ACOTH in Excel: Invert Hyperbolic Cotangent with a Strict Domain

• Introduced: Microsoft marks the function as introduced in Excel 2013. Version-marker reference.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Checked: October 14, 2026; official ACOTH reference. Documentation and independently calculated examples; no native Excel execution or broader platform certification.

=ACOTH(number)

The required argument follows the explicit syntax constraint ABS(number)>1. Thus 2 and −2 are in the real domain, while −1, 0 and 1 are not.

Microsoft’s current remarks contain conflicting error prose: one sentence uses a simple less-than-one condition, and another names ACOT rather than ACOTH. This guide follows the unambiguous absolute-value domain and the inverse definition. It does not use those inconsistent sentences to certify a native error-code matrix for rejected inputs.

Enter −4, −2, 2, 4, 10 in A2:A6. In B2 put =ACOTH(A2) and fill down. Expected values are approximately −0.255412812, −0.549306144, 0.549306144, 0.255412812, 0.100335348.

The output preserves the input’s sign. Increasing a positive input from 2 to 10 moves its inverse closer to zero; the inverse is not an increasing function on that positive branch.

For ordinary permitted x, the real mathematical inverse is 0.5*LN((x+1)/(x-1)). Compare:

=0.5*LN((2+1)/(2-1))
=0.5*LN((-2+1)/(-2-1))

Expected results are approximately 0.549306144 and −0.549306144. Both logarithm arguments are positive: 3 and 1/3. Negative x outside −1 is therefore compatible with the explicit real domain.

This check explains the signs without treating every sentence of a flawed error table as established behavior. It is an independent mathematical model, not native Excel execution.

Suppose a fictional response is y=3*COTH(x/2) with positive x. For target responses 6 and 12, solve with:

=2*ACOTH(6/3)
=2*ACOTH(12/3)

Expected solutions are approximately 1.098612289 and 0.510825624. Substituting each unrounded value into the original COTH expression reconstructs 6 and 12.

The positive branch requires y>3. A target of exactly 3 is an asymptotic limit rather than a finite x, while a positive target below 3 is outside this selected model. Choosing the branch is part of the problem statement.

With ordinary numeric inputs in D2:D6 containing −2, −1, 0, 1, 2, the policy =IF(ABS(D2)>1,ACOTH(D2),"Review domain") is intended to give −0.549306144, Review domain, Review domain, Review domain, 0.549306144.

The strict inequality deliberately excludes both endpoints. The wrapper does not establish a universal short-circuit guarantee for arbitrary arrays or imported expressions.

Inputs close to ±1 can generate large-magnitude inverses. Very large permitted inputs have inverses close to zero. Inspect stored precision before assuming a displayed endpoint or rounded zero is an exact source value.

Do not silently clamp an invalid source into the domain. Preserve it for review and test the actual target workbook if exact native error codes or extreme-input accuracy are required.

• Microsoft ACOTH reference.
• Editorial testing standards.

Worked inputs are fictional. Decimal answers are independently calculated mathematical expectations rounded to nine places, not captured native results. Extreme magnitudes, complete coercion matrices and cross-version numerical accuracy remain untested.