MPSI Colle 2 (21 au 25 septembre 2015) : Trigonométrie Nombres

(x, y, A)∈R2×R∗x x A
k y =x+kA
x≡y[A]
x≡x[A]
(x≡y[A]) ⇐⇒ (y≡x[A]) (x≡y[A]∧y≡z[A]) =⇒
x≡z[A]
n nx ≡y[A]
x≡y
nA n
R−1 1 2π
y= 1
Csin
y=xCsin
R+R−
tan
Dtan =x∈R/ x ̸=π
2[π]π
Dtan
∀x∈Dtan,tan′(x) = 1
cos2(x)= 1 + tan2(x)
Ctan
y=xCtan
0; π
2 −π 2; 0
äcos(x) = cos(y)y=±x[2π]
äsin(x) = sin(y)y=x[2π]y=π−x[2π]
äcos(x) = sin(y)
sin(y) = cos x+π
2
äAcos(x) + Bsin(x) = C
Acos(x) + Bsin(x) = √A2+B2cos (x−θ)
C
z
¯z
z¯z=z¯z=−z
|z|=√z¯z z =a+b|z|=√a2+b2∀z∈C,|z| ∈
R+|z|= 0 ⇐⇒ z= 0
z∈C|z|OM(z)
A B zAzBAB
zB−zA
ÝM z |z−a|=R
|z−a|6R|z−a|< R
a R
ÝM z |z−zA|=|z−zB|
[AB]
|zz′| |1/z′| |z/z′| |¯z| |zn|
∀(z, z′)∈C2,|z+z′|6|z|+|z′|
U
U U
xix = cos x+isin x∀x∈R,
ix
= 1
ix =−ix =ix−1i(x+y)=ix iy ixn=inx

cos3θ
n
k=0
cos (kθ)
n
k=0
sin (kθ)θ
z
|z|iθ θ2π
zarg(z)θ
[−π;π[
arg (zz′) arg (1/z′) arg (z/z′)
arg (zn) arg (¯z) arg (−z)z z′n
än>2S0=
n
k=0 n
kS1=
n
k=0
kn
kS2=
n
k=0
k2n
k
ä
ä
ä
n
k=0
cos (kθ)
n
k=0
sin (kθ)n∈Nθ∈R
äcos (3a) cos(a) cos π
9
ä