Formulaire - Pédago`Tech de l`INP Toulouse

INSTITUT NATIONAL POLYTECHNIQUE DE TOULOUSE
MATHS Rappels
Formulaire
Jean-Pierre Bourgade
Pascal Floquet
Xuân Meyer
Année Probatoire A Distance
Septembre 2012
II.1 Fonctions trigonométriques
II.1
1
Fonctions trigonométriques
sinus
sin(π/2) =
√1
sin(π/3) = √3/2
sin(π/4) = 2/2
sin(π/6) = 1/2
sin(−a) = − sin(a)
sin(π − a) = sin(a)
sin(π + a) = − sin(a)
sin(π/2 − a) = cos(a)
sin(π/2 + a) = − cos(a)
cosinus
cos(π/2) = 0
cos(π/3) =√1/2
cos(π/4) = √2/2
cos(π/6) = 3/2
cos(−a) = cos(a)
cos(π − a) = − cos(a)
cos(π + a) = − cos(a)
cos(π/2 − a) = sin(a)
cos(π/2 + a) = − sin(a)
Formules d’addition
tangente
tan(π/2) = √
∞
tan(π/3) = 3
tan(π/4) =
√1
tan(π/6) = 3/3
tan(−a) = − tan(a)
tan(π − a) = − tan(a)
tan(π + a) = tan(a)
tan(π/2 − a) = 1/ tan(a)
tan(π/2 + a) = −1/ tan(a)
Relations usuelles :
sin2 (x) + cos2 (x) = 1
sin(a + b) = sin(a) cos(b) + sin(b) cos(a)
cos(a + b) = cos(a) cos(b) − sin(a) sin(b)
tan(a + b) =
tan(a) + tan(b)
1 − tan(a) tan(b)
sin(a − b) = sin(a) cos(b) − sin(b) cos(a)
cos(a − b) = (cos(a) cos(b) + sin(a) sin(b)
tan(a − b) =
tan(a) − tan(b)
1 + tan(a) tan(b)
sin(2a) = 2 sin(a) cos(a)
cos(2a) = cos2 (a) − sin2 (a)
tan(2a) =
sin2 (x) =
tan2 (x)
1 − cos(2x)
=
2
1 + tan2 (x)
cos2 (x) =
1 + cos(2x)
1
=
2
1 + tan2 (x)
transformation : t = tan( a2 )
2t
1 + t2
sin(a) =
1 − t2
1 + t2
2t
tan(a) =
1 − t2
cos(a) =
2 tan(a)
1 − tan2 (a)
transformation somme-produit :
Transformation produit-somme :
a+b
a−b
sin(a) + sin(b) = 2 sin(
) cos(
)
2
2
sin(a) sin(b) =
cos(a − b) − cos(a + b)
2
a+b
a−b
cos(a) + cos(b) = 2 cos(
) cos(
)
2
2
cos(a) cos(b) =
cos(a − b) + cos(a + b)
2
sin((a + b)
tan(a) + tan(b) =
cos(a) cos(b)
sin(a) cos(b) =
sin(a + b) + sin(a − b)
2
sin(a) − sin(b) = 2 cos(
a+b
a−b
) sin(
)
2
2
cos(a) − cos(b) = −2 sin(
tan(a) − tan(b) =
a+b
a−b
) sin(
)
2
2
sin(a − b)
cos(a) cos(b)
Maths Rappels
II.2 Fonctions hyperboliques
II.2
2
Fonctions hyperboliques
Définitions
ex + e−x
2
ex − e−x
sinh(x) =
2
sinh(x)
ex − e−x
e2x − 1
tanh(x) =
= x
= 2x
−x
cosh(x)
e +e
e +1
cosh(x) =
cosh(x) + sinh(x) = ex
cosh(x) − sinh(x) = e−x
transformation : t = tanh( a2 )
Formules d’addition
cosh(a + b) = cosh(a) cosh(b) + sinh(b) sinh(a)
sinh(a + b) = sinh(a) cosh(b) + sinh(b) cosh(a)
tanh(a) + tanh(b)
tanh(a + b) =
1 + tanh(a) tanh(b)
sinh(a) =
1 + t2
1 − t2
2t
tanh(a) =
1 + t2
cosh(a) =
cosh(a − b) = cosh(a) cosh(b) − sinh(a) sinh(b)
sinh(a − b) = (sinh(a) cosh(b) − cosh(a) sinh(b)
tanh(a − b) =
tanh(a) − tanh(b)
1 − tanh(a) tanh(b)
sinh(2a) = 2 sinh(a) cosh(a) =
2 tanh a
1 − tanh2 a
cosh(2a) = cosh2 (a) + sinh2 (a) = 2 cosh2 a − 1 = 2 sinh2 a + 1 =
tanh(2a) =
2t
1 − t2
2 tanh(a)
1 + tanh2 (a)
1 + tanh2 (a)
1 − tanh2 (a)
transformation somme-produit :
Relations usuelles :
cosh(−x) = cosh(x)
sinh(−x) = − sinh(x)
tanh(−x) = − tanh(x)
coth(−x) = − coth(x)
cosh2 (x) − sinh2 (x) = 1
sinh(a) + sinh(b) = 2 sinh(
a−b
a+b
) cosh(
)
2
2
cosh(a) + cosh(b) = 2 cosh(
a+b
a−b
) cosh(
)
2
2
tanh2 (x)
1 − tanh2 (x)
1
cosh2 (x) =
1 − tanh2 (x)
cosh(x) + 1
x
cosh2 ( ) =
2
2
sinh(a) − sinh(b) = 2 cosh(
a+b
a−b
) sinh(
)
2
2
cosh(a) − cosh(b) = 2 sinh(
a+b
a−b
) sinh(
)
2
2
x
cosh(x) − 1
sinh2 ( ) =
2
2
tanh(a) − tanh(b) =
sinh2 (x) =
sinh(x)
cosh(x) − 1
x
=
=±
tanh( ) =
2
cosh(x) + 1
sinh(x)
tanh(a) + tanh(b) =
s
sinh(a + b)
cosh(a) cosh(b)
sinh(a − b)
cosh(a) cosh(b)
cosh(x) − 1
cosh(x) + 1
Maths Rappels
II.3 Développements limités
II.3
3
Développements limités
Table II.1: Développement limités de fonctions usuelles
f onction
ex
Développement limité d’ordre n au voisinage de 0
ex = 1 +
x
x2
xn
+
+ ··· +
+ xn ǫ(x)
1!
2!
n!
sin(x)
sin(x) = x −
x3
x5
x2p+1
+
+ · · · + (−1)p
+ x2p+2 ǫ(x)
3!
5!
(2p + 1)!
cos(x)
cos(x) = 1 −
x4
x2p
x2
+
+ · · · + (−1)p
+ x2p+1 ǫ(x)
2!
4!
(2p)!
tan(x)
tan(x) = x +
x5
x7
x3
+ 2 + 17
+ x8 ǫ(x)
3
15
315
sh(x)
sh(x) = x +
x5
x2p+1
x3
+
+ ··· +
+ x2p+2 ǫ(x)
3!
5!
(2p + 1)!
ch(x)
ch(x) = 1 +
x4
x2p
x2
+
+ ··· +
+ x2p+1 ǫ(x)
2!
4!
(2p)!
th(x)
th(x) = x −
x3
x5
x7
+ 2 − 17
+ x8 ǫ(x)
3
15
315
1
1+x
1
1−x
1
(1 + x)2
1
= 1 − x + x2 + · · · + (−1)n xn + +xn ǫ(x)
1+x
1
= 1 + x + x2 + · · · + xn + +xn ǫ(x)
1−x
1
= 1 − 2x + 3x2 + · · · + (−1)n (n + 1)xn + xn ǫ(x)
(1 + x)2
√
1+x
√
√
1
1+x
√
1+x=1+
x2
1.3.5 · · · (2n − 3) n
x
−
+ · · · + (−1)n+1
x + xn ǫ(x)
2 2.4
2.4 · · · (2n)
1
1.3.5 · · · (2n − 1) n
x 1.3 2
x + · · · + (−1)n+1
x + xn ǫ(x)
=1− +
2 2.4
2.4 · · · (2n)
1+x
α
α(α − 1) 2
α(α − 1) · · · (α − n + 1) n
x+
x + ··· +
x + xn ǫ(x)
1!
2!
n!
(1 + x)α , α ∈ Q
(1 + x)α = 1 +
ln(1 + x)
ln(1 + x) = x −
arcsin(x)
arcsin((x) = x +
Argsh(x)
Argsh(x) = x −
arctan(x)
Arctan(x) = x −
Argth(x)
Argth(x) = x +
ln(
1+x
)
1−x
x2
xn
+ · · · + (−1)n+1
+ xn ǫ(x)
2
n
1.3.5 · · · (2p − 1)
1 3
x + ··· +
x2p+1 + x2p+2 ǫ(x)
2.3
2.4.6 · · · (2p)(2p + 1)
1 3
1.3.5 · · · (2p − 1)
x + · · · + (−1)p
x2p+1 + x2p+2 ǫ(x)
2.3
2.4.6 · · · (2p)(2p + 1)
x3
x5
x2p+1
+
+ · · · + (−1)p
+ x2p+2 ǫ(x)
3
5
2p + 1
x3
x5
x2p+1
+
+ ··· +
+ x2p+2 ǫ(x)
3
5
2p + 1
1+x
x3
x5
x2p+1
ln(
+ x2p+2 ǫ(x)
)=2 x+
+
+ ··· +
1−x
3
5
2p + 1
Maths Rappels
II.4 Dérivation-intégration
II.4
4
Dérivation-intégration
Table II.2: Dérivées et primitives de fonctions usuelles
f onction
Dérivée
P rimitive
Conditions
a
0
xα
αxα−1
ax
xα+1
α+1
α 6= −1, x > 0
1
x
−
1
x2
1
+ x2
−
(a2
1
− x2
(a2
a2
a2
1
x 2 − a2
√
√
√
−
ln(x)
2x
+ x 2 )2
x
1
Arctan
a
a
a + x
1
ln 2a
a − x
x − a
1
ln
2a x + a 2x
− x 2 )2
2x
(x2 − a2 )2
x 2 + a2
x
√
2
x + a2
x 2 − a2
x
√
2
x − a2
x 6= 0
a 6= 0
a 6= 0, x 6= ±a
a 6= 0, x 6= ±a
√
a2
x√ 2
x + a2 +
ln(x + x2 + a2 )
2
2
a 6= 0
√
a2 x√ 2
ln x + x2 − a2 x − a2 −
2
2
a 6= 0, |x| > |a|
a2
x√ 2
x
a − x2 +
arcsin( )
2
2
a
√
ln(x + x2 + a2 )
ex
x
−√
2
a − x2
x
−p
(x2 + a2 )3
x
−p
(x2 − a2 )3
x
p
(a2 − x2 )3
ax
ax ln(a)
ln(x)
1
x
x ln(x) − x
x>0
loga (x)
1
x ln(a)
x(loga (x) − loga (e))
a > 0, a 6= 1, x > 0
sin(x)
cos(x)
cos(x)
− sin(x)
− cos(x)
sin(x)
a2 − x 2
1
x 2 + a2
1
√
x 2 − a2
1
√
a2 − x 2
√
√
ln x + x2 − a2 ex
1
cos2 (x)
tan(x)
1 + tan2 (x) =
cotan(x)
−(1 + cotan2 (x)) = −
1
sin(x)
−
1
cos(x)
sin(x)
cos2 (x)
cos(x)
sin2 (x)
1
sin2 (x)
a > 0, |x| < a
a 6= 0
a 6= 0, |x| > |a|
x
arcsin( )
a
a 6= 0, |x| < a
ex
ax
ln(a)
a > 0, a 6= 1
π
+ kπ, k ∈ Z
2
− ln |cos(x)|
x 6=
ln |sin(x)|
x 6= kπ, k ∈ Z
ln tan( x2 )
x 6= kπ, k ∈ Z
ln tan( x2 + π4 )
x 6=
π
2
+ kπ, k ∈ Z
Maths Rappels
II.5 Binôme de newton
5
f onction
Dérivée
P rimitive
Conditions
1
cos2 (x)
−2 tan(x)(1 + tan2 (x))
tan(x)
x 6=
1
√
1 − x2
−1
√
1 − x2
1
1 + x2
−1
1 + x2
arcsin((x)
arccos(x)
arctan(x)
arccotan(x)
th(x)
1 − th2 (x) =
coth(x)
1 − coth2 (x) = −
Argch(x)
Argth(x)
Argcoth(x)
II.5
√
1 − x2
|x| < 1
x arccos(x) −
√
1 − x2
|x| < 1
ch(x)
sh(x)
1
ch2 (x)
ln(ch(x))
1
sh2 (x)
ln |sh(x)|
th(x)
ch2 (x)
1
√
1 + x2
1
√
x2 − 1
1
1 − x2
1
1 − x2
−2
Argsh(x)
x arcsin(x) +
√
xarccotan(x) + ln( 1 + x2 )
ch(x)
sh(x)
1
+ kπ, k ∈ Z
√
x arctan(x) − ln( 1 + x2 )
sh(x)
ch(x)
ch2 (x)
π
2
x 6= 0
th(x)
xArgsh(x) −
xArgch(x) −
√
1 + x2
1
√
x>1
x2 − 1
√
xArgth(x) + ln( 1 − x2 )
|x| < 1
√
xArgcoth(x) + ln( x2 − 1)
|x| > 1
Binôme de newton
Le binôme de Newton est une formule mathématique donnée par Isaac Newton pour le développement
d’une puissance entière quelconque d’un binôme.
(x + y)n =
n
X
Cnk xn−k y k
k=0
où les nombres :
Cnk =
sont appelés coefficients binomiaux.
n!
k!(n − k)!
k+1
= Cnk + Cnk+1
Cn+1
k
Cnk = Cn−k
Cnk =
n k−1
C
k n−1
Maths Rappels