Mathc initiation/Fichiers h : c78a5

.

Définitions des fonctions hyperboliques par rapport à la fonction exponentielle

               exp(x) + exp(-x)                          exp(x) - exp(-x)
     cosh(x) = ----------------                sinh(x) = ----------------
                      2                                         2
                      
                      
               sinh(x)   exp(x) - exp(-x)                cosh(x)   exp(x) + exp(-x)
     tanh(x) = ------- = ----------------      coth(x) = ------- = ---------------- 
               cosh(x)   exp(x) + exp(-x)                sinh(x)   exp(x) - exp(-x)
               
               

                  1             2                           1              2       
     csch(x) = ------- = ----------------       sech(x) = ------- = ----------------
               sinh(x)   exp(x) - exp(-x)                 cosh(x)   exp(x) + exp(-x)

.

Liste des équations étudiées

cosh(x)**2 - sinh(x)**2 = 1
sech(x)**2 + tanh(x)**2 = 1
coth(x)**2 - csch(x)**2 = 1

           *****************

sinh(x+y) = cosh(x) sinh(y) + sinh(x) cosh(y)
cosh(x+y) = cosh(x) cosh(y) + sinh(x) sinh(y)

sinh(x-y) = sinh(x)cosh(y) - cosh(x)sinh(y)
cosh(x-y) = cosh(x) cosh(y) - sinh(x) sinh(y)

tanh(x+y) = [tanh(x)+tanh(y)] / [1+tanh(x)*tanh(y)]
tanh(x-y) = [tanh(x)-tanh(y)] / [1-tanh(x)*tanh(y)]

           *****************

cosh(2x) = cosh(x)**2 - sinh(x)**2
......... = 1 + 2*sinh(x)**2
......... = 2*cosh(x)**2 - 1
cosh(3x) = 4*cosh(x)**3 - 3*cosh(x)
cosh(4x) = 8 cosh(x)**4 - 8 cosh(x)**2 + 1


sinh(2x) = 2 cosh(x) sinh(x)
sinh(3x) = 3 sinh(x) + 4*sinh(x)**3
sinh(4x) = 4 sinh(x) cos(x) + 8 sinh(x)**3 cos(x)

tanh(2x) = (2 tanh(x))/(1 + tanh(x)**2)
tanh(3x) = (3 tanh(x) + tanh(x)**3) / (1+3 tan(x)**2)
 
           *****************

cosh(x)**2 = 1/2 + 1/2 cosh(2x)
cosh(x)**3 = 3/4 cosh(x) + 1/4 cosh(3x)
cosh(x)**4 = 3/8 + 1/2 cosh(2x) + 1/8 cosh(4x)

sinh(x)**2 = 1/2 cosh(2x) - 1/2
sinh(x)**3 = 1/4 sinh(3x) - 3/4 sinh(x)
sinh(x)**4 = 3/8 - 1/2 cosh(2x) + 1/8 cosh(4x)

           *****************

sinh(x) sinh(y) = 1/2 [cosh(x+y) - cosh(x-y)]
cosh(x) cosh(y) = 1/2 [cosh(x-y) + cosh(x+y)]
sinh(x) cosh(y) = 1/2 [sinh(x-y) + sinh(x+y)]

           *****************

sinh(x) + sinh(y) = 2 sinh( (x+y)/2 ) cosh( (x-y)/2 )
sinh(x) - sinh(y) = 2 sinh( (x-y)/2 ) cosh( (x+y)/2 )

cosh(x) + cosh(y) = 2 cosh( (x+y)/2 ) cosh( (x-y)/2 )
cosh(x) - cosh(y) = 2 sinh( (x+y)/2 ) sinh( (x-y)/2 )

           *****************
sinh(acosh(x)) = sqrt(x**2 - 1)
cosh(asinh(x)) = sqrt(x**2 + 1)

 
tanh(asech(x)) = sqrt(1 - x**2)
coth(acsch(x)) = sqrt(1 + x**2)