Angles et mesures des angles

I I+
M N M ≡N
f f(M) = N
A
A/≡
M N M ∼
=N
f f(M) = N
≡
A/∼
=
B
B/≡ B/∼
=
f
~u ~ u0
(O, ~u, ~v) (O0,~ u0,~ v0)
M(x, y) (O, ~u, ~v)M0=f(M) (x, y)
(O0,~ u0,~ v0)
φBO+
2
[OA) [OA0)vff
(s1, s2) (s3, s4)B

(s1, s2)≡(s3, s4)⇔φ((s1, s2)) = φ((s3, s4))
(s1, s2) (s3, s4)vfvg
φ h h(s1) = s3h(s2) = s4
g(h(s1)) = h(f(s1)) g◦h h ◦f
vg◦h=vh◦fvg◦vh=vh◦vg
O+
2vf=vg
vf=vg
h(s1) = s3h(s2) = s3
O O0(s1, s2) (s3, s4)h(s2) = h(f(s1)) = [O0h(f(A)))
s1= [OA)
O0h(f(A)) = vh(Of(A)) = vh(~ f(O)f(A)) = vh(vf(~ OA)) = vh(vg(~ OA)) = vg(vh(~ OA)) = ~ O0g(A)
h(s2) = s4
O+
2
(s1, s2) + (s2, s3) = (s1, s3)si
f g h f(s1) = s2g(s2) = s3h(s1) = s3
g◦f=h g(f(s1)) = h(s1)
α= (s1, s2)β= (s3, s4)
f(s2) = s3α= (s1, s2) = (f(s1), f(s2)) = (f(s1), s3)
α+β= (f(s1), s4)
α= (s0
1, s0 2)g g(s1) = s0
1g(s2) = s0 2
β= (s0
3, s0 4)h h(s3) = s0
3h(s4) = s0 4
f0f0f0(s0
2) = s0 3
(f0(s0
1), s0 4) (h−1(f0(s0 1)), s4)
h−1(f0(s0
1)) = f(s1)
h−1(f0(s0
2)) = h−1(s0 3) = s3=f(s2) = f(g−1(s0 2)) h−1◦f0
f◦g−1h−1(f0(s0
1)) = f(g−1(s0 1)) = f(s1)

ez=P+∞
n=0
zn
n!
cos(x) = eix+e−ix
2sin(x) = eix−e−ix 2ix∈R
x∈Reix =cos(x) + isin(x)
ez+z0=ezez0
cos(a+b) = cos(a)cos(b)−sin(a)sin(b)sin(a+b) = cos(a)sin(b) + sin(a)cos(b)
sin0=cos cos0=−sin
x∈Rcos2(x) + sin2(x) = 1
f(x) = cos2(x) + sin2(x)R
x
x7−→ eix R
cos(0) = 1 α > 0
[0; α]
sin(α)>0
[0; +[ y>0
Zy
0
sin(t)dt >(y−α)sin(α)
cos(α)−cos(y)>(y−α)sin(α)y−α6cos(α)
sin(α)−cos(y) sin(α)cos(y)>0
y−α6cos(α)
sin(α)
y+
ω=inf{y > 0, cos(y) = 0}
4ω
π= 2ω2π eix = 1
a b a2+b2= 1 θ
a=cos(θ)b=sin(θ)
z=a+ib
eiθ
R
2πZ

fR R
αZα=inf{x > 0, f(x)=0}
f:x7−→ eix αZker(f) = 2πZ
f
R
2πZO+
2
a2+b2= 1
θθ θ
θ θ 
RO+
2
2πZ
f G G0f(G)G/Kerf
f(G)G/Kerf
g g G/Kerf
f g f(g)f
g=g0⇔g−1g0=eG⇔f(g−1g0) = eG0⇔f(g) = f(g0)
f f
f
f(g−1g0) = f(g−1g0)
=f(g−1g0)f
=f(g)−1f(g0)f
=f(g)−1f(g0)f
π