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Universit Mohammed V-Agdal Filires SM-SMI / Module M6 / Analyse 2

Facult des Sciences Semestre S2/2005-2006

Dpartement de Mathmatiques

et dInformatique

FORMULAIRELES FONCTIONS CIRCULAIRES ET LEURS FONCTIONS RECIPROQUES

cos2a sin 2b 1

cosa b cos a.cos b sin a sin b cosa b cos a.cos b sin a sin b

sina b sin a.cos b sin b.cos a sina b sin a.cos b sin b.cos a

tana b tan atan b1tan a.tan b

tana b tan atan b1tan a.tan b

cosp cos q 2.cos pq

2 .cos

pq

2 cosp cos q 2.sin

pq

2 .sin

pq

2

sinp sin q 2.sin pq

2 .cos

pq

2 sinp sin q 2.sin

pq

2 .cos

pq

2

cosp.cos q 1

2cosp q cosp q sinp.sin q 1

2cosp q cosp qsinp.cos q 1

2sinp q sinp q

Changement de variable :

Soit t tan x2

, on a : cosx 1t2

1t2 , sinx 2t

1t2, tanx 2t

1t2

Drives :

cosx sinx, sinx cosx, tanx 1 tan 2x, cot anx 1 cot an2x

Arc cosx 11x2

( |x| 1) , Arc sinx 11x2

( |x| 1)

Arc tanx 11x2

, Arcco tanx 11x2

LES FONCTIONS HYPERBOLIQUES ET LEURS FONCTIONS RECIPROQUESchx e

xex

2 , shx e

xex

2 , tanhx e

xex

exex

Argchx lnx x2 1 x 1, Argshx lnx x2 1

Argthx 12

ln 1x1x

|x| 1, Arg cothx 12

ln 1xx1

|x| 1

ch2a sh 2b 1

cha b cha. chb sha. shb cha b cha. chb sha. shb

sha b sha. chb shb. cha sha b sha. chb shb. cha

tanha b tanh atanh b1tanh a.tanhb

tanha b tanh atanh b1tanh a.tanh b

chp chq 2. ch pq

2 . ch

pq

2 chp chq 2. sh

pq

2 . sh

pq

2

chp.

chq

1

2chp

q

chp

q shp.

shq

1

2chp

q

chp

qshp. chq 1

2shp q shp q

Changement de variable :

Soit t tanh x2

, on a : chx 1t2

1t2 , shx 2t

1t2, tanhx 2t

1t2

Drives :

ch x shx, shx chx, tanhx 1 tanh2x, cot anhx 1 cot anh2x

Argch x 1x21

( x 1) , Argshx 11x2

Arg tanhx 11x2

( |x| 1) , Argco tanhx 11x2

( |x| 1)