Résumé de mathématiques élémentaires

a b c d
m n
a
b=c d⇐⇒ a.d =b.c −a −b=a b−a b=−a b=a −b
ac
d=ac d
a
b.c d=ac bd
a
b+c d6=a+c b+d
a+c
b+c6=a b
am.an=am+n; (am)n=amn;am.bm= (a.b)mambn6= (a.b)m+n
a
bm
=am
bm;1 an
=a−n;am
an=am−nam bn6=a bm−n
(a+b)2=a2+ 2a.b +b2(a−b)2=a2−2a.b +b2(a−b).(a+b) = a2−b2
x
x>0|x|=x x 60|x|=−x
|x|x
|x|=max(x, −x)
x y
m
|x.y|=|x|.|y|
x
y
=|x|
|y||xm|=|x|m
| |x|−|y| | 6|x+y|6|x|+|y| | |x|−|y| | 6|x−y|6|x|+|y|
|x±y| 6=|x|±|y|
a
|x|6a−a6x6a|x|6a[−a;a]
|x|>a x 6−a x >a|x|>a]−∞;−a]∪[a; +∞[
a|x|6a|x|6a
|x|>a|x|>aR
a√a
a√a>0√a2=a
a√a

x√x2=|x|
a b m
√a.b =√a.√bra
b=√a
√b√am=√am
√a+b6=√a+√b
Rax2+bx +c= 0 a b c a 6= 0
∆ = b2−4ac
∆>0−b+√∆
2a−b−√∆ 2a∆=0
∆<0
ax2+bx = 0 ax2+c= 0
Rax2+bx +c a b c a 6= 0
∆ = b2−4ac
∆>0x1x2
x16x2ax2+bx +c a x ∈]−∞;x1[∪]x2; +∞[
−a x ∈]x1;x2[
∆<0ax2+bx +c a
a
x26a−√a6x6√a x ∈[−√a;√a])
x2>a x 6−√a x >√a x ∈]−∞;−√a]∪[√a; +∞[)
z=a+ib a b i
i2=−1
z=a+ib
M(a;b)
M z
z|z|
OM arg(z)
−→
u , −−→ OMi
z z0
|z.z0|=|z|.|z0|arg (zz0) = arg(z) + arg(z0) (mod.2π)
z
z0
=|z|
|z0|arg z z0=arg(z)−arg(z0) (mod.2π)
|z+z0| 6=|z|+|z0|arg(z+z0)6=arg(z) + arg(z0)
Arg(a+ib) = Arctan b
a

cos(a) = AB
AC
sin(a) = BC
AC
tan(a) = BC
AB
x
cos(x)M
cos(x) = OP OP
sin(x)M
sin(x) = OQ OQ
x6=π/2 + k.π k ∈Ztan(x) = sin(x)
cos(x)
T tan(x) = AT
AT
x
(cos(x))2+ (sin(x))2= 1 cos2(x) + sin2(x)=1
a b
cos(a+b) = cos(a)cos(b)−sin(a)sin(b)sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
cos(2a) = cos2(a)−sin2(a)=2cos2(a)−1=1−2sin2(a)sin(2a) = 2sin(a)cos(a)
x
eix =cos(x) + isin(x)cos(x) = eix +e−ix
2sin(x) = eix −e−ix 2i

x7→ x2x7→ 1
x
x7→ √xx7→ |x|

x7→ exx7→ ln(x)
(x, y)p
exp(x+y) = exp(x).exp(y)ex+y=ex.eyexp(x−y) = exp(x)
exp(y)ex−y=ex ey
exp(px)=(exp(x))pepx = (ex)p
x∈]0; +∞[y∈]0; +∞[p
ln(x.y) = ln(x) + ln(y)ln x
y=ln(x)−ln(y)ln(xp) = p.ln(x)
lim
x→+∞ln (x)=+∞lim
x→0ln (x) = −∞ lim
x→+∞ex= +∞lim x→−∞ ex= 0
n p
lim
x→+∞
(ex)p
xn= +∞lim
x→+∞
(ln(x))p
xn= 0 lim
x→−∞ xn.(ex)p= 0 lim
x→0xn.(ln(x))p= 0
f(x)Zg(x)f0(x)g(x)f(x)Zg(x)f0(x)g(x)
tpp∈N∗p.xp−11
x−1 x2
√x1
2√xln |x|1
x
exexsin(x)cos(x)
cos(x)−sin(x)tan(x)1
cos2(x)= 1 + tan2(x)