Hyperbolic 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 for the hyperbola rather than the circle. The basic hyperbolic functions are sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent).

2. How Does the Calculator Work?

The calculator computes standard 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)

3. Applications of Hyperbolic Functions

Details: Hyperbolic functions appear in solutions of differential equations, calculations of angles in hyperbolic geometry, and descriptions of hanging cables (catenary curves).

4. Using the Calculator

Tips: Enter a numeric value (in radians) and select the hyperbolic function to compute. The calculator will return the result (unitless).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between trigonometric and hyperbolic functions?
A: Trigonometric functions relate to circular functions, while hyperbolic functions relate to hyperbolas, though they share many analogous properties.

Q2: Are hyperbolic functions periodic?
A: Unlike trigonometric functions, only cosh and sech are even functions, and sinh, tanh, coth, and csch are odd functions, but they're not periodic.

Q3: What are inverse hyperbolic functions used for?
A: They're useful for solving equations involving hyperbolic functions and appear in integral calculus solutions.

Q4: Why are they called "hyperbolic" functions?
A: They parameterize the unit hyperbola (x² - y² = 1) just as trigonometric functions parameterize the unit circle.

Q5: What's the range of acosh function?
A: The acosh function is only defined for x ≥ 1, with output range [0, +∞).