Fonctions trigonométriques

π−a π +a−aπ
2−aπ 2+a a
cos(π−a) = −cos(a) cos(π
2−a) = sin(a) cos(−a) = cos(a)
sin(π−a) = sin(a) sin(π
2−a) = cos(a) sin(−a) = −sin(a)
cos(π+a) = −cos(a) cos(π
2+a) = −sin(a)
sin(π+a) = −sin(a) sin(π
2+a) = cos(a)
ABCD b h A(ABE)
A(ABCD) = bh H E (AB)
AHE ADE BHE BCE
A(ABE) = A(AHE) + A(BHE) = (A(AHE) + A(ADE) + A(BHE) + A(BCE))/2
=A(ABCD)/2 = bh/2.
a=AE b =BE c =AB ABE EF
c2= (b−a)2+ 2ab
A(ABE) = ab/2EF =b−a
c2=A(ABCD) = 4 × A(ABE) + EF 2= 2ab + (b−a)2=a2+b2.
ABE E
Y AA1Y BB1
AA1
Y A1=BB1
Y B1y Y
y a b
a AA1=y−sin(a)Y A1= cos(a)BB1= sin(b)−y
Y B1= cos(b)
y−sin(a)
cos(a)=sin(b)−y cos(b)⇐⇒ y×(cos(a) + cos(b)) = sin(a) cos(b) + sin(b) cos(a).
OY A(OY A)A(OY B) cos(a) cos(b)y
A(OY A) = ycos(a)/2A(OY B) = ycos(b)/2
OAB OB
OAB OB OA ×sin(a+b)OA =OB = 1
sin(a+b) = 2 × A(OAB) = 2(A(OY A) + A(OY B))
=y(cos(a) + cos(b)) = sin(a) cos(b) + sin(b) cos(a)
sin(x) 0 x0
cos(x) 1 x0
0≤ |sin(x)| ≤ |x|x∈]−π
2,π 2[ limx→0|sin(x)|= 0
limx→0sin(x) = 0 (cos x)2= 1 −(sin x)2
limx→0(cos x)2= 1 cos x x ∈[−π
2,π 2] cos x=p(cos x)2
limx→0cos(x) = √12= 1

sin(x)/x 1x0
sin 0 sin0(0)
x∈]−π
2,π 2[x6= 0
|sin(x)| ≤ |x| ≤ 
sin(x)
cos(x)
⇐⇒ 1≤
x
sin(x)
≤1
cos(x)⇐⇒ |cos(x)| ≤ 
sin(x)
x
≤1.
limx→0cos(x) = 1
|sin(x)/x|1x0 sin(x)x]−π
2,π 2[
|sin(x)/x|= sin(x)/x
lim
x→0
sin(x)−sin(0)
x−0= lim
x→0
sin(x)
x= 1.
sin 0 sin0(0) = 1
(1 −cos(x))/x 0x0
cos 0 cos0(0)
(1 + cos x)
1−cos x
x=1−(cos x)2 x(1 + cos x)=sin x x
sin x
1 + cos x
1
0 (cos x−1)/x 0x0 cos
0 cos0(0) = 0
f:R→R, x 7→ sin(a+x)
f0f0(0)
f(x) = sin(a) cos(x) + cos(a) sin(x) sin(a) cos(a)
f0f0(0) = sin(a) cos0(0) + cos(a) sin0(0) = cos(a)
tan0(x) = 1
(cos(x))2= 1 + (tan(x))2
tan0
tan0=sin
cos0
=sin0cos −sin cos0
cos2=cos2+ sin2 cos2=1 cos21 + tan2.
s= sin(π
3)c= cos(π 3) sin(3 ×π 3)s c
s c
3×π
3=π 3+2π 3
sin(3 ×π
3) = s×cos(2π 3) + c×sin(2π 3).
2π
3=π 3+π 3
sin(3 ×π
3) = s(c2−s2) + c(sc +cs) = 3sc2−s3.
c21−s2sin(3 ×π
3) = sin(π) = 0
0 = 3s−4s3π
3∈]0,π 2[s > 0s
s=√s2=p3/4 = √3/2c= 1/2