Hyperbolic Functions

In this tutorial we shall study certain combinations of ${e^x}$ and ${e^{ – x}}$, which are called hyperbolic functions. These functions have numerous applications and arise naturally in many mathematical problems. It will become evident as we progress that hyperbolic functions have many properties in common with trigonometric functions.

Hyperbolic Sine Function
The hyperbolic sine function, denoted by $\sinh x$, is defined as
$\sinh x = \frac{{{e^x} – {e^{ – x}}}}{2}$

Hyperbolic Cosine Function
The hyperbolic cosine function, denoted by $\cosh x$, is defined as
$\cosh x = \frac{{{e^x} + {e^{ – x}}}}{2}$

Hyperbolic Tangent Function
The hyperbolic tangent function, denoted by $\tanh x$, is defined as
$\tanh x = \frac{{{e^x} – {e^{ – x}}}}{{{e^x} + {e^{ – x}}}}$

Hyperbolic Cotangent Function
The hyperbolic cotangent function, denoted by $\coth x$, is defined as
$\coth x = \frac{{{e^x} + {e^{ – x}}}}{{{e^x} – {e^{ – x}}}}$

Hyperbolic Secant Function
The hyperbolic secant function, denoted by $\operatorname{sech} x$, is defined as
$\cosh x = \frac{2}{{{e^x} + {e^{ – x}}}}$

Hyperbolic Cosecant Function
The hyperbolic cosecant function, denoted by $\operatorname{csch} x$, is defined as
$\operatorname{csch} x = \frac{2}{{{e^x} – {e^{ – x}}}}$