Understanding the Hyperbolic Sine Function

What is the Hyperbolic Sine Function?

The hyperbolic sine function, denoted as sinh(x), is one of the hyperbolic functions in mathematics. It is analogous to the trigonometric sine function but is defined in terms of exponential functions rather than angles. The hyperbolic sine function has applications in various fields, including physics, engineering, and signal processing.

Formula and Definition

The hyperbolic sine function is defined as:

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

Where:

• \(x\) is any real number
• \(e\) is Euler's number (approximately 2.71828)

Properties of sinh(x)

• Domain: All real numbers
• Range: All real numbers
• sinh(x) is an odd function: sinh(-x) = -sinh(x)
• sinh(0) = 0
• As x approaches infinity, sinh(x) approaches positive infinity
• As x approaches negative infinity, sinh(x) approaches negative infinity

Calculation Steps

1. Input any real number x.
2. Calculate \(e^x\) and \(e^{-x}\).
3. Subtract \(e^{-x}\) from \(e^x\).
4. Divide the result by 2.

Example Calculation

Let's calculate sinh(1):

1. Input: x = 1
2. Calculate: \(e^1 \approx 2.71828\) and \(e^{-1} \approx 0.36788\)
3. Subtract: \(2.71828 - 0.36788 = 2.35040\)
4. Divide by 2: \(2.35040 / 2 = 1.17520\)

Therefore, sinh(1) ≈ 1.17520

Visual Representation

This graph illustrates the hyperbolic sine function. The point (1, 1.18) corresponds to sinh(1).