Understanding the Inverse Hyperbolic Sine Function

What is the Inverse Hyperbolic Sine Function?

The inverse hyperbolic sine function, denoted as asinh(x) or sinh⁻¹(x), is the inverse function of the hyperbolic sine. It returns the hyperbolic angle whose hyperbolic sine is the input value. This function is important in various fields of mathematics, physics, and engineering, particularly in problems involving exponential growth or decay.

Formula and Definition

The inverse hyperbolic sine function is defined as:

\[asinh(x) = \ln(x + \sqrt{x^2 + 1})\]

Where:

• \(x\) is any real number
• \(\ln\) is the natural logarithm function

Properties of asinh(x)

• Domain: All real numbers
• Range: All real numbers
• asinh(x) is an odd function: asinh(-x) = -asinh(x)
• asinh(0) = 0
• The graph of asinh(x) is symmetric about the origin
• asinh(x) is a strictly increasing function

Calculation Steps

1. Input any real number x.
2. Calculate \(x^2 + 1\).
3. Calculate the square root of the result from step 2.
4. Add x to the result from step 3.
5. Calculate the natural logarithm of the result from step 4.

Example Calculation

Let's calculate asinh(2):

1. Input: x = 2
2. Calculate: \(2^2 + 1 = 5\)
3. Calculate: \(\sqrt{5} \approx 2.236068\)
4. Add: \(2 + 2.236068 \approx 4.236068\)
5. Calculate: \(\ln(4.236068) \approx 1.443635\)

Therefore, asinh(2) ≈ 1.443635

Visual Representation

This graph illustrates the inverse hyperbolic sine function. The point (2, 1.44) corresponds to asinh(2).