

  ∫ f (x ) dx ≥ 0 WOTU
a

W9 #‫ א‬07‫↖א‬
  [a,b ]3 2^4‫ (א‬f =
 

b
1
f (x ) dx WD// ‫(א‬#‫א‬D.3 d‫א^א‬#0@‫א‬,/‫א‬
b − a ∫a

WF‫ \א‬c8 #‫↖א‬
I 3 d‫^א‬L4 v ′ u ′ L‫א‬#‫א‬Mf I 3 2^6 /%*L gL‫ (א‬v u =
I 3 d‫!==א‬4b a
b
b
  ∫ u (x )v ′ (x )dx = [u (x ) v (x )]ba − ∫ u ′ (x )v (x )dx
a
a

\) )↖



  (o, i, j ) # ok& ‚‫=א‬

 

  j  i L"|‫  א‬o 0/ (#ƒ‫א‬+0‫א‬$ D. u.A $ ‫ א‬#$


 



  1.u . A = i × j
22

  [a,b ]3 2^L4L‫ (א‬g  f = 

  [a,b ]3 2^4‫ (א‬f =

7 C g  C f Ln‫א‬L 4ƒ‫א א‬$ 

+„> U‫א‬7 C f ^n‫א‬L 4ƒ‫א א‬$ 

W ,. ( =Z‫א‬L,/‫ א‬+> U‫א‬

 W ,. ( =Z‫א‬L,/‫ א‬

 WD. x = b x = a

 x = b x = a

 b


(x ) − g (x ) dx 
f
.u.A
 ∫a


 

 b

  ∫ f (x ) dx .u.A

WD.

 a



WD.

 WNH J)
 WD.o@!‫א‬aD|<‫א א‬$ 

  €$*

 b

  ∫ f (x ) dx .u.A

 < f

 a

  [a,b ]3 d‫^א‬



 b



 < @ f

 a



  [a,b ]3 d‫^א‬

  ∫ −f (x ) dx .u.A


 DnA- o@

 

 

< f  •
 c

b



 a

c



[a, c ] 3 d‫^א‬

  ∫ f (x ) dx + ∫ −f (x ) dx .u.A


< @ f  •
  [c,b ] 3 d‫^א‬

 b

  ∫ ( f (x ) − g (x )) dx .u.A
 a

  (C g ) 6U# (C f )



 

  [a,b ]3 d‫^א‬
 
(C g ) 6U (C f ) •

 c

b
 ∫ ( f (x ) − g (x )) dx + ∫ (g (x ) − f (x ))dx .u.A
c
 a


  [a, c ] 3 d‫^א‬

(C f ) 6U (C g ) •

 

  [c,b ] 3 d‫^א‬

 

:O) ^)↖

73$ (C f ) ^n‫א‬O‫ א‬# #‫א‬od‫א‬o|$
  [a;b ]3 2a 8  (+> U‫א‬
 b

V =  ∫ π ( f (x ))²dx  u.v 
 a


W.

 | ‫ א‬#$ W uv 

23

4‫( א‬05 K2 0g ‫ א‬4‫א(א‬


> ‫ א‬0g ‫↖א(א א‬
W:1
]0; +∞[ 3 d‫ ^א‬x 

1
x

‫א‬#>‫אא‬#‫א‬D.vK<‫א‬o ?‫(אא‬
  ln W!  C! 1a#-r‫ א‬
WNH  ‫א‬

∀x ∈ ]0; +∞[

∀y ∈ ]0; +∞[

ln 1 = 0

ln (xy ) = ln x + ln y

ln e = 1

∀x ∈ ]0; +∞[
∀y ∈ ]0; +∞[
ln x = ln y ⇔ x = y
ln x > ln y ⇔ x > y

(r ∈ ) ln (x r ) = r ln x
1
ln   = − ln x
x 
x 
ln   = ln x − ln y
 y 

∀x ∈ ]0; +∞[

∀y ∈ 

ln x = y ⇔ x = e y

  ∀x ∈  * ln (x n ) = n ln x WOTU   X‫(א‬# n O 8‫א‬S&
W:‫  א‬0;

 WD. f ‫א‬#‫א‬1!-,2
D f = {x ∈  / x ∈ Du ‫ و‬u (x ) > 0}
D f = {x ∈  / x ∈ Du

 WD ,8U! f ‫א‬#‫א‬
f (x ) = ln [u (x )]
f (x ) = ln (u (x ))2 



‫ و‬u (x ) ≠ 0}



  f (x ) = ln u (x )

WA =

ln x
n =0
x →+∞ x
lim

( n ∈ » *)

lim (x n ln x ) = 0

x →0
>

ln (x + 1)
=1
x →0
x
lim

lim (ln x ) = +∞

x →+∞

lim (ln x ) = −∞

x →0
>

ln x
=1
x →1 x − 1
lim

W41J‫א‬

  ]0; +∞[ 3 d‫^א‬4 x

 ln x ‫א‬#‫א‬


  I 3 2^U!‫ (א‬u =

  I 3 d‫^א‬4 x

 ln [u (x )] ‫א‬#‫א‬OTU I 3 2^4  0g< u c 8‫א‬S&
24

WS7J‫א‬
  I 3 2^U!‫ (א‬u =

  I 3 2^6 /%* g  0g<‫ (א‬u c 8‫א‬S&
  I 3 d‫^א‬6 /%* g x
∀x ∈ I

'
)]

 ln [u (x )] ‫א‬#‫א‬WOTU

(ln [u (x ) =

u ' (x )
u (x )



W #

  ]0; +∞[ ^6 /%* g x

: #



1
∀x ∈ ]0; +∞[ (ln x )′ =
x

W ln *'

ln x

-

WQ #‫ א‬/0‫א‬


0 1 +∞

x

 ln x ‫א‬#‫א‬

+

a ∈ » *+ − {1} Wa) a Uh Og ‫↖א(א א‬

  loga W!  C!r‫אא‬#‫א‬D.a q @…o ?‫אא‬#‫ א‬W:1
∀x ∈ ]0; +∞[
∀x ∈ ]0; +∞[

loga (x ) =

∀y ∈ ]0; +∞[

loga (xy ) = loga x + logay

(r ∈  )

ln x
WM$
ln a
WNH  ‫א‬
loga 1 = 0

logaa = 1

loga (x r ) = r loga x

∀x ∈ ]0; +∞[

1
loga   = −loga x
x 

l oga x = l oga y ⇔ x = y

x 
loga   = loga x − logay
 y 



∀y ∈ ]0; +∞[

∀r ∈ 

l oga x = r ⇔ x = a r

:1&  =
0 <a <1
loga x < logay ⇔ x < y
lim loga x = −∞

a >1
loga x > logay ⇔ x > y
lim loga x = +∞

x →+∞

x →+∞

lim loga x = +∞

lim loga x = −∞

x →0+

x →0+

W7X#‫א‬
∀x ∈ ]0, +∞[

( loga x ) ' =
25

1
x ln a

 

4‫( א‬05 K2 c‫ א‬4‫א(א‬


> ‫ א‬0g ‫↖א(א א‬
W:1

K<‫א‬, ?‫אא‬#‫אא‬#‫א‬D.K<‫אא@א‬#‫א‬
  exp W!  C! 
exp (x ) = e x   = x +HB
WNH   ‫א‬
∀x ∈ 

∀x ∈  e x > 0
∀x ∈  ln ( e x ) = x

∀y ∈ 

e x ×ey = e x +y

∀x ∈ 

1
−x
x =e
e
ex
x −y
y =e
e

 

∀y ∈ ]0; +∞[



e x = y ⇔ x = ln y
∀ (x ; y ) ∈  ²
e x = ey ⇔ x = y
e x > ey ⇔ x > y
W:‫  א‬0;

 WD. f ‫א‬#‫א‬1!-,2
 

e ln x = x

∀x ∈ ]0, +∞[

r
(e x ) = erx

(r ∈ )

 WD ,8U! f ‫א‬#‫א‬

Df = 

 

D f = {x ∈  / x ∈ Du }

 

f (x ) = e

f (x ) = e

x

u (x )

WA =
lim e x = +∞

x →+∞

lim e x = 0

( n ∈ » *)

x →−∞
 x

e 

lim  n  = +∞
x →+∞  x 
lim (x ne x ) = 0
x →−∞

ex − 1
lim
=1
x →0
x

W41J‫א‬

   ^4 x

e

x

‫א‬#‫א‬


  I 3 2^U!‫ (א‬u =

  I 3 d‫^א‬4 x

e

u (x )

‫א‬#‫א‬OTU I 3 d‫^א‬4 u c 8‫א‬S&
26

WS7J‫א‬

(ex )′ = ex W #   ^6 /%*  g x

  ∀x ∈ 

e

x

‫א‬#‫א‬

  I 3 2^U!‫ (א‬u =

  I 3 d‫^א‬6 /%* g x

e

u (x )

‫א‬#‫א‬WOTU I 3 d‫^ א‬6 /%* g u c 8‫א‬S&
'

(eu x )
( )

∀x ∈ I

u (x )
= u ′ (x ) ×e


W # 

W ln ‫ (א‬Q #‫ א‬/0‫א‬



a ∈  ∗+ − {1} Wa) a Uh c‫↖א(א א‬

  expa W!  C!  a q @…@‫אא‬#‫^א‬,- loga ‫א‬#‫אא‬#‫א‬

W:1


expa (x ) = a x   = x +HB
WNH  ‫א‬
∀ (x ; y ) ∈  2
a x × a y = a x +y
r
(a x ) = a r x
1
−x
x =a
a
ax
x −y
y =a
a

(r ∈ )

a x = e x ln a
loga (a x ) = x

∀x ∈ 

∀x ∈ ]0; +∞[
∀ (x ; y ) ∈  2

a

loga (x )

=a

a x = ay ⇔ x = y

∀x ∈ 
∀y ∈ ]0; +∞[
x
a = y ⇔ x = loga (y )
W1&  =

0 <a <1
a < ay ⇔ x < y

a >1
a > ay ⇔ x > y

lim a x = 0

lim a x = +∞

x

x

x →+∞

x →+∞

x

lim a x = 0

lim a = +∞

x →−∞

x →−∞
x

a −1
= ln a
x →0
x

lim

(a x )′ = (ln a ) ×a x
27

W 7X#‫א‬

4‫( א‬05 K2

(7‫ א‬+‫(א‬c‫א‬
   = {z = a + ib / (a;b ) ∈ ²   i ² = −1} WD.#/‫א(א‬#‫א‬,2

Wi(7 +( P@‫ א‬8‫↖א‬
  (a;b ) ∈ ² WM$ #/‫(א‬# z = a + ib =
z v#/‫(א‬#Y;‫^א א‬,- a + ib



Re (z ) W! j!  z (#D// ‫^א;†א‬, a (#‫א‬



  Im (z ) W! j!  z (#D‡‫^א;†א‬, b (#‫• א‬
D//$(#. z OTU Im (z ) = 0 WO 8‫א‬S& • WjNH j)
  U!> ˆ‫(א‬#^, z OTU Im (z ) ≠ 0  Re (z ) = 0 WO 8‫א‬S& •

WE(7 +( i1↖
 L#/=(# z ′ z =
  z = z ′ ⇔ Re (z ) = Re (z ′)  Im (z ) = Im (z ′)

Wi(7 +( Q #‫ א‬/0‫↖א‬
 

(o,e1,e2 ) o€J# ok& v#/‫=א‚א‬
 

  (a;b ) ∈ ² WM$ #/‫(א‬# z = a + ib =
  M (a, b ) 0/  z v#/‫(א‬#‫! ‰א‬
M (z ) Wt  z (#‫^> א‬,- M 0/‫ א‬M 0/‫א‬e ^, z (#‫• א‬
 

  z = Aff (OM ) POM (z ) Wt OM "|‫א‬e Z8^, z (#‫• א‬


Wi(7 +( bk‫↖א‬
  (a;b ) ∈ ² WM$ #/‫(א‬# z = a + ib =

z = a − ib Wv#/‫(א‬#‫א‬. z (#‫א‬eU‫!א‬


 D// ‫א‬n,< O*) , M ′ (z )  M (z )

 

 D//$(# z ⇔ z = z



 9!>Dˆ(# z ⇔ z = −z
  z + z = 2 Re (z )
  z − z = 2i Im (z )
  zz = [Re (z )]² + [ Im (z )]²





 z + z ' = z + z ' •
 z ×z ' = z ×z ' •
  (n ∈  *)  z n = z n •
1

1

    =  •
z ' z '



z  z
  (z ' ≠ 0)    =
z '

z'



Wi(7 +( ↖
  (a;b ) ∈ ² WM$ #/‫(א‬# z = a + ib =
  z = zz = a ² + b ² Wt‫א‬D// ‫(א‬#‫א‬. z v#/‫(א‬#‫ א‬
28

z ×z′ = z × z′

zn = z n

z = z

−z = z

1
1
=
z′
z′

z
z
=
z′
z′

(n ∈ * )
(z ' ≠ 0)

W( >g i(7 +( c‫ א‬8‫  א‬Q#‫ א‬/X‫↖א‬
  M j->#K? #/‫(א‬# z =

 (e
1, OM ) W"‫ @ אא א‬g#$P θ . z v#/‫(א‬#‫ א‬#,
 

  arg z W! j!
  arg z = θ [2π ] Wt

 
WNH J)

 #K? #/‫(א‬# z =

 #K? a D//$(#EE‫א א‬
a <0
a >0

  arg z = θ [2π ]  r = z HB

 

 

W. z v#/‫(א‬#DEE‫א‬+p‫• א‬

 

a = [−a, π ]

a = [a, 0 ]

π

ai = −a, − 
2 


π

ai = a, + 
2 


 

z = r (cos θ + i sin θ ) = [r , θ ]

  z = reiθ WD. z v#/‫(א‬#@‫• א א‬
 

reiθ × r ' eiθ ' = rr ' ei (θ +θ ') • [r , θ ] × [r ', θ '] = [rr '; θ + θ ']
[r , θ ] = [r, −θ ]
reiθ = re−iθ •
−[r, θ ] = [r, π + θ ]
−reiθ = re i(π +θ ) •
n
[r , θ ]n = r n ; n θ 
rei θ = r nein θ •
1
1

1
1 −iθ '
=  ; −θ '
=
e

[r '; θ ']  r '

r'
r ' eiθ '

( )

re



r 'e

iθ '

=

[r ; θ ]
=
[r '; θ ']

r i (θ−θ ')
e

r'



arg (zz ') ≡ (arg z + arg z ') [2π ] •



arg z ≡ − arg z [2π ] •
− arg z ≡ (π + arg z )[2π ] •




arg

r

 ; θ − θ ' •
 r '


 ‫
د‬z ⇔ arg z = k π •

( k ∈ )

arg z n ≡ n arg z [2π ] •
1
arg ≡ − arg z [2π ] •
z




د   ف‬z ⇔ arg z =

π
+ kπ •
2

z
≡ (arg z − arg z ') [2π ] •
z'

∀k ∈ 

[r, θ + 2k π ] = [r, θ ]


W>A YN↖
∀θ ∈ 

(
(

Wk‫  א‬YN↖

)
)

1 iθ
e + e −iθ   ∀n ∈ 
2
1 iθ
(cos θ + i sin n θ )n = cos (n θ ) + i sin (n θ )
sin θ =
e − e−i θ
2i
cos θ =

W (a ∈  ) a) z ∈  z ² = a +#‫ א‬/)↖
 W( ‫א‬3$,2

 W( ‫א‬

S = {− a ; a }
S = {0}
S = {−i −a ; i −a }

a>0
a=0
a <0

29

z ∈  z² = a

(a ≠ 0) 77) +‫(א‬Ac  b  a Wa) z ∈ 
 W( ‫א‬3$,2
−b − ∆ −b + ∆ 
S = 
;

 2a

2a
−b
S=
2a
−b − i −∆ −b + i −∆ 
S = 
;



2a
2a

{ }

az ² + bz + c = 0 W+#‫ א‬/)↖
 W( ‫א‬

∆>0
z ∈
∆=0

az 2 + bz + c = 0

(∆ = b2 − 4ac )

∆<0

W(7‫ א‬+‫(א‬c‫ א‬%9  (6 O6&↖
 #/‫א‬g*‫א‬

 D@#C‫א"א‬

AB = z B − z A
z + zB
zI = A
2
 
 z − z A 
[2π ]
AB; AC ≡ arg  c
 z B − z A 
zC − z A
∈
zB − zA
z −z
z −z
z −z
z −z
  D A × D C ∈   P D A × B C ∈ 
z B − z A z B − zC
z B − z A z D − zC

  AB U ‫א‬

(

)

 D@#C‫א"א‬

  [A; B ] 0/‫א‬14 I

(

)


 

  AB
; AC  ‫אא‬q g

 ,/‰/C B A
   ‫א‬#‰/ D  C B A
 #/‫א‬g*‫א‬

AM = r
  r " % A  .8!r‫א! א‬#‫א‬k&D,- M




z − zA = r
(r > 0)

AM = BM
  [AB ] ‰@‫ א‬k&D,- M




z − zA = z − zB

  A a ‫אא‬o gME ABC
  A aLg ‫א‬v ME ABC
  A a ‫אא‬o g Lg ‫א‬v ME ABC
 :*A‫א‬v ME ABC

zC − z A
π

= r ; ± 

zB − zA
2 
zC − z A
= [1; θ ]
zB − zA
zC − z A
π

= 1; ± 

zB − zA
2 

zC − z A
π

= 1; ± 

zB − zA
3 

W+J‫ א‬. %‫(  < א‬7 .l↖
 W.v#/‫א‬jEx

 +n‫א‬





  u "|‫א‬e  b M$ z ′ = z + b

  u "|‫אא‬S t $‫א‬X{‫א‬

  Ω 0/‫א‬e ω M$ z ′ − ω = k (z − ω )

  k j< Ω ]8!vZ‫ א‬h D8 n‫א‬

  Ω 0/‫א‬e ω M$ z ′ − ω = ei θ (z − ω )

  θ j ‫א‬X Ω ]8!vZ‫ א‬r O‫ א‬#‫א‬

30

y (x ) = αeax −

(α ∈  )

 WA ‫ (א‬, ‫א‬+ ‫א‬
  y (x ) = αer1x + βer2x
(α, β ) ∈ ² WM$
  y (x ) = (αx + β )erx
  (α, β ) ∈ ² WM$

 WA ‫א (א‬

b
a

y ' = ay + b
(a ≠ 0)

 W+</- ,‫א (א‬
 L//$L$
  r2   r1 LŠ
  r ‫א‬#$  //$*$

 W ,‫ (" א‬

∆>0
∆=0

  r1 = p − iq
 

r ² + ar + b = 0

(∆ = a ² − 4b )

 WL/U‫א‬QL#/L$
  y (x ) = (α cos qx + β sin qx )e px
  (α, β ) ∈ ² WM$

 WA ‫א (א‬

∆<0

  r2 = p + iq

31

y ''+ ay '+ by = 0

4‫( א‬05 K2 "&‫ א‬J+#‫א‬

 WA ‫ (א‬, ‫א‬+ ‫א‬

4‫( א‬05 K2

Mm&‫( א‬Z‫א‬

  

  (o, i , j , k ) !% <o€J# ok& † B‫אא‡‹=א‬Z.6 @a
 Q=#‫ א‬F‫ א@(א‬J= OI JQ0‫ א‬F‫ א@(א‬W4 %‫ א‬Y‫↖א‬




  ϑ3 =L"| v (a ',b ', c ')   u (a,b, c ) =




u .v = aa '+ bb '+ cc ' 

  •


u = a ² + b ² + c ² 

 •


i a a'
b b'  a a'  a a' 

 u ∧ v = j b b ' =
i −
j +
k
c
c
'
c
c
'
b
b
'

k c c'



Wk#‫↖א‬
WD. B   A L0/L U ‫א‬
AB = (x B − x A ) ² + (yB − yA ) ² + (z B − z A ) ²

WD. ax + by + cz + d = 0 j(  (P ) ‚= M 0/U ‫א‬
  d (M , (Ρ)) =
 
AM ∧ u
  d (A, (∆)) =

u

ax M + byM + cz M + d
a ² + b² + c²



WD. ∆ (A, u ) o/= M 0/U ‫א‬
W`  +↖


  (P ) ‚‫^א‬,€"| n (a,b, c ) ⇔ (Ρ) : ax + by + cz + d = 0




  (ABC ) ‚‫^א‬,€"| AB ∧ AC OTU,/K?‰/ C B A c 8‫א‬S&
WD ‫~א‬U   @5 (ABC ) ‚‫ (א‬##y=m
 



  M ∈ (ABC ) ⇔ AM .(AB ∧ AC ) = 0

Wk +↖


WD. R  " %  Ω (a,b, c )  .8!U( 


  (x − a ) ² + (y − b ) ² + (z − c ) ² = R²


32

 @5  .##y=m [AB ]  . 0gP#$P (S ) U( 
 

 

 

 
M ∈ (S ) ⇔ AM . BM = 0  WD ‫~א‬U 

AB
 " % [AB ] 14 Ω  .8! (S ) ‫א‬W€$*
2

(Ρ) : ax + by + cz + d = 0 `   S (Ω, R) k nR71↖
  (Ρ) ‚‫ ^א‬Ω 8!,v(,‫‰א‬/‫ א‬H =
d = ΩH = d (Ω; (Ρ)) WHB

 

  (S ) ‫א‬H0/ (P ) ‚‫  א‬

 

 

  (C ) !‫(א‬eU
  H W .8!

  (S ) q J (P ) ‚‫א‬

  (S ) ‫( א‬P ) ‚‫א‬

  H 0/‫א‬a

 O h /5

  r = R2 − d 2 W " %

 

 

W (∆) O7  S (Ω, R) k nR71↖
  (∆) o/‫ ^א‬Ω 8!,v(,‫‰א‬/‫ א‬H =
d = ΩH = d (Ω; (∆)) WHB

 

  (S ) ‫א‬H0/ (∆) o/‫א‬

   (S ) q J (∆) o/‫א‬

    (S ) ‫ )∆( א‬o/‫א‬

 LŠL0/a

  H 0/‫א‬a

 O h /5

33

4‫( א‬05 K2 +‫א(א‬


W 0; QM↖
W:1

 CardE W! j!  E ,d‫( >!א‬#. E ",2D
Card∅ = 0

WNH )

WNH
 O "O ,2 B A
Card (A ∪ B ) = CardA + CardB − Card (A ∩ B )

W 0; O0↖
W:1

  E ",2=‫ †א‬A =
  A W!  C!r‫א‬,d‫א‬D. E ,|,< A o,
  A = {x ∈ E / x ∉ A} M$
WI).
A∩A = ∅
A∪A = E
cardA = cardE − cardA





W+‫ (א‬Qc‫ א‬A( #‫↖א‬
  (p ∈  *)

‫ א\ א‬p "| t0- !}Y

 Š8 n1 Fo3 ‫\ א‬5‫א‬O 8‫א‬S&
 Š8 n2 FoD E‫\ א‬5‫א‬O 8
.........................................
 Š8 n p Fo p  \5‫א‬O 8
  n1 × n2 × n3 × ... × n p

W†‫א‬#;‫א‬.,‫(א א‬#OTU
W‫א‬8 j(8  1C‫ א‬- ‫א‬8  1C‫↖א‬
W‫א‬8  1C‫א‬

  (p ≤ n ) » * ==!4 p n =
  n p W.!4 n L =!4 p 3‫<  !א‬-Q‫(א‬#
34

W‫א‬8 j(8  1C‫א‬

  (p ≤ n ) » * ==!4 p n =
 W.!4 n L =!4 p 3‫!א‬-O #  <-Q‫(א‬#
  Anp = n × (n − 1) × (n − 2) × ... × (n − p + 1)

 

 +‫ =אא‬p 
WNH )

 !4 n 3#<-Z8^,-!4 n L =!4 n 3‫!א‬-O # <-!-+8
  n ! = n × (n − 1) × (n − 2) × ... × 2 × 1 W .(#
W&[‫↖א‬
  n  .!> (#",2 E =
  (p ≤ n )  p ]!> (# E = A †+8
 !4 n L =!4 p 3l-^,
 C np =

Anp
 W. l‫]א‬Z.(#
p!

C np  Anp  n ! W+‫(א‬c‫↖א‬
n ∈ ∗

n ! = n × (n − 1) × (n − 2) × ... × 2 × 1
0! = 1
n!
n!
Anp =
C np =
p ! (n − p ) !
(n − p ) !
n −1
0
1
C nn = 1
Cn = n
Cn = 1
Cn = n
C np = C nn −p
C np−1 + C np = C np+1

WB%‫ א! א‬A <8↖
  (p ≤ n ) !4 n L =!4 p tn 
 WD ‫א‬3 #;‫א‬a ‫ ‡‹א‬
 t-Q‫א‬

 W.,‫< א‬n‫(א‬#

 Wtn‫א‬:

 o"K?

 C np

 DŒ

 o"

 np
  Anp

 3*$T  H 

 o"

35

 3*$&O #  H 

4‫( א‬05 K2 J0)J‫א‬


%9↖
 W] 

 WD ,$5‫א‬i04‫א‬

 |=!E8P+</- !}+8

 ‫א‬p !}

 ‫א‬p !|,‫א{  א‬,2D.

   {‫א‬O8 Ω

  Ω   {‫א‬O8=‫ †א‬A

A #$

 ‫א‬#$ ‫!א‬4=,B#$+8

 D‫א‬# ‫א‬#$

 #$‫ א‬OŒa B   A O )# ‫א‬e/y‫א‬S&

  A ∩ B # ‫א‬e/y

   ,. P B  P A e/y‫א‬S&

  A ∪ B # ‫א‬e/y

  (A ∩ A = ∅‫و‬A ∪ A = Ω)  A # ‫א‬.

  A #n( B‫א‬# ‫א‬

A∩B = ∅

 L,|K?O )#$ B A

Wo() 40)‫ א‬J o() ‫א‬7‫↖א‬
 ‫א‬p !}  &O8 Ω = W:1


  pi W. {ωi } # ‫א‬3 ,$‫א‬OP3/ pi j,ga {ωi } D‫א‬# ‫א‬#$3 ,$‫!א‬/ # •
  P ({ωi }) = pi Wt
# ‫אא‬Z.O-r‫אא‬# 5‫א‬5 ,$5‫א‬:,2.#$3 ,$‫• א‬
 W. A # ‫א‬3 ,$‫א‬OTU Ω = )#$ A = {ω1; ω2 ; ω3 ;...; ωn } O 8‫א‬S&vP
p (A) = p (ω1 ) + p (ω2 ) + p (ω3 ) + ... + p (ωn )
 ‫א‬p !}  &O8 Ω =

WNH


p (Ω) = 1   p (∅) = 0 •
Ω = A #$+ 0 ≤ p (A) ≤ 1 •

WL)#$( y‫א‬3 ,$‫• א‬
  Ω = B  A L)#$+
  p (A ∪ B ) = p (A) + p (B ) − p (A ∩ B )
 L,|K? B   A O 8‫א‬S& p (A ∪ B ) = p (A) + p (B )
W( B‫א‬# ‫א‬3 ,$‫• א‬

p (A) = 1 − p (A)

W Ω = A #$+

WJ0)J‫ א‬i1 "k↖
  Ω  " &O8‫א‬p !}a3 ,$5‫א א‬# 5‫אא‬#$‫א‬Hzc 8‫א‬S&
cardA
p (A) =

W. Ω = A #$+83 ,$‫א‬OTU
card Ω
36

W:1


WE-() .7‫ א‬JQRX‫ א‬40)J‫↖א‬
  p (A) ≠ 0 WMf‫א‬p‫א|! א‬G L0<-!L)#$ B   A = W:1
p (A ∩ B )
p (B ) = p (B A) =
W(#‫א‬.e/7 A # ‫א‬OP , B #$3 ,$‫א‬
p
A
(
)
A
  p (A) × p (B ) ≠ 0 WMf‫א‬p‫א|! א‬G L0<-! B   A L)#$+
p (A ∩ B ) = p (A) × p (B A) = p (B ) × p (A B ) W #

W

W:1

 ‫א‬p‫א|! א‬G L0<-! B   A L)#$+
O*/O )#$ B  A ⇔ p (A ∩ B ) = p (A) × p (B )



  Ω 3 Ž} Ω2   Ω1  ‫א‬p !}  &O8 Ω =

WNH

  (Ω1 ∩ Ω2 = ∅  Ω1 ∪ Ω2 = Ω)

( )



( )

  p (A) = p (Ω1 ) × p A Ω + p (Ω2 ) × p A Ω W Ω = A #$+
1
2

W*#‫א א‬HJ‫↖ א‬
  p j ,$‫אא‬p !}a )#$ A =
 W.‰<B  ! k , A # ‫א‬e/y3 ,$‫א‬O U ! n  !|‫]א‬Z.#P‫א‬S&
C nk (p )k (1 − p )n −k

  (k ≤ n )

WQM‫ א‬X >Y 40)‫ א‬j T↖
 ‫א‬p !}  &O8 Ω ^ ‫א‬p‫א‬KV=
 WL ‫א‬L$!‫א‬H< X D‫א‬p‫א‬KV‫א‬3 ,$‫א‬O g##n
X KV‫ א‬.Z\lr‫א‬o/‫א‬,2W X (Ω) = {x1; x 2; x 3 ;...; x n } ##y •

  {1;2;...;n } ,d‫ =א‬i + p (X = x i ) 3 ,$5‫א‬tw •

WQM‫ א‬X >Y# ip‫א‬9‫א] א‬qJ‫ א‬J*Y#‫ א‬JQ"‫ א‬/c‫↖א‬
 j g ‫א‬p‫א‬KV X =

  xi

  x1   x 2   x 3   ...   x n
  p (X = x i )  p1   p2   p3   ...   pn

 WD ‫א‬3 #; 9!

  E (X ) = x1 × p1 + x 2 × p2 + x 3 × p3 + ... + x n × pn  W X KV,DA !‫א‬+‫א‬
 V (X ) = E (X ²) − [E (X )] ²

 W X KV, ! V‫א‬

  σ (X ) = V (X )

 W X KV,vX‫!א‬0‫א‬9‫!א‬w5‫א‬

W:1


WQ‫(א‬,‫ א‬j 7‫↖א‬
  ! n  !|‫]א‬Z.#{‫א‬p !}a A #$3 ,$‫ א‬p =
  p  n ] 0@  ‫א‬#$ X-^, A # ‫" א‬Ue/nr‫(א!אא‬# |+8‰ !vZ‫ א‬X D‫א‬p‫א‬KV‫א‬
  ∀k ∈ {0;1;2;...; n }
 V (X ) = np (1 − p ) 

p (X = k ) = C nk × p k × (1 − p )n −k



E (X ) = n × p 

#

 

37

4‫( א‬05 K2 (>
?1) Q#‫^ א‬,‫א‬


W#‫* א‬M‫  א(א‬+J‫ א‬O7‫ א‬4( ↖

x

0

sin x

0

cosx

1

tan x

0

π

π

π

π

6
1
2
3
2
3
3

4
2
2
2
2

3
3
2
1
2

2

1

3

1

0

W#‫ א‬B‫ א‬E8 T.‫↖א‬

−x

cos (x + 2k π ) = cos x
sin (x + 2k π ) = sin x
tan (x + k π ) = tan x

π − x π +x

π
2

−x

π
2

+x

sin

- sin x

sin x

- sin x

cos x

cos x

cos

cos x

- cos x

- cos x

sin x

− sin x

tan x =

sin x
cos x

1
1 + tan ²x =
cos ²x

-1 ≤ cos x ≤ 1
-1 ≤ sin x ≤ 1
cos ²x + sin ²x = 1

WA  J+↖
cos x = cos a ⇔ x = a + 2k π  P x = -a + 2k π
sin x = sin a ⇔ x = a + 2k π P x = (π - a ) + 2k π
tan x = tan a ⇔ x = a + k π
( k ∈ )
38

W! 0; / _ eN↖
cos (a - b ) = cos a × cos b + sin a × sin b

cos (a + b ) = cos a × cos b - sin a × sin b

sin (a - b ) = sin a × cos b - cos a × sin b

sin (a + b ) = sin a × cos b + cos a × sin b

tan (a - b ) =

tan a - tan b
1 + tan a × tan b

tan (a + b ) =

tan a + tan b
1 - tan a × tan b

WLM↖
t = tan

cos 2a = cos ² a - sin ² a

a
WHA
2

= 2 cos ² a - 1

2t
1 + t²
1 - t²
cos a =
1 + t²
2t
tan a =
1 - t²

= 1 - 2 sin ² a

sin a =

sin 2a = 2 sin a × cos a
2 tan a
1 - tan ² a
1 + cos 2a
cos ² a =
2
1 - cos 2a
sin ² a =
2

tan 2a =

W F‫ (א‬r' ! 0; / _↖ W ! 0; r' F‫ (א‬/ _↖
 p + q 
 p − q 
cos p + cos q = 2 cos 
cos 

 2 
 2 
 p + q   p − q 
cos p − cos q = −2 sin 
sin 
 2   2 
 p + q 
 p − q 
sin p + sin q = 2 sin 
cos 

 2 
 2 
 p + q   p − q 
sin p − sin q = 2 cos 
sin 
 2   2 

1
[ cos (a + b ) + cos (a - b )]
2
1
sin a × sin b = − [ cos (a + b ) − cos (a − b )]
2
1
sin a × cos b = [ sin (a + b ) − sin (a − b )]
2
1
cos a × sin b = [ sin (a + b ) - sin (a − b )]
2
cos a × cos b =



(a,b ) ≠ (0,0) a cos x + b sin x W/ _↖


a
b
a cos x + b sin x = a ² + b ² 
cos x +
sin x 
 a ² + b ²

a ² + b²
= a ² + b ² cos (x − α)

 We/D//$(# α M$
cos α =

a
b
  sin α =
a ² + b²
a ² + b²
39


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