1. What are Hyperbolic Functions?

Hyperbolic functions sinh (hyperbolic sine) and cosh (hyperbolic cosine) are analogs of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. They appear frequently in solutions to differential equations and in engineering applications.

2. How Does the Calculator Work?

The calculator uses the following definitions:

Where:

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

Explanation: The functions combine exponential growth (e^x) and decay (e^-x) in different ways to produce their characteristic shapes.

3. Applications of Hyperbolic Functions

Details: Hyperbolic functions are used in:

• Calculating catenary curves (shape of hanging cables)
• Special relativity equations
• Heat transfer calculations
• Electrical engineering (transmission line theory)
• Solutions to Laplace's equation

4. Using the Calculator

Tips: Enter any real number value in radians. The calculator will compute both sinh and cosh of the input value.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between sinh and cosh?
A: They satisfy the identity cosh²(x) - sinh²(x) = 1, analogous to the trigonometric identity cos²(x) + sin²(x) = 1.

Q2: What are the ranges of these functions?
A: sinh(x) can be any real number, while cosh(x) ≥ 1 for all real x.

Q3: How do these relate to trigonometric functions?
A: sinh(ix) = i sin(x) and cosh(ix) = cos(x), where i is the imaginary unit.

Q4: What are their derivatives?
A: d/dx sinh(x) = cosh(x) and d/dx cosh(x) = sinh(x).

Q5: Are there inverse functions?
A: Yes, they are called arsinh (or sinh⁻¹) and arcosh (or cosh⁻¹).