Sinh Cosh Tanh Calculator

Hyperbolic Functions:

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

sinh(x):

cosh(x):

tanh(x):

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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 sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent).

2. How Does the Calculator Work?

The calculator uses the following definitions:

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

Where:

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

3. Applications of Hyperbolic Functions

Details: Hyperbolic functions appear in solutions of 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 any real number in radians. The calculator will compute sinh(x), cosh(x), and tanh(x) values.

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 different identities and properties.

Q2: What is the range of tanh(x)?
A: tanh(x) has a range of (-1, 1), approaching ±1 as x approaches ±∞.

Q3: What is cosh(0)?
A: cosh(0) = 1, since (e⁰ + e⁻⁰)/2 = (1 + 1)/2 = 1.

Q4: Are hyperbolic functions periodic?
A: Unlike trigonometric functions, sinh and cosh are not periodic. However, they have symmetry properties.

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