Fonctions Trigonométriques et hyperboliques

Fonctions Trigonométriques et hyperboliques

cosα = OP 1≤≤≤≤cosαααα≤≤≤≤1 sinα = OQ 1≤≤≤≤sinαααα≤≤≤≤1 tgα = AA′ =

= α α α

g t o c

1 cos

sin

cotgα = BB′ =

= α α α

tg 1 sin cos

α

= α α

= α

sin ec 1 cos cos ec 1

s

;

2 π 3π 2 π 3 π 4 π 6

0

π α

-1

0 1 2

3 2

2

2 0 1

sinα

-1

0

2

0

1

2

2 2

1

3 cosα

∞

∞ 0 1 3

3

0

1 tgα

cos(α + 2kπ) = cosα sin(α + 2kπ) = sinα tg(α + kπ ) = tgα cos(α) =cosα sin(α) =sinα tg(α) =tgα

•k nombre relatif•

cos(π + α) = cosα sin(π + α) = sinα tg(π + α) = tgα cos(π α) = cosα sin(π α) = sinα tg(π α) =tgα

cos(

2

π α ) = sinα sin(

2

π α ) = cosα tg( 2

π α ) = _tg¹_α cos(

2

π+ α ) = sinα sin(

2

π+ α ) = cosα tg( 2

π+ α ) = _tg¹_α

sinα = 0 ⇔ α = kπ cosα = 0 ⇔ α =

2 π + kπ sinα = 1 ⇔ α =

2 π + 2kπ cosα = 1 ⇔ α = 2kπ sinα = 1 ⇔ α =

2 3π

+ 2kπ cosα = 1 ⇔ α = π + 2k π

•k nombre relatif• cos(α+β) = cosα cosβ sinα sinβ

cos(αβ) = cosα cosβ + sinα sinβ sin(α+β) = sinα cosβ + cosα sinβ sin(αβ) = sinα cosβ cosα sinβ

β α

− β +

= α β +

α 1 tg tg tg ) tg

( tg

β + α

β

−

= α β

−

α 1 tg tg

tg ) tg

( tg

α

−

= α α 1 tg2

tg 2 2

tg cos2α = cos²α sin²α = 2cos²α 1 = 1 2sin²α =

α +

α

−

2 2

tg 1

tg

1

sin2α = 2sinα cosα =

α +

α tg2

1 tg 2

2cos²α = 1 + cos2α 2sin²α = 1 cos2α 4cos³α = cos3α + 3cosα 4sin³α = sin3α + 3sinα

sin

²

α α α α + cos

²

α α α α = = = = 1

= α α

+ ² ₂

cos tg 1

1

= α α

+ ² ₂

sin g 1

cot 1

sinα sinβ = 2

1[ cos(αβ) cos(α+β) ] cosα cosβ =

2

1[ cos(αβ) +cos(α+β) ] sinα cosβ =

2

1 [ sin(α+β) + sin(αβ) ]

− β α

α

−

= β β + α

β +

= α β

α cotg cotg

tg tg g

cot g cot

tg tg tg

tg

sinα + sinβ = 2 sin

(

^α₂^+β

)

^cos

(

^α₂^−β

)

sinα sinβ = 2 sin

(

^α₂^−β

)

cos

(

^α₂^+β

)

cosα + cosβ = 2 cos

(

^α₂^+β

)

^cos

(

^α₂^−β

)

cosα cosβ = 2 sin

(

^α₂^+β

)

sin

(

^α₂^−β

)

β α

β +

= α β +

α cos cos ) tg sin(

tg

β α

β

−

= α β

−

α cos cos ) tg sin(

tg

2 e cos e

i

iα −α

= + α

i 2

e sin e

i

iα −α

= − α

e^iα = cosα + i sinα e^-iα = cosα i sinα e =2,718281828...

i nbre imag: i² = -1

2 e ch e

α

− α +

=

α

;

2 e sh e

α

− α−

= α

α

− α

α

− α

+

= − α

= α α

e e

e e ch th sh

α

− α

α

− α

−

= + α

= α α

e e

e e sh cth ch

chα + shα = e^α

;

chα shα = e^α ch²α sh²α = 1

;

thα cthα = 1 sh(α) =shα

;

_ch(α) =chα th(α) =thα

;

cth(α) =cthα

a cosα + b sinα = a²+b² sin (α+θ)

2

2 b

a sin a

+

=

θ

;

2

2 b

a cos b

+

= θ

a cosα + b sinα = a²+b² cos (αθ)

2

2 b

a cos a

+

=

θ

;

2

2 b

a sin b

+

= θ

O P

Q ^A′

A

B B′

α πα

-α π+α

M

π

π α 2

α = β + 2kπ cosα = cosβ ⇔ V

α = β + 2kπ α = β + 2kπ sinα = sinβ ⇔ V

α = πβ + 2kπ tgα = tgβ ⇔ α = β + kπ

• k nombre relatif •

{ {

sh(α ± β) = shα chβ ± chα shβ ch(α ± β) = chα chβ ± shα shβ

β α

± β

±

= α β

±

α 1 th th th ) th

( th

sh2α = 2shα chα =

− α α th2

1 th 2

ch2α = ch²α + sh²α = 2sh²α + 1 = 2ch²α 1 =

α

− α +

2 2

h t 1

th

1

th2α =

α +

α h2

t 1

th

2

;

cth2α =

α α +

cth 2

cth

1 ²

shα ± shβ = 2 sh

(

^α₂^±β

)

^ch

(

^α₂^mβ

)

chα + chβ = 2 ch

(

^α₂^+β

)

^ch

(

^α₂^−β

)

chα chβ = 2 sh

(

^α₂^+β

)

sh

(

^α₂^−β

)

β α

β

±

= α β

±

α ch ch

) ( th sh th

(chα ± shα)ⁿ = chnα ± shnα π +α

2

M^r:ABDELMOUTALEB M^r:ABDELMOUTALEB