Hyperbolic Function Calculator

Hyperbolic Functions:

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

Input Value (x):

radians

Function:

Result:

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1. What Are Hyperbolic Functions?

Hyperbolic functions are analogs of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. The basic hyperbolic functions are the hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh).

2. How Does the Calculator Work?

The calculator computes various hyperbolic functions and their inverses:

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

Where:

\( x \) — Input value in radians
\( e \) — Euler's number (~2.71828)

Explanation: These functions are computed using exponential functions and have properties similar to trigonometric functions but for hyperbolas.

3. Applications of Hyperbolic Functions

Details: 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

Tips: Enter a numeric value for x (in radians), select the hyperbolic function you want to compute, and click "Calculate". For inverse functions, note the input restrictions.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between trigonometric and hyperbolic functions?
A: Trigonometric functions relate to circles, while hyperbolic functions relate to hyperbolas. They have similar properties but different geometric interpretations.

Q2: Can I input degrees instead of radians?
A: No, this calculator requires input in radians. Convert degrees to radians first (radians = degrees × π/180).

Q3: What are the ranges of hyperbolic functions?
A: sinh(x) and cosh(x) range from -∞ to +∞, while tanh(x) ranges between -1 and 1.

Q4: Why do some inverse functions have input restrictions?
A: For example, acosh(x) requires x ≥1 because cosh(x) never produces values less than 1.

Q5: Where are hyperbolic functions used in real life?
A: They're used in physics (special relativity), engineering (catenary arches), and mathematics (complex analysis).