Sinh Calculator

Hyperbolic Sine Formula:

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

Hyperbolic Sine (sinh):

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1. What Is Hyperbolic Sine?

The hyperbolic sine (sinh) is a mathematical function that is part of the family of hyperbolic functions, analogous to the ordinary trigonometric functions but for a hyperbola rather than a circle.

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, calculations of catenary curves, special relativity, and complex analysis.

4. Using the Calculator

Tips: Enter any real number value in radians. The result is dimensionless and can range from negative to positive infinity.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between sin and sinh?
A: sin is a trigonometric function for circular relationships, while sinh is a hyperbolic function for hyperbola relationships.

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

Q3: Is sinh(x) always increasing?
A: Yes, the hyperbolic sine function is strictly increasing throughout its domain.

Q4: What is sinh(0)?
A: sinh(0) = 0, as both exponential terms become 1 and cancel each other out.

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