Hyperbolic Functions in Scientific 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

Hyperbolic 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 Hyperbolic Functions Work

The fundamental hyperbolic functions are defined as:

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

Where:

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

Key Properties: Unlike trigonometric functions, hyperbolic functions are not periodic. They relate to the standard exponential function.

3. Applications of Hyperbolic Functions

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

4. Using the Calculator

Tips: Enter any real number value (in radians) and select the hyperbolic function you want to compute. The calculator will return the result (unitless).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between sin and sinh?
A: sin is a periodic trigonometric function (relates to circles), while sinh is a hyperbolic function (relates to hyperbolas) that grows exponentially.

Q2: Are hyperbolic functions periodic?
A: No, unlike trigonometric functions, hyperbolic functions are not periodic.

Q3: What are inverse hyperbolic functions?
A: They are the inverse operations (asinh, acosh, atanh) that return the value whose hyperbolic function is the input.

Q4: Why are they called "hyperbolic" functions?
A: They relate to the geometry of hyperbolas, just as trigonometric functions relate to circles.

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