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# FormulaireTrigo

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```Formulaire de trigonom&eacute;trie circulaire
B
Kb
cotan(x)
b
bH
M
b
sin(x)
b
x
cos(x) = abscisse de M
tan(x)
sin(x) = ordonn&eacute;e de M
b b1
cos(x) A
tan(x) = AH
cotan(x) = BK
eix = zM
sin(x)
cos(x)
π
1
π
+ πZ, tan(x) =
et pour x ∈
/ πZ, cotan(x) =
. Enfin pour x ∈
/ Z, cotan(x) =
.
2
cos(x)
sin(x)
2
tan(x)
Valeurs usuelles.
Pour x ∈
/
x en
◦
x en rd
0
30
45
60
90
0
π
6
π
4
π
3
√
3
2
π
2
sin(x)
0
cos(x)
1
tan(x)
0
cotan(x)
∞
√
1
2
√ =
2
2
√
1
2
√ =
2
2
1
2
√
3
2
1
√
3
√
3
1
1
1
2
√
3
1
√
3
1
0
∞
0
∀x ∈ R, cos2 x + sin2 x = 1
1
π
.
∀x ∈
/ + πZ, 1 + tan2 x =
2
cos2 x
1
.
∀x ∈
/ πZ, 1 + cotan2 x =
sin2 x
angle oppos&eacute;
angle suppl&eacute;mentaire
cos(x + 2π) = cos x
sin(x + 2π) = sin x
tan(x + 2π) = tan x
cotan(x + 2π) = cotan x
cos(x + π) = − cos x
sin(x + π) = − sin x
tan(x + π) = tan x
cotan(x + π) = cotan x
cos(−x) = cos x
sin(−x) = − sin x
tan(−x) = − tan x
cotan(−x) = − cotan x
cos(π − x) = − cos x
sin(π − x) = sin x
tan(π − x) = − tan x
cotan(π − x) = − cotan x
angle compl&eacute;mentaire
π
cos( − x) = sin x
2
π
sin( − x) = cos x
2
π
tan( − x) = cotan x
2
π
cotan( − x) = tan x
2
quart de tour direct
π
cos(x + ) = − sin x
2
π
sin(x + ) = cos x
2
π
tan(x + ) = − cotan x
2
π
cotan(x + ) = − tan x
2
quart de tour indirect
π
cos(x − ) = sin x
2
π
sin(x − ) = − cos x
2
π
tan(x − ) = − cotan x
2
π
cotan(x − ) = − tan x
2
c Jean-Louis Rouget, 2008. Tous droits r&eacute;serv&eacute;s.
1
http ://www.maths-france.fr
Formules de duplication
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 + sin b cos a
sin(a − b) = sin a cos b − sin b cos a
cos(2a) = cos2 a − sin2 a
= 2 cos2 a − 1
= 1 − 2 sin2 a
sin(2a) = 2 sin a cos a
tan a + tan b
1 − tan a tan b
tan a − tan b
tan(a − b) =
1 + tan a tan b
tan(2a) =
tan(a + b) =
2 tan a
1 − tan2 a
Formules de lin&eacute;arisation
1 + cos(2a)
2
1
−
cos(2a)
sin2 a =
2
1
(cos(a − b) + cos(a + b))
2
1
sin a sin b = (cos(a − b) − cos(a + b))
2
1
sin a cos b = (sin(a + b) + sin(a − b))
2
cos a cos b =
cos2 a =
Formules de factorisation
cos x, sin x et tan x
en fonction de t=tan(x/2)
1 − t2
1 + t2
2t
sin x =
1 + t2
2t
tan x =
1 − t2
p−q
p+q
cos
2
2
p−q
p+q
sin
cos p − cos q = −2 sin
2
2
p−q
p+q
cos
sin p + sin q = 2 sin
2
2
p−q
p+q
sin p − sin q = 2 sin
cos
2
2
x
2
x
1 − cos x = 2 sin2
2
1 + cos x = 2 cos2
cos x =
cos p + cos q = 2 cos
Divers
cos(3x) = 4 cos3 x − 3 cos x
sin(3x) = 3 sin x − 4 sin3 x
R&eacute;solution d’&eacute;quations
cos x = cos a ⇔
∃k ∈ Z/ x = a + 2kπ
ou
∃k ∈ Z/ x = −a + 2kπ
sin x = sin a ⇔
∃k ∈ Z/ x = a + 2kπ
ou
∃k ∈ Z/ x = π − a + 2kπ
tan x = tan a ⇔
∃k ∈ Z/ x = a + kπ
Exponentielle complexe
ix
∀x ∈ R, e
= cos x + i sin x.
Valeurs usuelles
√
1
3 √ iπ/4
, 2e
e0 = 1, eiπ/2 = i, eiπ = −1, e−iπ/2 = −i, e2iπ/3 = j = − + i
= 1 + i.
2
2
Propri&eacute;t&eacute;s alg&eacute;briques
∀x ∈ R, |eix | = 1.
1
eix
= ei(x−y) ,
= e−ix = eix
iy
e
eix
Formules d’Euler
∀(x, y) ∈ R2 , eix &times; eiy = ei(x+y) ,
eix + e−ix
et eix + e−ix = 2 cos x.
2
eix − e−ix
∀x ∈ R, sin x =
et eix − e−ix = 2i sin x.
2i
∀x ∈ R, cos x =
Formule de Moivre
∀x ∈ R, ∀n ∈ Z, (eix )n = einx .
c Jean-Louis Rouget, 2008. Tous droits r&eacute;serv&eacute;s.
2
http ://www.maths-france.fr
```