INSTITUT NATIONAL POLYTECHNIQUE DE TOULOUSE MATHS Rappels Formulaire Jean-Pierre Bourgade Pascal Floquet Xuân Meyer Année Probatoire A Distance Septembre 2012 II.1 Fonctions trigonométriques II.1 1 Fonctions trigonométriques sinus sin(π/2) = √1 sin(π/3) = √3/2 sin(π/4) = 2/2 sin(π/6) = 1/2 sin(−a) = − sin(a) sin(π − a) = sin(a) sin(π + a) = − sin(a) sin(π/2 − a) = cos(a) sin(π/2 + a) = − cos(a) cosinus cos(π/2) = 0 cos(π/3) =√1/2 cos(π/4) = √2/2 cos(π/6) = 3/2 cos(−a) = cos(a) cos(π − a) = − cos(a) cos(π + a) = − cos(a) cos(π/2 − a) = sin(a) cos(π/2 + a) = − sin(a) Formules d’addition tangente tan(π/2) = √ ∞ tan(π/3) = 3 tan(π/4) = √1 tan(π/6) = 3/3 tan(−a) = − tan(a) tan(π − a) = − tan(a) tan(π + a) = tan(a) tan(π/2 − a) = 1/ tan(a) tan(π/2 + a) = −1/ tan(a) Relations usuelles : sin2 (x) + cos2 (x) = 1 sin(a + b) = sin(a) cos(b) + sin(b) cos(a) cos(a + b) = cos(a) cos(b) − sin(a) sin(b) tan(a + b) = tan(a) + tan(b) 1 − tan(a) tan(b) sin(a − b) = sin(a) cos(b) − sin(b) cos(a) cos(a − b) = (cos(a) cos(b) + sin(a) sin(b) tan(a − b) = tan(a) − tan(b) 1 + tan(a) tan(b) sin(2a) = 2 sin(a) cos(a) cos(2a) = cos2 (a) − sin2 (a) tan(2a) = sin2 (x) = tan2 (x) 1 − cos(2x) = 2 1 + tan2 (x) cos2 (x) = 1 + cos(2x) 1 = 2 1 + tan2 (x) transformation : t = tan( a2 ) 2t 1 + t2 sin(a) = 1 − t2 1 + t2 2t tan(a) = 1 − t2 cos(a) = 2 tan(a) 1 − tan2 (a) transformation somme-produit : Transformation produit-somme : a+b a−b sin(a) + sin(b) = 2 sin( ) cos( ) 2 2 sin(a) sin(b) = cos(a − b) − cos(a + b) 2 a+b a−b cos(a) + cos(b) = 2 cos( ) cos( ) 2 2 cos(a) cos(b) = cos(a − b) + cos(a + b) 2 sin((a + b) tan(a) + tan(b) = cos(a) cos(b) sin(a) cos(b) = sin(a + b) + sin(a − b) 2 sin(a) − sin(b) = 2 cos( a+b a−b ) sin( ) 2 2 cos(a) − cos(b) = −2 sin( tan(a) − tan(b) = a+b a−b ) sin( ) 2 2 sin(a − b) cos(a) cos(b) Maths Rappels II.2 Fonctions hyperboliques II.2 2 Fonctions hyperboliques Définitions ex + e−x 2 ex − e−x sinh(x) = 2 sinh(x) ex − e−x e2x − 1 tanh(x) = = x = 2x −x cosh(x) e +e e +1 cosh(x) = cosh(x) + sinh(x) = ex cosh(x) − sinh(x) = e−x transformation : t = tanh( a2 ) Formules d’addition cosh(a + b) = cosh(a) cosh(b) + sinh(b) sinh(a) sinh(a + b) = sinh(a) cosh(b) + sinh(b) cosh(a) tanh(a) + tanh(b) tanh(a + b) = 1 + tanh(a) tanh(b) sinh(a) = 1 + t2 1 − t2 2t tanh(a) = 1 + t2 cosh(a) = cosh(a − b) = cosh(a) cosh(b) − sinh(a) sinh(b) sinh(a − b) = (sinh(a) cosh(b) − cosh(a) sinh(b) tanh(a − b) = tanh(a) − tanh(b) 1 − tanh(a) tanh(b) sinh(2a) = 2 sinh(a) cosh(a) = 2 tanh a 1 − tanh2 a cosh(2a) = cosh2 (a) + sinh2 (a) = 2 cosh2 a − 1 = 2 sinh2 a + 1 = tanh(2a) = 2t 1 − t2 2 tanh(a) 1 + tanh2 (a) 1 + tanh2 (a) 1 − tanh2 (a) transformation somme-produit : Relations usuelles : cosh(−x) = cosh(x) sinh(−x) = − sinh(x) tanh(−x) = − tanh(x) coth(−x) = − coth(x) cosh2 (x) − sinh2 (x) = 1 sinh(a) + sinh(b) = 2 sinh( a−b a+b ) cosh( ) 2 2 cosh(a) + cosh(b) = 2 cosh( a+b a−b ) cosh( ) 2 2 tanh2 (x) 1 − tanh2 (x) 1 cosh2 (x) = 1 − tanh2 (x) cosh(x) + 1 x cosh2 ( ) = 2 2 sinh(a) − sinh(b) = 2 cosh( a+b a−b ) sinh( ) 2 2 cosh(a) − cosh(b) = 2 sinh( a+b a−b ) sinh( ) 2 2 x cosh(x) − 1 sinh2 ( ) = 2 2 tanh(a) − tanh(b) = sinh2 (x) = sinh(x) cosh(x) − 1 x = =± tanh( ) = 2 cosh(x) + 1 sinh(x) tanh(a) + tanh(b) = s sinh(a + b) cosh(a) cosh(b) sinh(a − b) cosh(a) cosh(b) cosh(x) − 1 cosh(x) + 1 Maths Rappels II.3 Développements limités II.3 3 Développements limités Table II.1: Développement limités de fonctions usuelles f onction ex Développement limité d’ordre n au voisinage de 0 ex = 1 + x x2 xn + + ··· + + xn ǫ(x) 1! 2! n! sin(x) sin(x) = x − x3 x5 x2p+1 + + · · · + (−1)p + x2p+2 ǫ(x) 3! 5! (2p + 1)! cos(x) cos(x) = 1 − x4 x2p x2 + + · · · + (−1)p + x2p+1 ǫ(x) 2! 4! (2p)! tan(x) tan(x) = x + x5 x7 x3 + 2 + 17 + x8 ǫ(x) 3 15 315 sh(x) sh(x) = x + x5 x2p+1 x3 + + ··· + + x2p+2 ǫ(x) 3! 5! (2p + 1)! ch(x) ch(x) = 1 + x4 x2p x2 + + ··· + + x2p+1 ǫ(x) 2! 4! (2p)! th(x) th(x) = x − x3 x5 x7 + 2 − 17 + x8 ǫ(x) 3 15 315 1 1+x 1 1−x 1 (1 + x)2 1 = 1 − x + x2 + · · · + (−1)n xn + +xn ǫ(x) 1+x 1 = 1 + x + x2 + · · · + xn + +xn ǫ(x) 1−x 1 = 1 − 2x + 3x2 + · · · + (−1)n (n + 1)xn + xn ǫ(x) (1 + x)2 √ 1+x √ √ 1 1+x √ 1+x=1+ x2 1.3.5 · · · (2n − 3) n x − + · · · + (−1)n+1 x + xn ǫ(x) 2 2.4 2.4 · · · (2n) 1 1.3.5 · · · (2n − 1) n x 1.3 2 x + · · · + (−1)n+1 x + xn ǫ(x) =1− + 2 2.4 2.4 · · · (2n) 1+x α α(α − 1) 2 α(α − 1) · · · (α − n + 1) n x+ x + ··· + x + xn ǫ(x) 1! 2! n! (1 + x)α , α ∈ Q (1 + x)α = 1 + ln(1 + x) ln(1 + x) = x − arcsin(x) arcsin((x) = x + Argsh(x) Argsh(x) = x − arctan(x) Arctan(x) = x − Argth(x) Argth(x) = x + ln( 1+x ) 1−x x2 xn + · · · + (−1)n+1 + xn ǫ(x) 2 n 1.3.5 · · · (2p − 1) 1 3 x + ··· + x2p+1 + x2p+2 ǫ(x) 2.3 2.4.6 · · · (2p)(2p + 1) 1 3 1.3.5 · · · (2p − 1) x + · · · + (−1)p x2p+1 + x2p+2 ǫ(x) 2.3 2.4.6 · · · (2p)(2p + 1) x3 x5 x2p+1 + + · · · + (−1)p + x2p+2 ǫ(x) 3 5 2p + 1 x3 x5 x2p+1 + + ··· + + x2p+2 ǫ(x) 3 5 2p + 1 1+x x3 x5 x2p+1 ln( + x2p+2 ǫ(x) )=2 x+ + + ··· + 1−x 3 5 2p + 1 Maths Rappels II.4 Dérivation-intégration II.4 4 Dérivation-intégration Table II.2: Dérivées et primitives de fonctions usuelles f onction Dérivée P rimitive Conditions a 0 xα αxα−1 ax xα+1 α+1 α 6= −1, x > 0 1 x − 1 x2 1 + x2 − (a2 1 − x2 (a2 a2 a2 1 x 2 − a2 √ √ √ − ln(x) 2x + x 2 )2 x 1 Arctan a a a + x 1 ln 2a a − x x − a 1 ln 2a x + a 2x − x 2 )2 2x (x2 − a2 )2 x 2 + a2 x √ 2 x + a2 x 2 − a2 x √ 2 x − a2 x 6= 0 a 6= 0 a 6= 0, x 6= ±a a 6= 0, x 6= ±a √ a2 x√ 2 x + a2 + ln(x + x2 + a2 ) 2 2 a 6= 0 √ a2 x√ 2 ln x + x2 − a2 x − a2 − 2 2 a 6= 0, |x| > |a| a2 x√ 2 x a − x2 + arcsin( ) 2 2 a √ ln(x + x2 + a2 ) ex x −√ 2 a − x2 x −p (x2 + a2 )3 x −p (x2 − a2 )3 x p (a2 − x2 )3 ax ax ln(a) ln(x) 1 x x ln(x) − x x>0 loga (x) 1 x ln(a) x(loga (x) − loga (e)) a > 0, a 6= 1, x > 0 sin(x) cos(x) cos(x) − sin(x) − cos(x) sin(x) a2 − x 2 1 x 2 + a2 1 √ x 2 − a2 1 √ a2 − x 2 √ √ ln x + x2 − a2 ex 1 cos2 (x) tan(x) 1 + tan2 (x) = cotan(x) −(1 + cotan2 (x)) = − 1 sin(x) − 1 cos(x) sin(x) cos2 (x) cos(x) sin2 (x) 1 sin2 (x) a > 0, |x| < a a 6= 0 a 6= 0, |x| > |a| x arcsin( ) a a 6= 0, |x| < a ex ax ln(a) a > 0, a 6= 1 π + kπ, k ∈ Z 2 − ln |cos(x)| x 6= ln |sin(x)| x 6= kπ, k ∈ Z ln tan( x2 ) x 6= kπ, k ∈ Z ln tan( x2 + π4 ) x 6= π 2 + kπ, k ∈ Z Maths Rappels II.5 Binôme de newton 5 f onction Dérivée P rimitive Conditions 1 cos2 (x) −2 tan(x)(1 + tan2 (x)) tan(x) x 6= 1 √ 1 − x2 −1 √ 1 − x2 1 1 + x2 −1 1 + x2 arcsin((x) arccos(x) arctan(x) arccotan(x) th(x) 1 − th2 (x) = coth(x) 1 − coth2 (x) = − Argch(x) Argth(x) Argcoth(x) II.5 √ 1 − x2 |x| < 1 x arccos(x) − √ 1 − x2 |x| < 1 ch(x) sh(x) 1 ch2 (x) ln(ch(x)) 1 sh2 (x) ln |sh(x)| th(x) ch2 (x) 1 √ 1 + x2 1 √ x2 − 1 1 1 − x2 1 1 − x2 −2 Argsh(x) x arcsin(x) + √ xarccotan(x) + ln( 1 + x2 ) ch(x) sh(x) 1 + kπ, k ∈ Z √ x arctan(x) − ln( 1 + x2 ) sh(x) ch(x) ch2 (x) π 2 x 6= 0 th(x) xArgsh(x) − xArgch(x) − √ 1 + x2 1 √ x>1 x2 − 1 √ xArgth(x) + ln( 1 − x2 ) |x| < 1 √ xArgcoth(x) + ln( x2 − 1) |x| > 1 Binôme de newton Le binôme de Newton est une formule mathématique donnée par Isaac Newton pour le développement d’une puissance entière quelconque d’un binôme. (x + y)n = n X Cnk xn−k y k k=0 où les nombres : Cnk = sont appelés coefficients binomiaux. n! k!(n − k)! k+1 = Cnk + Cnk+1 Cn+1 k Cnk = Cn−k Cnk = n k−1 C k n−1 Maths Rappels