1. What is Hyperbolic Sine?

The hyperbolic sine (sinh) is a mathematical function related to the regular sine function but for hyperbolas rather than circles. It's defined using exponential functions and appears in various areas of mathematics and physics.

2. How Does the Calculator Work?

The calculator uses the hyperbolic sine formula:

Where:

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

Explanation: The function calculates the difference between exponential growth and decay at the given input value, divided by 2.

3. Applications of Hyperbolic Sine

Details: Hyperbolic sine appears in solutions to differential equations, special relativity, catenary curves (hanging cables), and electrical engineering.

4. Using the Calculator

Tips: Enter any real number (positive or negative) in radians. The calculator will compute the corresponding hyperbolic sine value.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between sin(x) and sinh(x)?
A: sin(x) is the circular trigonometric function, while sinh(x) is the hyperbolic trigonometric function. They have different properties and applications.

Q2: What is the range of sinh(x)?
A: The range of sinh(x) is all real numbers (-∞, ∞). It grows exponentially in both positive and negative directions.

Q3: Can I input degrees instead of radians?
A: The formula requires radians. Convert degrees to radians first (radians = degrees × π/180).

Q4: What are some important identities for sinh?
A: Key identities include: sinh(-x) = -sinh(x), sinh(x+y) = sinh(x)cosh(y) + cosh(x)sinh(y), and d/dx sinh(x) = cosh(x).

Q5: How is sinh(x) related to other hyperbolic functions?
A: It's related to cosh(x) via cosh²(x) - sinh²(x) = 1, analogous to the Pythagorean identity for trigonometric functions.