Understanding the Inverse Hyperbolic Cosine Function

What is the Inverse Hyperbolic Cosine Function?

The inverse hyperbolic cosine function, denoted as acosh(x) or cosh⁻¹(x), is the inverse function of the hyperbolic cosine. It returns the hyperbolic angle whose hyperbolic cosine 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 cosine function is defined as:

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

Where:

• \(x\) is any real number greater than or equal to 1
• \(\ln\) is the natural logarithm function

Properties of acosh(x)

• Domain: x ≥ 1
• Range: [0, ∞)
• acosh(1) = 0
• The graph of acosh(x) is symmetric about the y-axis
• acosh(x) is a strictly increasing function for x ≥ 1

Calculation Steps

1. Input any real number x ≥ 1.
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 acosh(2):

1. Input: x = 2
2. Calculate: \(2^2 - 1 = 3\)
3. Calculate: \(\sqrt{3} \approx 1.732051\)
4. Add: \(2 + 1.732051 \approx 3.732051\)
5. Calculate: \(\ln(3.732051) \approx 1.316958\)

Therefore, acosh(2) ≈ 1.316958

Visual Representation

This graph illustrates the inverse hyperbolic cosine function. The point (2, 1.32) corresponds to acosh(2).