1. What Are Hyperbolic Functions?

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

2. How Does the Calculator Work?

The calculator computes various hyperbolic functions and their inverses using PHP's built-in mathematical functions:

Explanation: These functions are defined in terms of exponential functions and have properties similar to trigonometric functions but with important differences.

3. Applications of Hyperbolic Functions

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

4. Using the Calculator

Tips: Enter a numeric value and select the hyperbolic function you want to compute. For inverse functions, note the input restrictions (acosh requires x≥1, atanh requires -1

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 identities but different signs in some cases.

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

Q3: Where are hyperbolic functions used in real life?
A: They're used in physics (special relativity), engineering (catenary arches), and mathematics (complex analysis).

Q4: What are the ranges of hyperbolic functions?
A: sinh(x) has range (-∞,∞), cosh(x) has range [1,∞), and tanh(x) has range (-1,1).

Q5: How are inverse hyperbolic functions defined?
A: They're the inverse functions of the hyperbolic functions, often expressed in terms of logarithms.