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"

~	$sinh x = u$	$cosh x = u$	$tanh x = u$	$coth x = u$	$sech x = u$	$csch x = u$
$sinh x$	$u$	$\sqrt{u^2 - 1}$	$\frac{u}{\sqrt{1 - u^2}}$	$\frac{1}{\sqrt{u^2 - 1}}$	$\frac{\sqrt{1 - u^2}}{u}$	$\frac{1}{u}$
$cosh x$	$\sqrt{1 + u^2}$	$u$	$\frac{1}{\sqrt{1 - u^2}}$	$\frac{u}{\sqrt{u^2 - 1}}$	$\frac{1}{u}$	$\frac{\sqrt{1 + u^2}}{u}$
$tanh x$	$\frac{u}{\sqrt{1 + u^2}}$	$\frac{\sqrt{u^2 - 1}}{u}$	$u$	$\frac{1}{u}$	$\sqrt{1 - u^2}$	$\frac{1}{\sqrt{1 + u^2}}$
$coth x$	$\frac{\sqrt{1 + u^2}}{u}$	$\frac{u}{\sqrt{u^2 - 1}}$	$\frac{1}{u}$	$u$	$\frac{1}{\sqrt{1 - u^2}}$	$\sqrt{1 + u^2}$
$sech x$	$\frac{1}{\sqrt{1 + u^2}}$	$\frac{1}{u}$	$\sqrt{1 - u^2}$	$\frac{\sqrt{u^2 - 1}}{u}$	$u$	$\frac{u}{\sqrt{1 + u^2}}$
$csch x$	$\frac{1}{u}$	$\frac{1}{\sqrt{u^2 - 1}}$	$\frac{\sqrt{1 - u^2}}{u}$	$\sqrt{u^2 - 1}$	$\frac{u}{\sqrt{1 - u^2}}$	$u$

GRAPHS OF HYPERBOLIC FUNCTIONS

y = sinh x

y = cosh x

y = tanh x

y = coth x

y = sech x

y = csch x

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

y = sinh^-1x

y = cosh^-1x

y = tanh^-1x

y = coth^-1x

y = sech^-1x

y = csch^-1x

RELATIONSHIP BETWEEN HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS

sin(ix) = i sinh x	cos(ix) = cosh x	tan(ix) = i tanh x
csc(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 csc x	sech(ix) = sec x	coth(ix) = -i cot x

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

RELATIONSHIP BETWEEN INVERSE HYPERBOLIC AND INVERSE TRIGONOMETRIC FUNCTIONS

sin^-1 (ix) = isinh^-1x	sinh^-1(ix) = i sin^-1x
cos^-1 x = ±i cosh^-1 x	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^-1 x = ±i sec^-1x
csc^-1(ix) = -i csch^-1x	csch^-1(ix) = -i csc^-1x