Trigonometry/Cosh, Sinh and Tanh

The functions cosh x, sinh x and tanh x have 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 'coshine', and $\displaystyle \sinh$ is an abbreviation for shine. $\displaystyle \sinh$ is pronounced shine, $\displaystyle \cosh$ is pronounced 'cosh', as you'd expect, and $\displaystyle \tanh$ is pronounced than, with the th soft as in three. At one time, the pronunciation tanch was used.

[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(i x) = cosh(x)
• sin(i x) = i sinh(x)
• tan(i x) = i tanh(x)

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

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