Graphs of sinh(x), cosh(x) and tanh(x)

Introduction to Hyperbolic Functions

The hyperbolic functions sinh(x), cosh(x) and tanh(x), pronounced ‘shine’, ‘cosh’, and ‘than’, are similar to the trigonometric sine, cosine and tangent functions.
These functions are defined using exponential functions, which means they have unique properties and uses.
The hyperbolic functions are a fundamental part of advanced mathematics subjects like calculus, complex numbers, and differential equations.

The hyperbolic sine function, sinh(x), is defined as (e^x - e^-x) / 2. Its graph is a curve that passes through the origin (0,0) and increases without bound as x goes to positive or negative infinity.
The hyperbolic cosine function, cosh(x), is defined as (e^x + e^-x) / 2. Its graph starts from point (0,1) and increases without bound as x goes to positive infinity, and approaches 0 as x goes to negative infinity.
The hyperbolic tangent function, tanh(x), is defined as sinh(x) / cosh(x). Its graph is a curve that passes through the origin and asymptotically approaches 1 and -1 as x goes to positive and negative infinity respectively.

sinh(-x) = -sinh(x), cosh(-x) = cosh(x), and tanh(-x) = -tanh(x). This means sinh and tanh are odd functions, while cosh is an even function.
The identity cosh²(x) - sinh²(x) = 1 is equivalent to the familiar Pythagorean identity for sine and cosine, but note the difference in the sign.
The derivative of sinh(x) is cosh(x), the derivative of cosh(x) is sinh(x), and the derivative of tanh(x) is sech²(x), where sech is the hyperbolic secant function.

Hyperbolic functions have various applications in physics, geometry, and engineering. For example, they describe the shape of a hanging cable (catenary), describe the distribution of heat or the flow of current in certain systems.
They are also used in complex number theory and in the solution of differential equations.

Mastering the definitions, graph characteristics, properties and identities of sinh(x), cosh(x) and tanh(x) is essential for understanding hyperbolic functions. Moreover, appreciating their practical applications can reveal the beauty and power of mathematics.