Hyperbolic functions - sinh, cosh, tanh, coth, sech, csch

DEFINITION OF HYPERBOLIC FUNCTIONS

Hyperbolic sine of x = sinh x = (ex - e-x)/2

Hyperbolic cosine of x = cosh x = (ex + e-x)/2

Hyperbolic tangent of x = tanh x = (ex - e-x)/(ex + e-x)

Hyperbolic cotangent of x = coth x = (ex + e-x)/(ex - e-x)

Hyperbolic secant of x = sech x = 2/(ex + e-x)

Hyperbolic cosecant of x = csch x = 2/(ex - e-x)

RELATIONSHIPS AMONG HYPERBOLIC FUNCTIONS

tanh x = sinh x/cosh x

coth x = 1/tanh x = cosh x/sinh x

sech x = 1/cosh x

csch x = 1/sinh x

cosh2x - sinh2x = 1

sech2x + tanh2x = 1

coth2x - csch2x = 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

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