Formulaire de trigonométrie

Formulaire de trigonométrie
Trigonométrie de base
Cercle trigonométrique
Similitude
cos(θ ) = x
P(θ )
tan(θ )
A
tan(θ ) =
B
y
sin(θ )
tan(θ )
=
=
x cos(θ )
1
Cercle trigonométrique étendu
Thm. de Pythagore
C2 = A2 + B2
c(
θ)
C
sin(θ ) = y
cos2 (θ ) + sin2 (θ ) = 1
sin(θ )
se
P(θ )
1
A
Rapports trigonométriques
tan(θ )
a
y
θ
x
A B C
= =
a b c
a A b B
=
=
etc.
c C c c
B
C
b
c
1
Dans deux triangles semblables, les proportions des côtés correspondants sont toujours les mêmes.
tan(θ )
sin(θ
=
1
cos(θ )
sec(θ )
1
=
1
cos(θ )
1 + tan2 (θ ) = sec2 (θ )
θ
cos(θ )
A
Loi des sin
cos(θ )
Loi des cos
ctg2 (θ ) + 1 = csc2 (θ )
θ
B
h
A
A
B
C
=
=
sin(α) sin(β ) sin(γ)
B
C
γ
β
ctg(θ )
cos(θ
=
1
sin(θ )
csc(θ )
1
=
1
sin(θ )
Fonctions trigonométriques inverses
α
C
cs
c(
θ)
θ
P(θ )
1
O
ctg(θ )
sin(θ )
H
O
sin(θ ) =
H
A
cos(θ ) =
H
O
tan(θ ) =
A
θ
A
C2 = A2 + B2 − 2AB cos(θ )
cos(θ ) = x ⇐⇒ θ = acos(x)
θ
Triangles comportant des angles usuels
sin(θ ) = y ⇐⇒ θ = asin(y)
θ
π/4
1
√
2
2
1
π/3
π/6
√
3/2
1
tan(θ ) = m ⇐⇒ θ = atan(m)
1
1
5π/12
π/12
p
√
2 + 3/2
1
asin(x) ∈ [−π/2, π/2]
1
2
π/4
√
2
2
acos(x) ∈ [0, π]
θ
p
√
2− 3
2
atan(x) ∈] − π/2, π/2[
θ
)
(ϕ
s in
Symétries et rotations
P(θ )
θ
P(π − θ )
P(θ )
)
(ϕ
cos
cos(−θ ) = cos(θ )
ϕ
θ
sin(−θ ) = − sin(θ )
sin(θ ) sin(ϕ)
cos(π − θ ) = − cos(θ )
θ
cos(θ ) cos(ϕ)
sin(π − θ ) = sin(θ )
cos(θ ± π) = − cos(θ )
P(θ ± π)
P θ + π2
P(θ ± π)
P(−θ )
π
P
3π
2
−θ
P(θ ± π)
cos
− θ = sin(θ )
π2
sin
−
θ
= cos(θ )
π
P 2 −θ
2
3π
cos
− θ = − sin(θ )
2
3π
− θ = − cos(θ )
sin
2
P(θ )
cos(θ + ϕ) = cos(θ ) cos(ϕ) − sin(θ ) sin(ϕ)
sin(θ ± π) = − sin(θ )
π
P(θ )
cos θ +
= − sin(θ )
2
π
= cos(θ )
sin θ +
2
π
cos θ −
= sin(θ )
2
π
P θ − π2
= − cos(θ )
sin θ −
2
sin(θ ) cos(ϕ) cos(θ ) sin(ϕ)
P(θ + ϕ)
sin(θ + ϕ) = sin(θ ) cos(ϕ) + cos(θ ) sin(ϕ)
Somme d’angles
Identitiés trigonométriques
Multiples d’angles
sin(2θ ) = 2 sin(θ ) cos(θ )
cos(2θ ) = cos2 (θ ) − sin2 (θ )
Carrés de sinus et cosinus
sin2 (θ ) =
1 − cos(2θ )
2
cos2 (θ ) =
1 + cos(2θ )
2
Produits de sinus et cosinus
sin(θ − ϕ) + sin(θ + ϕ)
2
sin(θ − ϕ) + sin(θ + ϕ)
cos(θ ) sin(ϕ) =
2
cos(θ − ϕ) + cos(θ + ϕ)
cos(θ ) cos(ϕ) =
2
cos(θ − ϕ) − cos(θ + ϕ)
sin(θ ) sin(ϕ) =
2
sin(θ ) cos(ϕ) =