# 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

\[\operatorname{csch} x = \frac{2}{{{e^x} – {e^{ – x}}}}\]