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).

$\cosh$ is an abbreviation for 'cosine hyperbolic', and $\sinh$ is an abbreviation for 'sine hyperbolic'.

$\sinh$ is pronounced sinch,

$\cosh$ is pronounced 'cosh', as you'd expect,

and $\tanh$ 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,

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

Reciprocal functions may be defined in the obvious way:

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

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

It is easily shown that ${\cosh }^2(x)-{\sinh }^2(x)=1$, analogous to the result ${\cos }^2(x)+{\sin }^2(x)=1.$ 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,

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

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

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

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

If a > |b| then

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

If |a| < b then

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

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 Template:Fracπ as x tends to infinity, and -Template:Fracπ 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,

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

We also have

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