étude de la continuité et de la dérivabilité des fonctions circulaires

(O, i, j)CO
 x ∈RCM
x(\ i, −−→ OM)
 x
cos(x)xsin(x)
cos sin
x
2π
∀θ∈R,cos2(θ) + sin2(θ)=1

∀(a, b)∈R2,cos(a+b) = cos(a) cos(b)−sin(a) sin(b),
∀(a, b)∈R2,sin(a+b) = sin(a) cos(b) + cos(a) sin(b).
a b
(O, i, j)C
O A
B a
([ i, −→ OA)b(\
−→
OA, −−→ OB)
a+b(\ i, −−→ OB)
(O, i, j)B
(cos(a+b),sin(a+b))
k
−→
OA +π
2(O, −→
OA,  k)
B(cos(b),sin(b))
A(O, i, j) (cos(a),sin(a))
−→
OA = cos(a) i+ sin(a) j,
k= cos a+π
2
i+ sin a+π
2
j
cos a+π
2=−cos a+π 2−π=−cos a−π 2=−cos π 2−a=−sin(a),
sin a+π
2= cos(a)
k=−sin(a) i+ cos(a) j−−→
OB
−−→
OB =
cos(a+b) i+ sin(a+b) j
cos(b)−→ OA + sin(b) k= cos(b)cos(a) i+ sin(a) j+ sin(b)−sin(a) i+ cos(a) j.
cos(a+b) = cos(a) cos(b)−sin(a) sin(b)
sin(a+b) = sin(a) cos(b) + cos(a) sin(b)

R
x∈R|sin(x)|6|x|
x
•x∈R\i−π
2,π 2h
|x|>π
2>1>|sin(x)|.
•x= 0
•x∈i0,π
2h
OMI
[OI) [OM)
OMI OI ×sin(x)
2=sin(x) 2
x
2π π
xx×π
2π=x 2
06sin(x)6x
•x∈i−π
2,0h
|sin(x)|=−sin(x) = sin(−x)6−x
| {z }
0<−x< π
2
=|x|.
|sin(x)|6|x|x∈R
∀x∈R,06|sin(x)|6|x|,
sin(x)−→
x→00
x∈Rcos2(x) + sin2(x)=1 −π
2,π 2
R+x∈−π
2,π 2cos(x) = q1−sin2(x)
cos(x)−→
x→01
x0∈R
h∈Rsin(x0+h) = sin(x0) cos(h) + cos(x0) sin(h)
sin(x0+h)−→
h→0sin(x0)×1 + cos(x0)×0 = sin(x0).
x0
x0
R

x∈i−π
2,0h∪i0,π 2hcos(x)6sin(x) x61
•xi0,π
2h
OMI
[OI) [OM)
OT I
OI ×IT
2=IT 2IT
sin(x)
IT =cos(x) OI =OM OT .
IT =sin(x)
cos(x)
sin(x)
26x 261 2×sin(x) cos(x),sin(x)6x6sin(x) cos(x).
sin(x)6x x 6sin(x)
cos(x)cos(x)>0
xcos(x)6sin(x)
x > 0
cos(x)6sin(x)
x61.
•x7→ cos(x)x7→ sin(x)
x
x∈−π
2,0
sin(x)
x−→
x→01
sin(0) = 0
sin0(0) = 1
x7→ cos(x)−cos(0)
x−0
x∈R
cos(x)−cos(0) = cos(x)−1
= cos 2×x
2−1
=cos2x
2−sin2x 2−1 cos(a+b) = · · ·
=1−sin2x
2−sin2x 2−1
=1−2 sin2x
2−1
=−2 sin2x
2

x∈R∗
cos(x)−cos(0)
x−0=−2×sin2x
2
x=−2×sin2x
2
x
22×x
22
x=−x 2× sin x
2
x
2!2
.
sin x
2
x
2!2
−→
x→01,
cos(x)−cos(0)
x−0−→
x→00.
cos0(0) = 0
cos(x)−1
x2−→
x→0−1
2
x0∈R
h∈R∗
sin(x0+h)−sin(x0)
h=(sin(x0) cos(h) + cos(x0) sin(h)) −sin(x0) h= sin(x0)×cos(h)−1 h+ cos(x0)×sin(h) h.
sin(x0+h)−sin(x0)
h−→
h→0sin(x0)×0 + cos(x0)×1.
x0sin0(x0) = cos(x0)
x0∈R R sin0= cos
cos cos0=−sin