ACOTH in Zoho Sheet: Inverse Hyperbolic Cotangent Explained

=ACOTH(number)

The Zoho ACOTH documentation says the input cannot lie between −1 and +1, inclusive. Its published examples include =ACOTH(-2) → −0.549306144, =ACOTH(4) → 0.255412812, and =ACOTH(-3/2) → −0.804718956. The vendor also explicitly identifies #NUM! when the input is between −1 and +1. Those examples and the error rule are documentation evidence, not a new native execution conducted for this guide.

ACOTH is the inverse of the hyperbolic cotangent function on its real domain. It is odd: changing the input sign changes the output sign. As positive input becomes very large, the output approaches zero from above; near 1+, the positive output grows without bound. The corresponding behavior for negative inputs is symmetric. This explains why an input close to ±1 can produce a very large magnitude even when the input itself looks ordinary.

Enter the valid ACOTH inputs below in C2:C9, with examples on both sides of the forbidden interval. Enter =ACOTH(C2) into E2, fill down through E9 and display nine decimal places.

How to reproduce: (1) Enter the example values in column C of a new sheet. (2) Confirm values such as -1.5 and 1.5 are numeric, not text. (3) Enter =ACOTH(C2) in E2; fill to E9. (4) Verify the sign symmetry: row 5 −1.5 and row 6 +1.5 should have equal magnitudes and opposite signs. (5) Check the independently computed values rather than treating a formatted display as proof of exact equality. (6) Put a deliberate invalid test =ACOTH(1) in a separate scratch cell to confirm how the active Zoho editor reports the boundary; no such test was run here.

The table is independent mathematical reference data, computed using the identity ACOTH(x) = ATANH(1/x) for |x|>1, and cross-checked against a logarithmic identity. These are not Zoho Sheet execution results. In particular, a CSV import may use different decimal marks or interpret negative numbers as text depending on settings.

Consider an illustrative mathematical model in which a normalized response r is given by:

r = coth(t / tau)

Here t and tau are measured in the same unit, and tau>0. For t>0, the response r is greater than 1. To recover the elapsed time from a measured response, rearrange the equation:

t = tau * acoth(r)

Suppose a model parameter tau is 12 minutes and the normalized observed response is 2.0. Put 12 in B2 and 2 in C2 on a separate model worksheet. The corresponding candidate Zoho formula is:

=B2*ACOTH(C2)

The independently calculated result is 6.591673732 minutes. The units come entirely from tau, not from ACOTH itself. This is a mathematical model demonstration, not a claim that a particular physical instrument obeys such a response law; using it with real process data requires an appropriate justified model and calibration.

If a model’s measurement noise can produce r<=1, the formula should not silently clip values to 1.0001: doing so substitutes invented data and introduces large sensitivity near the boundary. Instead, flag the row for investigation. A useful guard is:

=IF(ABS(C2)<=1;"OUT OF DOMAIN";B2*ACOTH(C2))

For the positive-time model, also require C2>1; the absolute-value guard by itself checks only ACOTH’s mathematical domain. For B2=12,C2=2, the expected numeric output is 6.591674; for B2=12,C2=0.9, the intended output is the text OUT OF DOMAIN if the expression is accepted by the target Zoho engine. The guard is illustrative only—it was not run in native Zoho Sheet. Numeric-text handling, semicolon separators and blank values must be checked in the actual document. If tau<=0 is invalid for your model, validate that as well using a separate input check.

For valid real inputs:

ACOTH(x) = ATANH(1/x)
ACOTH(x) = 0.5 * LN((x+1)/(x-1))
COTH(ACOTH(x)) = x

The formulas can be used for theoretical cross-checks. For example, at x=2 the reciprocal is 0.5, whose inverse hyperbolic tangent is 0.549306144. The logarithmic ratio is (2+1)/(2-1)=3, so LN(3)/2 also gives 0.549306144. At x=−2, the ratio is (−1)/(−3)=1/3, and LN(1/3)/2 is negative. Each expression is real because |x|>1.

A proposed alternative Zoho formula is =ATANH(1/C2), which is algebraically equivalent on valid inputs. An alternative is not necessarily a superior computational implementation: near ±1, loss of numerical precision and huge intermediate derivatives may matter, and extremely large values can challenge finite-precision arithmetic. Never claim exact bit-for-bit agreement between separate implementations without native testing. Likewise, =COTH(ACOTH(C2)) is a useful inverse identity, but COTH must be documented and supported in the target sheet before treating it as a production check.

The Zoho help page also displays generic #N/A! and #REF! entries; these are not proof that ACOTH itself performs a lookup or produces those errors on valid numbers. Errors may propagate from formulas used to compute its argument. Never infer a particular platform-specific type coercion rule merely from the name ACOTH.

Microsoft documents Excel ACOTH with one argument requiring absolute value greater than 1. The domain and mathematical intent are common reference points, but the exact engine response to text, blanks, boundary roundoff, and numerical precision is not established to be identical. Microsoft’s help remarks contain apparent ACOT/ACOTH wording inconsistencies; do not replace Zoho’s own error table with those remarks.

ACOTH differs from ACOT (inverse ordinary cotangent), ATANH (inverse hyperbolic tangent for |x|<1), ACOSH (inverse hyperbolic cosine for x≥1) and ASINH (inverse hyperbolic sine over the real line).

• Primary vendor source: ACOTH — Zoho Sheet Help, directly checked 2026-10-11; syntax, strict domain and three published examples recorded.
• Cross-software sources: ACOTH — Microsoft Support and ACOTH — Google Docs Editors Help, directly checked 2026-10-11; no other spreadsheet engine was executed for this package.
• Independent evaluation: Eight rows computed in Python and cross-checked using atanh(1/x), log((x+1)/(x-1))/2, and reciprocal hyperbolic tangent identities. Native Zoho Sheet executions: 0. Explicit error strings, cross-platform availability, locale separators and special text inputs remain pending native verification; no first-support version is inferred from an undated function help page.