Hyperbolic Sine Calculator

Hyperbolic Sine Formula:

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

Hyperbolic Sine (sinh(x)):

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1. What is Hyperbolic Sine?

The hyperbolic sine (sinh) is a mathematical function related to the regular sine function but for hyperbolas instead of 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:

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

Where:

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

Explanation: The function calculates the difference between exponential growth and decay functions, divided by two.

3. Applications of Hyperbolic Sine

Details: Hyperbolic sine appears in solutions to differential equations, special relativity, catenary curves, and complex analysis.

4. Using the Calculator

Tips: Enter any real number value in radians. The result is dimensionless (unitless).

5. Frequently Asked Questions (FAQ)

Q1: How is hyperbolic sine different from regular sine?
A: While regular sine is periodic and bounded, hyperbolic sine grows exponentially in both directions.

Q2: What is the range of sinh(x)?
A: The range is all real numbers (-∞, ∞).

Q3: What is sinh(0)?
A: sinh(0) = 0.

Q4: How is sinh related to other hyperbolic functions?
A: It's related to cosh (hyperbolic cosine) via the identity cosh²(x) - sinh²(x) = 1.

Q5: Can I use degrees instead of radians?
A: No, hyperbolic functions always use radians as input.