Formulaire de Trigonométrie

Formulaire de Trigonométrie
Angles associés
Une lecture efficace du cercle trigonométrique permet de retrouver les relations suivantes :
´
´
³π
³π
+ x = − sin(x)
− x = sin(x)
cos
cos
π
π
³ π2
³ π2
´
´
+x
−x
2
2
sin
sin
+ x = cos(x)
− x = cos(x)
2
2
cos (π − x) = − cos(x)
sin (π − x) = sin(x)
π−x
x
cos (π + x) = − cos(x)
sin (π + x) = − sin(x)
π+x
−x
cos (−x) = cos(x)
sin (−x) = − sin(x)
Valeurs remarquables
0
cos
sin
tan
1
0
0
π
6
p
3
2
π
4
p
2
2
p
2
2
1
2
p
3
3
1
π
3
π
2
π
1
2
p
3
2
0
−1
1
0
p
3
0
Relations entre cos, sin et tan
cos2 (x) + sin2 (x) = 1
1 + tan2 (x) =
1
cos2 (x)
si x 6≡
π
[π]
2
Formules d’addition
cos(a + b) = cos(a) cos(b) − sin(a) sin(b)
sin(a + b) = sin(a) cos(b) + sin(b) cos(a)
cos(a − b) = cos(a) cos(b) + sin(a) sin(b)
sin(a − b) = sin(a) cos(b) − sin(b) cos(a)
tan(a) + tan(b)
tan(a + b) =
1 − tan(a) tan(b)
tan(a − b) =
tan(a) − tan(b)
1 + tan(a) tan(b)
Formules de duplication
cos(2a) = cos2 (a) − sin2 (a)
= 2cos2 (a) − 1
sin(2a) = 2sin(a) cos(a)
tan(2a) =
2tan(a)
1 − tan2 (a)
= 1 − 2sin2 (a)
Transformation de somme en produit
³p +q´
³p −q´
cos(p) + cos(q) = 2cos
cos
2
2
³p −q´
³p +q ´
cos
sin(p) + sin(q) = 2sin
2
2
³p +q ´
³p −q´
cos(p) − cos(q) = −2sin
sin
2
2
³p −q ´
³p +q ´
sin(p) − sin(q) = 2sin
cos
2
2
Transformation de produit en somme
1
[sin(a + b) + sin(a − b)]
2
1
cos(a) cos(b) = [cos(a + b) + cos(a − b)]
2
1
sin(a) sin(b) = − [cos(a + b) − cos(a − b)]
2
sin(a) cos(b) =
cos2 (a) =
1 + cos(2a)
2
sin2 (a) =
cos, sin et tan en fonction de l’angle moitié
³a´
1− t2
2t
Si t = tan
, on a : cos(a) =
; sin(a) =
2
1+ t2
1+ t2
Equations trigonométriques
cos(a) = cos(b) ⇐⇒
½
a
b = −a
a ≡ b[2π]
a ≡ −b[2π]
1 − cos(2a)
2
;
tan(a) =
sin(a) = sin(b) ⇐⇒
b = π−a
½
tan2 (a) =
2t
1− t2
a ≡ b[2π]
a ≡ π − b[2π]
a
1 − cos(2a)
1 + cos(2a)
tan(a) = tan(b) ⇐⇒ a ≡ b[π]