# Hyperbolic trigonometric functions

The hyperbolic trigonometric functions are analogs of the standard trigonometric functions using a hyperbola as the defining conic section rather than a circle. They are defined as:

• Hyperbolic sine: $sinh(x) = \frac{e^x - e^{-x}}{2}$
• Hyperbolic cosine: $cosh(x) = \frac{e^x + e^{-x}}{2}$
• Hyperbolic tangent: $tanh(x) = \frac{sinh(x)}{cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
• Hyperbolic cosecant: $csch(x) = \frac{1}{sinh(x)} = \frac{2}{e^x - e^{-x}}$
• Hyperbolic secant: $sech(x) = \frac{1}{cosh(x)} = \frac{2}{e^x + e^{-x}}$
• Hyperbolic cotangent: $coth(x) = \frac{cosh(x)}{sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$
 sinh and cosh tanh and coth