كتاب الدوال المثلثية ودوالها العكسية

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