How To Calculate Hyperbolic Functions

Hyperbolic Functions Formulas:

\[ \sinh(x) = \frac{e^x - e^{-x}}{2} \] \[ \cosh(x) = \frac{e^x + e^{-x}}{2} \] \[ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \] \[ \text{csch}(x) = \frac{1}{\sinh(x)} \] \[ \text{sech}(x) = \frac{1}{\cosh(x)} \] \[ \coth(x) = \frac{\cosh(x)}{\sinh(x)} \]

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1. What Are Hyperbolic Functions?

Hyperbolic functions are analogs of the ordinary trigonometric functions, but for the hyperbola rather than the circle. The basic hyperbolic functions are sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent), with their reciprocal functions csch, sech, and coth.

2. How Are Hyperbolic Functions Calculated?

The hyperbolic functions are defined using exponential functions:

\[ \sinh(x) = \frac{e^x - e^{-x}}{2} \] \[ \cosh(x) = \frac{e^x + e^{-x}}{2} \] \[ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \]

The reciprocal functions are:

\(\text{csch}(x) = \frac{1}{\sinh(x)}\)
\(\text{sech}(x) = \frac{1}{\cosh(x)}\)
\(\coth(x) = \frac{\cosh(x)}{\sinh(x)}\)

3. Applications of Hyperbolic Functions

Applications: Hyperbolic functions appear in solutions of linear differential equations, calculation of angles in hyperbolic geometry, Laplace's equation, and in the description of hanging cables (catenary).

4. Using the Calculator

Instructions: Enter any real number value and select which hyperbolic function you want to calculate. The calculator will compute the result using the exponential definitions.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between trigonometric and hyperbolic functions?
A: While trigonometric functions relate to the unit circle, hyperbolic functions relate to the hyperbola. They have similar but not identical properties.

Q2: Are hyperbolic functions periodic?
A: Unlike trigonometric functions, hyperbolic functions are not periodic.

Q3: What are the ranges of hyperbolic functions?
A: sinh(x) and cosh(x) have ranges (-∞, ∞) and [1, ∞) respectively. tanh(x) has range (-1, 1).

Q4: Can hyperbolic functions be expressed as series?
A: Yes, for example: \(\sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots\)

Q5: What are the derivatives of hyperbolic functions?
A: The derivative of sinh(x) is cosh(x), and the derivative of cosh(x) is sinh(x).