Hyperbolic Geometry

Overview

The hyperbolic functions are analogues of the trigonometric functions but defined using the hyperbola instead of the circle.

The hyperbolic sine function, denoted $\sinh$ , is defined as:

\sinh x = \frac{e^{x} - e^{- x}}{2}

The hyperbolic cosine function, denoted $\cosh$ , is defined as:

\cosh x = \frac{e^{x} + e^{- x}}{2}

The hyperbolic tangent function, denoted $\tanh$ , is defined as:

\tanh x = \frac{\sinh x}{\cosh x}

The hyperbolic secant function, denoted $sech$ , is defined as:

sech x = \frac{1}{\cosh x}

The hyperbolic cosecant function, denoted $csch$ , is defined as:

csch x = \frac{1}{\sinh x}

The hyperbolic cotangent function, denoted $coth$ , is defined as:

coth x = \frac{1}{\tanh x}

There exist a number of identities analogous to those of the trigonemetric functions.

The hyperbolic sine and hyperbolic tangent functions are odd whereas the hyperbolic cosine function is even.

For any real number $t$ , $\cosh^{2} (t) - \sinh^{2} (t) = 1$ .