 # 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 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