Hyperbolic functions are analogues of ordinary trigonometric and circular functions in mathematics, but they are defined using the hyperbola rather than the circle.

In hyperbolic geometry, these functions are used to calculate distances and angles. They can also be found in the solutions of many linear differential equations, cubic equations, and Laplace’s equation in Cartesian coordinates. We will discuss in detail the basic hyperbolic function, its properties, formulas, and hyperbolic function identities. The free online Hyperbolic function calculator can be used to compute the value of hyperbolic functions.

Below are the basic formulas for hyperbolic functions and their graph functions:

The formula defines the hyperbolic function \( f(x) = sinh \space x \)

\( sinh \space x = \dfrac{e^x – e^{-x}}{2} \)

The function meets these two conditions:

\( sinh \space 0 = 0 \) and \( sinh(−x) = − sinh \space x \)

The graph of sinh x is consistently between the graphs of \( \dfrac{e^x}{2} \) and \( \dfrac{e^{-x}}{2} \).

The hyperbolic Cosine function is defined by the formula

\( cosh \space x = \dfrac{e^x + e^{-x}}{2} \)

The function meets these two conditions:

\( cosh \space 0 = 1 \) and \( cosh \space x = cosh(−x) \)

The graph of cosh x is consistently above the graphs of \( \dfrac{e^x}{2} \) and \( \dfrac{e^{-x}}{2} \).

The formula defines the hyperbolic function f(x) = tanh x.

\( tanh \space x = \dfrac{e^x – e^{-x}}{2} ÷ \dfrac{e^x + e^{-x}}{2} \)

\( = \dfrac{e^x – e^{-x}}{e^x + e^{-x}} \)

Hyperbolic Function Identities are:

\( cosh^2 \space x – sinh^2 \space x = 1 \)

\( sinh ( – x) = – sinh \space x\)

\( cosh ( – x) = cosh \space x \)

\( tanh ( – x) = – tanh \space x \)

\( coth (– x) = – coth \space x\)

What are Inverse Hyperbolic Functions?

Inverse hyperbolic functions, also known as area hyperbolic functions, are multivalued functions that are inverse functions of hyperbolic functions. These functions return the hyperbolic angles corresponding to the given hyperbolic function value. For inverse hyperbolic function in complex plane we use these formulas:

\(sinh^{-1} \space x = ln(x + \sqrt{1+x^2}) \)

\( cosh^{-1} \space x = ln(x + \sqrt{x^2 –1}) \)

\( tanh^{-1} \space x = (\dfrac{1}{2})(ln(1 + x) – ln(1 – x)) \)

What are Hyperbolic Trig Functions?

Trigonometric functions relate the angle of a circle to its sides. The trig function captures the oscillatory movement of an angle around the circle, while the hyperbolic functions capture the oscillatory movements around a hyperbola. The basic trig functions include sin(x), tan(x), & cos(x). Their hyperbolic counterparts have sinh(x), cosh (x), & tanh(x).

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