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 exponential definitions:

Where:

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

Explanation: These functions combine exponential growth and decay to produce hyperbolic analogs of trigonometric functions.

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 (catenaries). They're also used in special relativity and complex analysis.

4. Using the Calculator

Tips: Enter any real number value for x (in radians). The calculator will compute sinh(x), cosh(x), and tanh(x) with high precision.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between hyperbolic and trigonometric functions?
A: They're related through complex numbers: sinh(ix) = i sin(x) and cosh(ix) = cos(x).

Q2: What are the ranges of these functions?
A: sinh(x) has range (-∞, ∞), cosh(x) has range [1, ∞), and tanh(x) has range (-1, 1).

Q3: Are there hyperbolic versions of other trig functions?
A: Yes, including coth, sech, and csch, which are hyperbolic analogs of cot, sec, and csc.

Q4: What are the derivatives of hyperbolic functions?
A: d/dx sinh(x) = cosh(x), d/dx cosh(x) = sinh(x), d/dx tanh(x) = sech²(x).

Q5: How are these functions implemented in programming?
A: Most programming languages provide sinh(), cosh(), and tanh() functions in their math libraries.