## coth x

### The function coth(x) is the hyperbolic cotangent of x

The function coth(x) is the hyperbolic cotangent of x. It is defined as the ratio of the hyperbolic cosine to the hyperbolic sine of x. The formula for coth(x) is:

coth(x) = cosh(x) / sinh(x)

where cosh(x) is the hyperbolic cosine function and sinh(x) is the hyperbolic sine function.

To understand the concept of hyperbolic functions, we need to briefly introduce hyperbolic trigonometry. Just as there are circular trigonometric functions (sin, cos, tan, etc.) that relate angles to the sides of a right triangle in a circle, there are also hyperbolic trigonometric functions that relate angles to the sides of a hyperbola.

The hyperbolic cosine function (cosh(x)) is defined as:

cosh(x) = (e^x + e^(-x)) / 2

where e is the base of the natural logarithm (approximately equal to 2.71828). The hyperbolic sine function (sinh(x)) is defined as:

sinh(x) = (e^x – e^(-x)) / 2

The coth(x) function is the reciprocal of the hyperbolic tangent function (tanh(x)). It can also be expressed in terms of exponentials as:

coth(x) = e^x / e^x – e^(-x)

Just like circular trigonometric functions, hyperbolic trigonometric functions have several properties and identities that can be used in various calculations and problem-solving. It is important to note that the hyperbolic trigonometric functions and their inverses (such as coth(x)) are used in many areas of mathematics and physics, particularly in areas involving exponential growth, electric circuits, and wave propagation.

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