# Cosinus – Sinus – Tangente dans le triangle rectangle

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```Cosinus – Sinus – Tangente dans le triangle rectangle
_____________
Soit le triangle ABC rectangle en A :
B
C&ocirc;t&eacute; oppos&eacute;
hypoth&eacute;nuse
A
C
I Cosinus C :
cosC =
hypoth&eacute;nuse
0 ≤ cosC ≤ 1 pour 0 ≤ C ≤ 90&deg;
A quoi &ccedil;a sert ?
a) Calcul de la mesure de l’angle C :
⎛ AC ⎞
cosC =
=
donc C = cos-1 ⎜
⎟
hypoth&eacute;nuse BC
⎝ BC ⎠
exemple : si AC = 12 et BC = 17 alors
⎛ 12 ⎞
cosC =
=
=
donc C = cos-1 ⎜ ⎟ ≈ 45&deg;
hypoth&eacute;nuse BC 17
⎝ 17 ⎠
b) Calcul du c&ocirc;t&eacute; AC :
cosC =
=
hypoth&eacute;nuse BC
donc AC = BC x cosC
exemple : si C = 25&deg; et BC = 15 alors
cosC =
=
donc AC = BC x cosC = 15 x cos 25&deg; ≈ 13,6
hypoth&eacute;nuse BC
c) Calcul de l’hypoth&eacute;nuse BC :
cosC =
=
hypoth&eacute;nuse BC
⎛ AC ⎞
donc BC = ⎜
⎟
⎝ cosC ⎠
exemple : si C = 35&deg; et AC = 12 alors
AC
⎛ 12 ⎞
cosC =
=
BC =
donc BC = ⎜
⎟ ≈ 14,6
hypoth&eacute;nuse BC
cos C
⎝ cos 35&deg; ⎠
II Sinus C :
sinC =
c&ocirc;t&eacute; oppos&eacute;
hypoth&eacute;nuse
0 ≤ sinC ≤ 1 pour 0 ≤ C ≤ 90&deg;
A quoi &ccedil;a sert ?
a) Calcul de la mesure de l’angle C :
c&ocirc;t&eacute; oppos&eacute; AB
⎛ AB ⎞
=
donc C = sin-1 ⎜
sinC =
⎟
hypoth&eacute;nuse BC
⎝ BC ⎠
exemple : si AB = 9 et BC = 17 alors
c&ocirc;t&eacute; oppos&eacute; AB 9
⎛ 9 ⎞
=
=
donc C = sin-1 ⎜ ⎟ ≈ 32&deg;
sinC =
hypoth&eacute;nuse BC 17
⎝ 17 ⎠
b) Calcul du c&ocirc;t&eacute; AB :
c&ocirc;t&eacute; oppos&eacute; AB
=
sinC =
hypoth&eacute;nuse BC
donc AB = BC x sin C
exemple : si C = 25&deg; et BC = 15 alors
c&ocirc;t&eacute; oppos&eacute; AB
=
donc AB = BC x sin C = 15 x sin 25&deg; ≈ 6,3
sinC =
hypoth&eacute;nuse BC
c) Calcul de l’hypoth&eacute;nuse BC :
c&ocirc;t&eacute; oppos&eacute; AB
=
sinC =
hypoth&eacute;nuse BC
⎛ AB ⎞
donc BC = ⎜
⎟
⎝ sinC ⎠
exemple : si C = 35&deg; et AB = 8 alors
AB
c&ocirc;t&eacute; oppos&eacute; AB
=
BC =
sinC =
hypoth&eacute;nuse BC
sin c
⎛ 8 ⎞
donc BC = ⎜
⎟ ≈ 13,9
⎝ sin 35&deg; ⎠
III Tangente C :
tanC =
c&ocirc;t&eacute; oppos&eacute;
0 ≤ tanC &lt; ∞
pour 0 ≤ C &lt; 90&deg;
A quoi &ccedil;a sert ?
a) Calcul de la mesure de l’angle C :
c&ocirc;t&eacute; oppos&eacute;
AB
⎛ AB ⎞
=
donc C = tan-1 ⎜
tanC =
⎟
⎝ AC ⎠
exemple : si AC = 12 et AB = 7 alors
c&ocirc;t&eacute; oppos&eacute;
AB 7
⎛ 7 ⎞
=
=
donc C = tan-1 ⎜ ⎟ ≈ 30&deg;
tanC =
⎝ 12 ⎠
b) Calcul du c&ocirc;t&eacute; AB :
c&ocirc;t&eacute; oppos&eacute;
AB
=
tanC =
donc AB = AC x tan C
exemple : si C = 20&deg; et AC = 11 alors
c&ocirc;t&eacute; oppos&eacute;
AB
=
donc AB = AC x tan C = 11 x tan 20&deg; ≈ 4
tanC =
c) Calcul du c&ocirc;t&eacute; AC :
c&ocirc;t&eacute; oppos&eacute;
AB
=
tanC =
⎛ AB ⎞
donc AC = ⎜
⎟
⎝ tanC ⎠
exemple : si C = 40&deg; et AB = 9 alors
c&ocirc;t&eacute; oppos&eacute;
AB
AB
=
AC =
donc AC =
tanC =
tan C
9
⎛
⎞
⎜ tan 40&deg; ⎟ ≈ 10,7
⎝
⎠
IV Formules utiles :
tanC =
sin C
cos C
sin&sup2; C + cos&sup2; C = 1
ce qui signifie
(sin C)&sup2; + (cos C)&sup2; = 1
```