How to Use the ASINH, ACOSH, and ATANH Functions in Microsoft Excel for Inverse Hyperbolic Modeling

While primary hyperbolic functions (SINH, COSH, TANH) describe hanging cable profiles, transmission line voltage distributions, and fluid dynamics, inverse problems require computing the underlying parameter from a known hyperbolic ratio. In electrical network engineering, quantitative risk analysis, and relativistic kinematic calculations, analysts must reverse these exponential combinations. Microsoft Excel provides three dedicated mathematical functions: ASINH (inverse hyperbolic sine), ACOSH (inverse hyperbolic cosine), and ATANH (inverse hyperbolic tangent).

Mathematical Definitions and Logarithmic Equivalents

Because hyperbolic equations are constructed from exponential components ($e^x$ and $e^{-x}$), their inverses can be expressed directly in terms of natural logarithms:

• Inverse Hyperbolic Sine: $\text{asinh}(x) = \sinh^{-1}(x) = \ln\left(x + \sqrt{x^2 + 1}\right)$
• Inverse Hyperbolic Cosine: $\text{acosh}(x) = \cosh^{-1}(x) = \ln\left(x + \sqrt{x^2 – 1}\right)$ for $x \ge 1$
• Inverse Hyperbolic Tangent: $\text{atanh}(x) = \tanh^{-1}(x) = \frac{1}{2} \ln\left(\frac{1 + x}{1 – x}\right)$ for $-1 < x < 1$

Unlike circular inverse trigonometric functions (such as ASIN, ACOS, ATAN) which return measurements in radians requiring conversion to degrees, inverse hyperbolic functions evaluate pure real numbers. No conversion using DEGREES() or RADIANS() is necessary.

Syntax and Parameters

The syntax across all three functions requires a single numerical argument:

ASINH(number)
ACOSH(number)
ATANH(number)

Where number is any real number satisfying the mathematical domain constraints of the respective curve.

Practical Example 1: Calculating ASINH across Positive and Negative Domains

The inverse hyperbolic sine function is strictly increasing and defined across the entire real number line from $-\infty$ to $+\infty$:

• Cell A2: -2.5
• Cell A3: 0
• Cell A4: 2.5

In cell B2, compute the inverse value:

=ASINH(A2)

Excel returns:

• ASINH(-2.5) ≈ -1.6472
• ASINH(0) = 0
• ASINH(2.5) ≈ 1.6472

Notice that ASINH is an odd function; changing the sign of the input inverts the sign of the result symmetrically ($f(-x) = -f(x)$).

Practical Example 2: Cable Sag Modeling with ACOSH

A civil engineer calculating the parameter $c$ for a suspension cable under uniform tension knows the ratio of maximum sag elevation $y$ to catenary apex is 1.45. To determine the inverse parameter:

In cell C2:

=ACOSH(1.45)

Excel computes the inverse hyperbolic cosine, returning approximately 0.9027.

To verify the result, compute the forward hyperbolic cosine: =COSH(0.9027), which returns 1.45, confirming reciprocal accuracy.

Practical Example 3: Relativistic Velocity Rapidity with ATANH

In special relativity, combining relativistic velocities without non-linear addition formulas uses “rapidity” $\theta$, where $\theta = \text{atanh}(v/c)$. If an astronomical particle moves at 0.85 times the speed of light ($v/c = 0.85$):

=ATANH(0.85)

Excel calculates the rapidity parameter, yielding approximately 1.2562.

Handling Domain Violations and Error Conditions

Unlike standard polynomial formulas, hyperbolic curves have strict domain boundaries that trigger calculation errors when exceeded:

• #NUM! Error in ACOSH: Occurs whenever number < 1. Because $\cosh(x) \ge 1$ for all real $x$, values below 1 have no real inverse. For example, =ACOSH(0.5) returns #NUM!.
• #NUM! Error in ATANH: Occurs whenever number <= -1 or number >= 1. Because $\tanh(x)$ is asymptotically bounded strictly between $-1$ and $+1$, evaluating =ATANH(1) or =ATANH(1.2) generates a #NUM! error.
• #VALUE! Error: Occurs if text strings or non-numeric cells are passed into any parameter.

To safeguard large automated calculation models against out-of-domain errors, validate inputs with IF logic or wrap formulas with IFERROR:

=IF(A2 >= 1, ACOSH(A2), "Value must be >= 1")