Hyperbolic functions - sinh, cosh, tanh, coth, sech, csch
DEFINITION OF HYPERBOLIC FUNCTIONS
Hyperbolic sine of x
$\text{sinh}\ x = \frac{e^{x} - e^{-x}}{2}$
Hyperbolic cosine of x
$\text{cosh}\ x = \frac{e^x + e^{-x}}{2}$
Hyperbolic tangent of x
$\text{tanh}\ x = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
Hyperbolic cotangent of x
$\text{coth}\ x = \frac{e^x + e^{-x}}{e^x - e^{-x}}$
Hyperbolic secant of x
$\text{sech}\ x = \frac{2}{e^x + e^{-x}}$
Hyperbolic cosecant of x
$\text{csch}\ x = \frac{2}{e^x - e^{-x}}$
RELATIONSHIPS AMONG HYPERBOLIC FUNCTIONS
$\text{tanh}\ x = \frac{\text{sinh}\ x}{\text{cosh}\ x}$
$\text{coth}\ x = \frac{1}{\text{tanh}\ x} = \frac{\text{cosh}\ x}{\text{sinh}\ x}$
$\text{sech}\ x = \frac{1}{\text{cosh}\ x}$
$\text{csch}\ x = \frac{1}{\text{sinh}\ x}$
$\text{cosh}^2x - \text{sinh}^2x = 1$
$\text{sech}^2x + \text{tanh}^2x = 1$
$\text{coth}^2x - \text{csch}^2x = 1$
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) = -coth x
ADDITION FORMULAS
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)
coth(x ± y) = (coth x coth y ± l)/(coth y ± coth x)
DOUBLE ANGLE FORMULAS
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh2x + sinh2x = 2 cosh2x — 1 = 1 + 2 sinh2x
tanh 2x = (2tanh x)/(1 + tanh2x)
HALF ANGLE FORMULAS
$\sinh \frac{x}{2} = \pm \sqrt{\frac{\cosh x - 1}{2}}$ [+ if x > 0, - if x < 0]
$\cosh \frac{x}{2} = \sqrt{\frac{\cosh x + 1}{2}}$
$\tanh \frac{x}{2} = \pm \sqrt{\frac{\cosh x - 1}{\cosh x + 1}}$ [+ if x > 0, - if x < 0]
$=\frac{sinh(x)}{1 + cosh(x)} = \frac{cosh(x) - 1}{sinh(x)}$
MULTIPLE ANGLE FORMULAS
sinh 3x = 3 sinh x + 4 sinh3 x
cosh 3x = 4 cosh3 x — 3 cosh x
tanh 3x = (3 tanh x + tanh3 x)/(1 + 3 tanh2x)
sinh 4x = 8 sinh3 x cosh x + 4 sinh x cosh x
cosh 4x = 8 cosh4 x — 8 cosh2 x + 1
tanh 4x = (4 tanh x + 4 tanh3 x)/(1 + 6 tanh2 x + tanh4 x)
POWERS OF HYPERBOLIC FUNCTIONS
sinh2 x = ½cosh 2x — ½
cosh2 x = ½cosh 2x + ½
sinh3 x = ¼sinh 3x — ¾sinh x
cosh3 x = ¼cosh 3x + ¾cosh x
sinh4 x = 3/8 - ½cosh 2x + 1/8cosh 4x
cosh4 x = 3/8 + ½cosh 2x + 1/8cosh 4x
SUM, DIFFERENCE AND PRODUCT OF HYPERBOLIC FUNCTIONS
sinh x + sinh y = 2 sinh ½(x + y) cosh ½(x - y)
sinh x - sinh y = 2 cosh ½(x + y) sinh ½(x - y)
cosh x + cosh y = 2 cosh ½(x + y) cosh ½(x - y)
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 — y))
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 in the section "Functions of Negative Arguments"
GRAPHS OF HYPERBOLIC FUNCTIONS






INVERSE HYPERBOLIC FUNCTIONS
If x = sinh y, then y = sinh-1 a 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.
$\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1})$ $-\infty < x < \infty$
$\cosh^{-1} x = \ln(x + \sqrt{x^2 - 1})$ $x \geq l$ [$\cosh^{-1} x > 0$ is principal value]
$\tanh^{-1} x = \frac{1}{2} \ln\frac{(1 + x)}{(1 - x)}$ $- 1 < x < 1$
$\coth^{-1} x = \frac{1}{2} \ln\frac{(x + 1)}{(x - 1)}$ $x > 1$ or $x < -1$
$\text{sech}^{-1} x = \ln(\frac{1}{x} + \sqrt{\frac{1}{x^2} - 1})$ $0 < x \leq l$ [$\text{sech}^{-1} x > 0$ is principal value]
$\text{csch}^{-1} x = \ln(\frac{1}{x} + \sqrt{\frac{1}{x^2} + 1})$ $x \neq 0$
RELATIONS BETWEEN INVERSE HYPERBOLIC FUNCTIONS
csch-1 x = sinh-1 (1/x)
sech-1 x = cosh-1 (1/x)
coth-1 x = tanh-1 (1/x)
sinh-1(-x) = -sinh-1x
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
PERIODICITY OF HYPERBOLIC FUNCTIONS
In the following k is any integer.
sinh (x + 2kπi) = sinh x csch (x + 2kπi) = csch x
cosh (x + 2kπi) = cosh x sech (x + 2kπi) = sech x
tanh (x + kπi) = tanh x coth (x + kπi) = coth x