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Formulaire de trigonométrie Page 1 G. COSTANTINI http://bacamaths.net/
TRIGONOMÉTRIE : FORMULAIRE
Angles associés
Une lecture efficace du cercle trigonométrique permet de retrouver les relations suivantes :
Relations entre cos, sin et tan
cos2
(x) + sin2
(x) = 1 1 + tan2
(x) = 2
1
cos ( )x
Formules d'addition
cos(a – b) = cos(a) cos(b) + sin(a) sin(b) cos(a + b) = cos(a) cos(b) – sin(a) sin(b)
sin(a – b) = sin(a) cos(b) – cos(a) sin(b) sin(a + b) = sin(a) cos(b) + cos(a) sin(b)
tan(a - b) =
tan( ) tan( )
1 tan( )tan( )
a b
a b
-
+
tan(a + b) =
tan( ) tan( )
1 tan( )tan( )
a b
a b
+
-
Formules de duplication
cos(2a) = cos2
(a) – sin2
(a) = 2 cos2
(a) - 1 = 1 - 2 sin2
(a) sin(2a) = 2 sin(a) cos(a) tan(2a) = 2
2tan( )
1 tan ( )
a
a-
Extensions : cos(3a) = 4cos3
(a) - 3cos(a) sin(3a) = 3sin(a) - 4sin3
(a) tan(3a) =
3
2
3tan( ) tan ( )
1 3tan ( )
a a
a
-
-
Au delà, utiliser la formule de Moivre.
Formules de linéarisation
cos2
(a) =
1 cos(2 )
2
a+
sin2
(a) =
1 cos(2 )
2
a-
tan2
(a) =
1 cos(2 )
1 cos(2 )
a
a
-
+
Extensions : cos3
(a) =
cos( 3 ) 3cos( )
4
a a+
sin3
(a) =
sin(3 ) 3sin( )
4
a a- +
tan3
(a) =
sin(3 ) 3sin( )
cos(3 ) 3cos( )
a a
a a
- +
+
Au delà, utiliser les formules d'Euler. Les formules d'Euler permettent également de montrer que :
cos(a) cos(b) =
2
1
[cos(a - b) + cos(a + b)] cos(a) sin(b) =
2
1
[sin(a + b) - sin(a - b)] sin(a) sin(b) =
2
1
[cos(a - b) - cos(a + b)]
cos(p) + cos(q) = 2 cos
2
p q+æ ö
ç ÷
è ø
cos
2
p q-æ ö
ç ÷
è ø
cos(p) - cos(q) = -2 sin
2
p q+æ ö
ç ÷
è ø
sin
2
p q-æ ö
ç ÷
è ø
sin(p) + sin(q) = 2 sin
2
p q+æ ö
ç ÷
è ø
cos
2
p q-æ ö
ç ÷
è ø
sin(p) - sin(q) = 2 sin
2
p q-æ ö
ç ÷
è ø
cos
2
p q+æ ö
ç ÷
è ø
Résolution d'équations trigonométriques
cos(U) = cos(V) Û (U = V [2p] ou U = -V [2p])
sin(U) = sin(V) Û (U = V [2p] ou U = p -V [2p])
tan(U) = tan(V) Û U = V [p]
Expression du cosinus, sinus et tangente en fonction de la tangente de l'angle moitié
Si t = tan
2
aæ ö
ç ÷
è ø
, on a : cos(a) =
1
1
2
2
-
+
t
t
; sin(a) =
2
1 2
t
t+
; tan(a) =
2
1 2
t
t-
2
x
p
+
cos(–x) = cos(x)
sin(–x) = –sin(x)
p–x
p+x
x
–x
cos(p – x) = –cos(x)
sin(p – x) = sin(x)
cos(p + x) = –cos(x)
sin(p + x) = –sin(x)
p
2
- x
cos
2
x
pæ ö+ç ÷
è ø
= -sin(x)
sin
2
x
pæ ö+ç ÷
è ø
= cos(x)
cos
2
x
pæ ö-ç ÷
è ø
= sin(x)
sin
2
x
pæ ö-ç ÷
è ø
= cos(x)
U = p -V V
U
V

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Trigo

  • 1. Formulaire de trigonométrie Page 1 G. COSTANTINI http://bacamaths.net/ TRIGONOMÉTRIE : FORMULAIRE Angles associés Une lecture efficace du cercle trigonométrique permet de retrouver les relations suivantes : Relations entre cos, sin et tan cos2 (x) + sin2 (x) = 1 1 + tan2 (x) = 2 1 cos ( )x Formules d'addition cos(a – b) = cos(a) cos(b) + sin(a) sin(b) cos(a + b) = cos(a) cos(b) – sin(a) sin(b) sin(a – b) = sin(a) cos(b) – cos(a) sin(b) sin(a + b) = sin(a) cos(b) + cos(a) sin(b) tan(a - b) = tan( ) tan( ) 1 tan( )tan( ) a b a b - + tan(a + b) = tan( ) tan( ) 1 tan( )tan( ) a b a b + - Formules de duplication cos(2a) = cos2 (a) – sin2 (a) = 2 cos2 (a) - 1 = 1 - 2 sin2 (a) sin(2a) = 2 sin(a) cos(a) tan(2a) = 2 2tan( ) 1 tan ( ) a a- Extensions : cos(3a) = 4cos3 (a) - 3cos(a) sin(3a) = 3sin(a) - 4sin3 (a) tan(3a) = 3 2 3tan( ) tan ( ) 1 3tan ( ) a a a - - Au delà, utiliser la formule de Moivre. Formules de linéarisation cos2 (a) = 1 cos(2 ) 2 a+ sin2 (a) = 1 cos(2 ) 2 a- tan2 (a) = 1 cos(2 ) 1 cos(2 ) a a - + Extensions : cos3 (a) = cos( 3 ) 3cos( ) 4 a a+ sin3 (a) = sin(3 ) 3sin( ) 4 a a- + tan3 (a) = sin(3 ) 3sin( ) cos(3 ) 3cos( ) a a a a - + + Au delà, utiliser les formules d'Euler. Les formules d'Euler permettent également de montrer que : cos(a) cos(b) = 2 1 [cos(a - b) + cos(a + b)] cos(a) sin(b) = 2 1 [sin(a + b) - sin(a - b)] sin(a) sin(b) = 2 1 [cos(a - b) - cos(a + b)] cos(p) + cos(q) = 2 cos 2 p q+æ ö ç ÷ è ø cos 2 p q-æ ö ç ÷ è ø cos(p) - cos(q) = -2 sin 2 p q+æ ö ç ÷ è ø sin 2 p q-æ ö ç ÷ è ø sin(p) + sin(q) = 2 sin 2 p q+æ ö ç ÷ è ø cos 2 p q-æ ö ç ÷ è ø sin(p) - sin(q) = 2 sin 2 p q-æ ö ç ÷ è ø cos 2 p q+æ ö ç ÷ è ø Résolution d'équations trigonométriques cos(U) = cos(V) Û (U = V [2p] ou U = -V [2p]) sin(U) = sin(V) Û (U = V [2p] ou U = p -V [2p]) tan(U) = tan(V) Û U = V [p] Expression du cosinus, sinus et tangente en fonction de la tangente de l'angle moitié Si t = tan 2 aæ ö ç ÷ è ø , on a : cos(a) = 1 1 2 2 - + t t ; sin(a) = 2 1 2 t t+ ; tan(a) = 2 1 2 t t- 2 x p + cos(–x) = cos(x) sin(–x) = –sin(x) p–x p+x x –x cos(p – x) = –cos(x) sin(p – x) = sin(x) cos(p + x) = –cos(x) sin(p + x) = –sin(x) p 2 - x cos 2 x pæ ö+ç ÷ è ø = -sin(x) sin 2 x pæ ö+ç ÷ è ø = cos(x) cos 2 x pæ ö-ç ÷ è ø = sin(x) sin 2 x pæ ö-ç ÷ è ø = cos(x) U = p -V V U V