Lettres grecques et symboles mathématiques
alpha
beta
gamma
delta
epsilon
zeta
eta
theta
iota
kappa
lambda
mu
nu
xi
o
omicron
pi
rho
sigma
tau
upsilon
phi
Theta
chi
psi
Lambda
omega
Xi
Gamma
Delta
Pi
∀ ∩ ⊂
Sigma
Pour tout
∃ ⇒ ∪
Intersection
Upsilon
Il existe
Réunion
Phi
Implique
vide
⇐⇒ ∈ Psi
Omega
Equivalent
appartient
est inclus
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
1
ﺗﻠﺨﯿﺺ ﻓﻲ ﻣﺎدة اﻟﺮﯾﺎﺿﯿﺎت اﻟﺪوال اﻟﻤﺜﻠﺜﯿﺔ ودواﻟﮭﺎ اﻟﻌﻜﺴﯿﺔ اﻟﺪاﺋﺮة اﻟﻤﺜﻠﺜﯿﺔ اﻟﻔﺎﺻﻠﺔ ھﻲ) ( cosو اﻟﺘﺮﺗﯿﺒﺔ ھﻲ ) (Sin
)) (A(cos( ),sin
اﻟﺪاﻟﺔ)Y= Sin(x اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ ∞<
< ∞), −
( +
) !)
( (
+⋯+
!
−
=
) !)
( (
∑= ) (sin
اﻟﺷﻛل اﻷﺳﻲ =) (sin اﻟﻣﺷﺗق ) (= cos اﻟرﺳم اﻟﺑﯾﺎﻧﻲ ]Sin(x) :[-][-1 ;1 )X sin(x
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
2
sin(x)
ﺑﻌض اﻟﻌﻼﻗﺎت اﻟﺗﻲ ﺗﺧص اﻟداﻟﺔ
sin ( ) = 1 − cos ( ) 1 = (1 − cos(2 )) 2 Sin (0) =0 sin(− ) =−sin( ) … … … … … … (fonction impair) Sin( ± )=sin( )cos( ) ±sin( )cos( ) Sin (2 ) =2sin( )cos( ) ( ) = ( )
Sin (3 ) = 3Sin ( )-4sin ( ) Sin( ∓ )= ±sin( ) Sin( )+sin( )=2sin( ) ( ) )
Sin( )-sin( )=2cos(
(
)
Sin( )sin( )= [cos( − ) −
( + )]
Y=sin ( )=Arcsin(x)داﻟﺘﮭﺎ اﻟﻌﻜﺴﯿﺔ ھﻲ اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ Arcsin(x)= −∞ <
+
. .
+
. . . .
+
. . . . .
+⋯+
. . …..( . . …..
) (
)
+ (
),
<∞
اﻟﻣﺷﺗق =
1 √1 − اﻟرﺳم اﻟﺑﯾﺎﻧﻲ
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
3
Arcsin(x): [-1 ;1] [-] X arcsin (x)
Y=Cos(x)اﻟﺪاﻟﺔ اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ cos( ) =∑
(
) !
=1−
!
+⋯+
(
) !
+ (
), −∞ <
<∞
اﻟﺷﻛل اﻷﺳﻲ cos( ) = اﻟﻣﺷﺗق = −sin ( ) اﻟرﺳم اﻟﺑﯾﺎﻧﻲ Cos(x):[ 0;][-1;1] X cos (x)
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
4
Cos(x)
ﺑﻌض اﻟﻌﻼﻗﺎت اﻟﺗﻲ ﺗﺧص اﻟداﻟﺔ
( )=1− ( ) 1 = (1 + (2 )) 2 cos (0) =1 cos(− ) = cos( ) … … … … … … (fonction pair) cos( ± ) =cos( )cos( ) ∓sin( )cos( ) cos (2 ) = ( )− ( ) ( )−1 =2 =1−2 ( ) =
( ) ( )
cos (3 ) = -3cos ( )+4cos ( ) cos( ∓ ) =− cos( ) cos( )+cos( ) =2cos( ) ( ) cos( )-cos( ) =−2si (
)
(
cos( )cos( )= [cos( − ) +
) ( + )] ﺑﻌﺾ اﻟﻌﻼﻗﺎت اﻟﺘﻲ ﺗﺮﺑﻄﮭﻤﺎsin وcos
sin ( ) + cos ( ) = 1 sin( )cos( )= [sin(α + β) + sin(α − β)] cos( ) sin ( )= [sin(α + β) − sin(α − β)] cos( ± ) =∓ sin( )
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
5
) (Sin( ± )= cos داﻟﺘﮭﺎ اﻟﻌﻜﺴﯿﺔ ھﻲ)( )=Arccos(x
Y=cos
اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ ( +
),
) )
(. . ….. (
. . …..
−⋯−
. . . . .
+
. . . .
−
. .
−
−
=)Arccos(x ∞<
< ∞−
اﻟﻣﺷﺗق −1 √1 −
=
اﻟرﺳم اﻟﺑﯾﺎﻧﻲ ]Arccos(x): [-1;1] [ 0; )X arccos (x
اﻟﺪاﻟﺔ)Y=Tan(x اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ
∞<
< ∞), −
( +
+
+
tan(x)= +
اﻟﺷﻛل اﻷﺳﻲ )
( tan(x)= −
اﻟﻣﺷﺗق
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
6
=
1 = 1 + tan ( ) cos ( ) اﻟرﺳم اﻟﺑﯾﺎﻧﻲ Tan(x):][ |R X tan (x)
tan(x)
ﺑﻌض اﻟﻌﻼﻗﺎت اﻟﺗﻲ ﺗﺧص اﻟداﻟﺔ
( ) ( )
( )=
= −1 + =
1− 1+
1 ( ) (2 ) (2 ) ( )
=
( )
tan (0) =0 tan(−θ) = −tan(θ) … … … … … … (fonction impair) ( )± ( ) tan( ± ) = ∓
tan (2 )
=
tan( ∓ ) = tan( ± ) =
( ) ( )
(
( ) ( ∓ ) ±
=
( ∓ ) ( ± )
=
( )
=∓ tan( )
( ) ( )
=∓ cot( )
=∓
( ± ) ∓ ( ) ( ± )
tan( ) ±tan( ) =
( )
( )
( )
Y=tan ( )=Arctan(x)داﻟﺘﮭﺎ اﻟﻌﻜﺴﯿﺔ ھﻲ
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
7
اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ ∞<
< ∞), −
( +
)
(
+⋯+
−
=
)
(
∑= )Arctan(x
اﻟﻣﺷﺗق = اﻟرﺳم اﻟﺑﯾﺎﻧﻲ Arctan(x): |R ][ X Arc tan (x)
اﻟﺪاﻟﺔ)=cotan(x اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ
اﻟﻣﺷﺗق ) ( = −1 − co tan
) (
=
اﻟرﺳم اﻟﺑﯾﺎﻧﻲ cotan(x): ]0; [R X cotan (x)
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
8
داﻟﺘﮭﺎ اﻟﻌﻜﺴﯿﺔ ھﻲ)Y=cotan ( )=Arccotang(x اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ ),
( +
)
(
−⋯+
+
−
=
)
(
∑ Arccotan(x) = + ∞<
< ∞−
اﻟﻣﺷﺗق −1 1+
=
اﻟرﺳم اﻟﺑﯾﺎﻧﻲ Arccotan(x): R]0; [ X Arccotan (x)
اﻟﺪاﻟﺔ)Y=ch(x اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
9
ch( ) =∑
!
= 1+
!
+ ⋯+
!
+ (
), −∞ <
<∞
اﻟﺷﻛل اﻷﺳﻲ ch( ) = اﻟﻣﺷﺗق = ℎ( ) اﻟرﺳم اﻟﺑﯾﺎﻧﻲ ch(x): R[1;+∞[ X ch (x)
chﺑﻌض اﻟﻌﻼﻗﺎت اﻟﺗﻲ ﺗﺧص ℎ(0) =1 ch(−θ) = ch(θ) … … … … … … (fonction pair) ℎ ( ) =1+ ℎ ( ) ch( ± ) =ch( )ch( ) ± sh( )sh( ) ch (2 ) = ℎ ( ) + ℎ ( ) =2 ℎ ( ) + 1 =2 ℎ ( )−1 =
( ) ( )
ch( )+ch( ) =2ch( ch( )-ch( ) =2sh(
) ℎ( ) ℎ(
) )
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
10
chو shﺑﻌﺾ اﻟﻌﻼﻗﺎت اﻟﺘﻲ ﺗﺮﺑﻄﮭﻤﺎ ch ( ) − sh ( ) = 1 ch( ) ±sh( ) = e± ) ([ ℎ( ) ± ℎ( ) ] = ch( ) ±sh = ± ) =( ± داﻟﺘﮭﺎ اﻟﻌﻜﺴﯿﺔ ھﻲ)Y=ch ( )=Argch(x اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ
اﻟﻣﺷﺗق √
=
اﻟﺪاﻟﺔ)=sh(x اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ ∞<
< ∞), −
( +
!)
( + ⋯+
!
+
=
!)
(
∑= ) (sh
اﻟﺷﻛل اﻷﺳﻲ =) (sh اﻟﻣﺷﺗق ) (= ℎ اﻟرﺳم اﻟﺑﯾﺎﻧﻲ sh(x): R R X sh (x)
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
11
ﺑﻌض اﻟﻌﻼﻗﺎت اﻟﺗﻲ ﺗﺧصsh )ℎ(0 =0 )sh(−θ )= − sh(θ) … … … … … … (fonction impair ) ( ℎ = ℎ ( )−1 ) (sh( ± ) =sh( )ch( ) ± sh( )ch ) sh (2 ) (=2sh( )ch ) ( = ∓
() ℎ ) داﻟﺘﮭﺎ اﻟﻌﻜﺴﯿﺔ ھﻲ)Y=sh ( )=Argsh(x
) ( ±
(sh( ) ±sh( ) =2sh
اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ ?∞ <
< ∞), −
( +
!)
( + ⋯+
!
+
=
!)
∑= ) (sh
(
اﻟﻣﺷﺗق 1 √1 +
=
اﻟﺪاﻟﺔ)=th(x اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ
∞<
( +
< ∞), −
−
−
+
=)Th(x
اﻟﺷﻛل اﻷﺳﻲ ) (
=
) (
= )th(x
اﻟﻣﺷﺗق ) (
= )
) (
(= ]
)
([ =
)
(=
اﻟرﺳم اﻟﺑﯾﺎﻧﻲ th(x): R ]-1;1[ X th (x)
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
12
thﺑﻌض اﻟﻌﻼﻗﺎت اﻟﺗﻲ ﺗﺧص ℎ(0) th(−θ)
=0 = − th(θ) … … … … … … (fonction impair) ℎ ( ) ℎ ( )= ℎ ( ) 1 =1− ( ) 1 − ℎ(2 ) =− 1 + ℎ(2 ) ( )
= th( ± ) = th (2 )
( ) ( )± ( ) ±
=
( ) ( )
th( ) ±th( ) =
( )
( ) ( ± ) ( )
( )
Y=th ( )=Argth(x)داﻟﺘﮭﺎ اﻟﻌﻜﺴﯿﺔ ھﻲ اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ Argth( ) =∑ Argth(x)=−
=
(
)
(
)=
+
(
+⋯+ (
)
+ (
), −∞ <
<∞
) اﻟﻣﺷﺗق
=
−1 1− =
th(x)اﻟﺪاﻟﺔ
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
13
اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ
اﻟﺷﻛل اﻷﺳﻲ اﻟﻣﺷﺗق ) (
= )
) (
([= −
(] = −
)
)
(
=
اﻟرﺳم اﻟﺑﯾﺎﻧﻲ coth(x): R\{0} R\{[-1 ;1]} X coth (x)
داﻟﺘﮭﺎ اﻟﻌﻜﺴﯿﺔ ھﻲ)Y=coth ( )=Argcoth(x اﻟﻧﺷر اﻟﻣﺣدود ﻟﻠداﻟﺔ ? )
(
=)Argcoth(x
اﻟﻣﺷﺗق = اﻟرﺳم اﻟﺑﯾﺎﻧﻲ [1;+∞[ R R R ]-1;1[ R
Argcosh : Argsinh: Argtanh:
}R\{[-1 ;1]} R\{0
Argcoth:
زوارق ﯾﺣﻲ
ZOUAREG YAHIA
14
ﺟﺪول ﻳﻮﺿﺢ دوال وﺑﻌﺾ ﻣﺸﺘﻘـﺎﺗﻬﺎ اﻟﺪاﻟﺔ Cos(x) Arccos(x)
ﻣﺠﻤﻮﻋﺔ اﻻﻧﻄﻼق [ 0;] [-1;1]
ﻣﺠﻤﻮﻋﺔ اﻟﻮﺻﻮل [-1;1] [ 0;]
Ch(x) Argch(x)
R [1;+∞[
[1;+∞[
Sin(x) Arcsin(x)
[-] [-1 ;1]
[-1 ;1] [-]
Sh(x) Argsh(x)
R R
R R
R
ﻣﺸﺘﻘﺘﮭﺎ -Sin(x) −1 √1 − Sh(x) 1 √ −1 Cos(x) 1 √1 − Ch(x) 1 √
زوارق ﯾﺣﻲ
+1
ZOUAREG YAHIA
15
sin cos Arctang(x)
][
R
R
][
R
]-1;1[
Argth(x)
]-1;1[
R
Cotang(x)
]0; [
R
Arccotang(x)
R
]0; [
Coth(x)
R\{0}
R\{[-1 ;1]}
tan
=
th(x)=
1 1+
1 = ℎ (
4 + ) −1 1− -1−co (x)= −1 1+ −1 −4 ℎ
Argcoth(x)
R\{[-1 ;1]}
R\{0}
زوارق ﯾﺣﻲ
(x)=
1+
=
(
− 1 1−
−
ZOUAREG YAHIA
16
)2