Learning Ashram

HYPERBOLIC FUNCTIONS

 

HYPERBOLIC FUNCTIONS

 
 
DEFINITION OF HYPERBOLIC FUNCTIONS
 
   
 

RELATIONSHIPS AMONG HYPERBOLIC FUNCTIONS

 
   
 

FUNCTIONS OF NEGATIVE ARGUMENTS

 
 
  • sinh(-x) = -sinh x
  • cosh(-x) = cosh x
  • tanh (-x)=-tanh x
  • csch(-x) = -csch x
  • sech (-x) = sech x
  • coth(-x) = -cothx
 
 

ADDITION FORMULAS

 
   
 

DOUBLE ANGLE FORMULAS

 
   
 

HALF ANGLE FORMULAS

 
   
 

MULTIPLE ANGLE FORMULAS

 
   
 

POWERS OF HYPERBOLIC FUNCTIONS

 
  sinh2 x = ½ cosh 2x —½
cosh2 x = ½ cosh 2x + ½
sinh3 x =¼ sinh 3x — ¾ sinh x
cosh3 x = ¼ cosh 3x + ¾ cosh x
sinh4 x = ⅜ — ½ cosh 2x + ⅛ cosh 4x
cosh4x = ⅜ + ½ cosh2x + ⅛ cosh 4x
 
 

SUM, DIFFERENCE AND PRODUCT OF HYPERBOLIC FUNCTIONS

 
  sinh x + sinh y = 2 sinh ½(x + y) cosh ½(x - w)
sinh x - sinh j/ = 2 cosh i(x + y) sinh ½(x - y)
cosh x + cosh y - 2 cosh i(x + y) cosh ½(x - j/)
cosh x - cosh y = 2 sinh ½(x + y) sinh ½(x — y)
sinh x sinh y = ½{cosh (x + y) - cosh (x - y)}
cosh x cosh y= ½{cosh (x + y) + cosh (x — j/)}
sinh x cosh y = ½{sinh (x + y) + sinh (x - y)}
 
 

EXPRESSION OF HYPERBOLIC FUNCTIONS IN TERMS OF OTHERS

 
 

In the following we assume x > 0. If x < 0 use the appropriate sign as indicated by formulas

 
 

GRAPHS OF HYPERBOLIC FUNCTIONS

 
   
 

INVERSE HYPERBOLIC FUNCTIONS

 
 

If x = sinh y, then y = sinh-1 x is called the inverse hyperbolic sine of x. Similarly we define the other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the case of inverse trigonometric functions.we restrict ourselves to principal values for which they can be considered as single-valued.
The following list shows the principal values [unless otherwise indicated] of the inverse hyperbolic functions expressed in terms of logarithmic functions which are taken as real valued.

 
 

RELATIONS BETWEEN INVERSE HYPERBOLIC FUNCTIONS

 
 
  • csch-1x = sinh-1 (l/x)
  • sech-1 x = cosh-1 (1/x)
  • coth-1 x = tanh-1 (1/x)
  • sinh-1(-x) =-sinh-1 x
  • tanh-1(-x) = - tanh-1x
  • coth-1 (-x) = -coth-1x
  • csch-1 (-x) =- csch-1x
 
 

GRAPHS OF INVERSE HYPERBOLIC FUNCTIONS

 
   
 

RELATIONSHIP BETWEEN HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS

 
 
  • sin (ix) = i sinh x
  • cos (ix) = cosh x
  • tan (ix) = i tanh x
  • esc (ix) =- i csch x
  • sec (ix)= sech x
  • cot (ix) =-i coth x
  • sinh (ix) = i sin x
  • cosh (ix) = cos x
  • tanh ((ix)=- i tan x
  • csch (ix)=-i esc x
  • sech (ix) = sec x
  • coth (ix) = - ieot x
 
 

PERIODICITY OF HYPERBOLIC FUNCTIONS

 
 

In the following k is any integer.

  • sinh (x + 2kπi) = sinh x
  • cosh (x + 2kπi) = cosh x
  • tanh (x + 2kπi) = tanh x
  • csch (x + 2kπi) = csch x
  • sech (x + 2kπi) = sech x
  • coth (x + 2kπi) = coth x
 
 
RELATIONSHIP BETWEEN INVERSE HYPERBOLIC AND INVERSE TRIGONOMETRIC FUNCTIONS
 
 
  • sin-1 (ix) = i sinh-1x
  • sinh-1(ix) = i sin-1x
  • cos-1x = ±i cosh-1x
  • cosh-1x = ±i cos-1x
  • tan-1(ix) = i tanh-1x
  • tanh-1(ix) = i tan-1x
  • cot-1 (ix) = - i coth-1x
  • coth-1(ix) = -i cot-1x
  • sec-1 x = ±i sech-1x
  • sech-1x = ±i sec-1x
  • csc-1 (ix) = - i csch-1x
  • csch-1(ix) = - i csc-1x