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).

$$\displaystyle \cosh$$ 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 $1/2$&pi; as x tends to infinity, and -$1/2$&pi; 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

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