Trigonometry/Cosh, Sinh and Tanh

The functions cosh x, sinh x and tanh xhave much the same relationship to the rectangular hyperbola y² = x² - 1 as the circular functions do to the circle y² = 1 - x². They are therefore sometimes called the hyperbolic functions (h for hyperbolic).

Notation and pronunciation $\displaystyle \cosh$ is an abbreviation for 'cosine hyperbolic', and $\displaystyle \sinh$ is an abbreviation for 'sine hyperbolic'. $\displaystyle \sinh$ is pronounced sinch, $\displaystyle \cosh$ is pronounced 'cosh', as you'd expect, and $\displaystyle \tanh$ is pronounced tanch.

[Diagram of rectangular hyperbola to illustrate]

Definitions

They are defined as

$\cosh(x)={\frac {1}{2}}(e^{x}+e^{-x});\,\,\sinh(x)={\frac {1}{2}}(e^{x}-e^{-x});\,\,\tanh(x)={\frac {\sinh(x)}{\cosh(x)}}$

Equivalently,

$\displaystyle e^{x}=\cosh(x)+\sinh(x);\,\,e^{-x}=\cosh(x)-\sinh(x)$

Reciprocal functions may be defined in the obvious way:

$\operatorname {sech} (x)={\frac {1}{\cosh(x)}};\,\,\operatorname {cosech} (x)={\frac {1}{\sinh(x)}};\,\,\coth(x)={\frac {1}{\tanh(x)}}$

1 - tanh²(x) = sech²(x); coth²(x) - 1 = cosech²(x)

It is easily shown that $\displaystyle \cosh ^{2}(x)-\sinh ^{2}(x)=1$, analogous to the result $\displaystyle \cos ^{2}(x)+\sin ^{2}(x)=1.$ In consequence, sinh(x) is always less in absolute value than cosh(x).

sinh(-x) = -sinh(x); cosh(-x) = cosh(x); tanh(-x) = -tanh(x).

Their ranges of values differ greatly from the corresponding circular functions:

• cosh(x) has its minimum value of 1 for x = 0, and tends to infinity as x tends to plus or minus infinity;
• sinh(x) is zero for x = 0, and tends to infinity as x tends to infinity and to minus infinity as x tends to minus infinity;
• tanh(x) is zero for x = 0, and tends to 1 as x tends to infinity and to -1 as x tends to minus infinity.

[Add graph]

Addition formulae

There are results very similar to those for circular functions; they are easily proved directly from the definitions of cosh and sinh:

sinh(x+y) = sinh(x)cosh(y) + cosh(x)sinh(y)

cosh(x+y) = cosh(x)cosh(y) + sinh(x)sinh(y)

Inverse functions

If y = sinh(x), we can define the inverse function x = sinh^-1y, and similarly for cosh and tanh. The inverses of sinh and tanh are uniquely defined for all x. For cosh, the inverse does not exist for values of y less than 1. For y = 1, x = 0. For y > 1, there will be two corresponding values of x, of equal absolute value but opposite sign. Normally, the positive value would be used. From the definitions of the functions,

$\displaystyle \sinh ^{-1}x=\ln(x+{\sqrt {x^{2}+1}})$

$\displaystyle \cosh ^{-1}x=\ln(x+{\sqrt {x^{2}-1}})$

$\tanh ^{-1}x={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)$

Simplifying a cosh(x) + b sinh(x)

If a > |b| then

$\displaystyle a\cosh(x)+b\sinh(x)={\sqrt {a^{2}-b^{2}}}\cosh(x+c)$ where :$\displaystyle \tanh(c)={\frac {b}{a}}$

If |a| < b then

$\displaystyle a\cosh(x)+b\sinh(x)={\sqrt {b^{2}-a^{2}}}\sinh(x+d)$ where :$\displaystyle \tanh(d)={\frac {a}{b}}$

Relations to complex numbers

• $\cos(ix)=\cosh(x)$
• $\cos(x)=\cosh(ix)$

• $\sin(ix)=i\sinh(x)$
• $i\sin(x)=\sinh(ix)$

• $\tan(ix)=i\tanh(x)$
• $i\tan(x)=\tanh(ix)$

The addition formulae and other results can be proved from these relationships.

The gudermannian

The gudermannian (named after Christoph Gudermann, 1798–1852) is defined as gd(x) = tan^-1(sinh(x)). We have the following properties:

• gd(0) = 0;
• gd(-x) = -gd(x);
• gd(x) tends to ¹⁄₂π as x tends to infinity, and -¹⁄₂π as x tends to minus infinity.

The inverse function gd^-1(x) = sinh^-1(tan(x)) = ln(sec(x)+tan(x)).

Differentiation

As can be proved from the definitions above,

${\frac {d}{dx}}\sinh(x)=\cosh(x);\,\,{\frac {d}{dx}}\cosh(x)=\sinh(x);\,\,{\frac {d}{dx}}\tanh(x)=sech^{2}(x);\,\,{\frac {d}{dx}}gd(x)=sech(x)$

We also have

${\frac {d}{dx}}\sinh ^{-1}(x)={\frac {1}{\sqrt {x^{2}+1}}};\,\,{\frac {d}{dx}}\cosh ^{-1}(x)={\frac {1}{\sqrt {x^{2}-1}}};\,\,{\frac {d}{dx}}\tanh ^{-1}(x)={\frac {1}{1-x^{2}}};\,\,{\frac {d}{dx}}gd^{-1}(x)=\sec(x)$.