ACOTH in Apple Numbers: Domain, Negative Inputs, and Examples

=ACOTH(2) has an expected result of about 0.549306. ACOTH is an inverse hyperbolic function, not a conversion between degrees and radians. The input must be less than −1 or greater than 1.

Purpose: Inverse hyperbolic cotangent supports mathematical, scientific, engineering, or geometry calculations where the formula’s input and permitted domain match this documented definition. The worked numbers on this page are independently calculated expectations and have not been observed running in Apple Numbers.

Apple’s English Formulas and Functions Help 15.4 lists the exact signature:

=ACOTH(num)

• num (required): hyperbolic cotangent ratio (unitless). Apple documents ACOTH(num) with a single numeric input strictly below −1 or strictly above 1. The boundaries −1 and 1 are excluded, as are all values between them. Apple gives ACOTH(2) as approximately 0.549306144334055 and ACOTH(COTH(3)) as approximately 3. No optional arguments/defaults are listed.
• Arguments and defaults: there is one required numeric input; there are no optional positions and no documented defaults. =ACOTH(B2) refers to a cell in the same table. For one argument there is no argument-order ambiguity.
• Regional input: the guide uses English function identifiers and period decimal separators, such as 0.5. If the document uses decimal commas, type numeric literals in its local format; Apple’s function overview says multi-argument formulas typed by hand use semicolons when the decimal separator is a comma. That separator rule does not change this function’s one-argument signature.
• Platforms/releases: the 15.4 number identifies the edition of Apple’s Functions Help linked here. It is not proof of the first Numbers release supporting this function or identical behaviour on every Mac, iPhone, iPad and web version. No native edition was executed for this guide.
• Aliases: only the English identifier ACOTH is established here. No additional aliases or historical equivalences were verified; translated labels in other locales are not automatically alternative identifiers.

Official documentation: Apple ACOTH function; Apple Functions overview. Consulted 2026-10-11 UTC.

1. Create a table named MathInputs in Numbers. Keep row 1 as headers, with the inputs in column B and calculated results starting in column C.
2. Type the B-column values from the table below as real numbers, not quoted strings or cells formatted as Text. Enter the row labels in column A; they are for the reader, not arguments.
3. For ACOTH, start cell C2 with =ACOTH(B2) and fill down to the last row. Use at least six decimal places in the result column so displayed rounding is easy to inspect.
4. Compare the formulas and values in C2 through C9 against the independently predicted figures. Formula fills should refer to B3, B4, etc.

Row: 2

A: description
negative 5

B: numeric input
-5

C: formula
=ACOTH(B2)

C: expected numeric result
-0.202733

Row: 3

A: description
negative 3

B: numeric input
-3

C: formula
=ACOTH(B3)

C: expected numeric result
-0.346574

Row: 4

A: description
negative 2

B: numeric input
-2

C: formula
=ACOTH(B4)

C: expected numeric result
-0.549306

Row: 5

A: description
negative 1.5

B: numeric input
-1.5

C: formula
=ACOTH(B5)

C: expected numeric result
-0.804719

Row: 6

A: description
positive 1.5

B: numeric input
1.5

C: formula
=ACOTH(B6)

C: expected numeric result
0.804719

Row: 7

A: description
positive 2

B: numeric input
2

C: formula
=ACOTH(B7)

C: expected numeric result
0.549306

Row: 8

A: description
positive 3

B: numeric input
3

C: formula
=ACOTH(B8)

C: expected numeric result
0.346574

Row: 9

A: description
positive 5

B: numeric input
5

C: formula
=ACOTH(B9)

C: expected numeric result
0.202733

Validation labels: The values above are expected, computed by Python standard-library mathematics, rounded to six decimals. They are not native Numbers test observations. Numbers may display fewer/more digits depending on cell format; small floating-point differences should be evaluated with a tolerance.

Use Numbers table references when inputs are elsewhere

Within MathInputs, B2 is sufficient. If the formula is in a different Numbers table, Apple’s documented qualified-reference pattern allows MathInputs::B2 (for example, =ACOTH(MathInputs::B2)). For a different sheet, the documented pattern is Sheet Name::MathInputs::B2 if disambiguation is required. Click the cell in the formula editor to obtain the correct token if your real sheet/table names contain spaces. Copying relative references normally changes the referenced row; lock $ components deliberately if you need a fixed source cell. Apple cell references.

Why the domain is two disconnected intervals

The mathematical inverse hyperbolic cotangent has a real output only where x < −1 or x > 1. Notice how this differs from ATANH, whose real-input interval is −1 < x < 1. The formulas may look similar, but swapping them changes the valid input region. In a data-import workflow, rows containing −1, 0 and 1 must be treated as invalid for ACOTH before calculation.

Reproduce the logarithmic identity on safe inputs

For B2=2, the mathematical identity acoth(x) = 0.5 ln((x+1)/(x-1)) predicts that =LN((B2+1)/(B2-1))/2 will give approximately 0.549306. For B2=-2 it predicts approximately −0.549306. This derivation is only for real abs(x) > 1; it may behave differently numerically extremely close to a boundary because a ratio and a logarithm amplify rounding. The built-in ACOTH is the simpler formula for normal use.

Negative inputs reflect the positive curve

acoth(-x) = -acoth(x) on the valid real domain. Accordingly, =ACOTH(-2) and =ACOTH(2) have equal expected magnitudes with opposite signs. If a copied formula shows both signs the same, check the source cell, absolute references and unintended ABS calls before concluding that the function is incorrect.

Boundary sensitivity and the inverse check

=ACOTH(1.0001) has an expected result around 4.951769; close-to-one inputs can produce unexpectedly large magnitudes. As x approaches 1 from above, the mathematical result grows without bound. =COTH(ACOTH(2)) is expected to return approximately 2, and =ACOTH(COTH(3)) is expected to return approximately 3. These are diagnostic identities, not observations from a native Numbers run.

Data typing and error messages

Apple describes a numeric input for ACOTH. The official page establishes the documented mathematical domain; it does not enumerate every error code, rounding threshold, overflow behaviour, or conversion rule for text, blanks, Boolean values and dates. Therefore this guide avoids asserting a specific Numbers error message for a nonnumeric cell or out-of-domain value. First test the direct numeric input recommended in Apple’s examples, then inspect any imported text or regional number formatting. For extreme magnitudes, don’t claim exact machine precision or Apple-specific limits without a native test.

• Apple Numbers: ACOTH() is documented in the linked 15.4 help. Numbers references can include both sheet and table names, separated with double colons.
• Microsoft Excel: ACOTH documentation supports the same named real-valued mathematical function. Excel also documents ACOTH(number) and the |number| > 1 restriction. Excel-specific error codes should not be attributed to Apple Numbers without testing.
• Google Sheets: the official Sheets function catalogue lists ACOTH under its Math functions, with one argument. That establishes existence and documented syntax in Sheets, not identical error coercion, floating-point precision, or release support.
• On migration: recalculate several positive, negative, and boundary-adjacent examples using the target application; convert Numbers table-qualified references into the destination’s own worksheet reference syntax. Distinguish a numeric expected value from a screen result seen in an app. Locale and sheet-name handling need target-file verification.

ACOTH is the inverse of COTH, not of ordinary trigonometric COT. Its output is a real hyperbolic parameter, not a degree angle.

• Apple ACOTH reference — exact signature, numeric argument, domain/meaning, and Apple’s own listed examples; Help 15.4, accessed 2026-10-11 UTC.
• Apple Functions overview — function naming, required arguments, and decimal-comma separator rule; Help 15.4, accessed 2026-10-11 UTC.
• Apple cell-reference formats — same-table and qualified references, and relative/absolute copying behaviour; Help 15.4, accessed 2026-10-11 UTC.
• Apple help welcome — high-level application/platform context only; accessed 2026-10-11 UTC.
• Google Sheets function list — corroborates the separate Sheets function name; accessed 2026-10-11 UTC.
• Microsoft Excel ACOTH function — corroborates the Excel name and documents Excel-specific bounds where stated; accessed 2026-10-11 UTC.

• ACOSH — continue with a related Apple Numbers calculation.
• ASINH — continue with a related Apple Numbers calculation.
• ATANH — continue with a related Apple Numbers calculation.
• COSH — continue with a related Apple Numbers calculation.
• SINH — continue with a related Apple Numbers calculation.

For a software-specific comparison: LibreOffice Calc ACOTH, Apache OpenOffice Calc ACOTH, Gnumeric ACOTH. A shared name does not establish identical argument, error or version behaviour.

The linked official function documentation was reviewed on October 12, 2026. Apple’s Formulas and Functions Help 15.4 identifies the documentation edition, not a minimum Numbers release or a guarantee of historical platform parity. Selected worked examples were checked using independent mathematical or text-processing models; results are expected values, not observations from the spreadsheet application. Native Apple Numbers execution was not performed. Undocumented coercion, exact error messages and platform-specific behaviour remain unverified.

View reproducible example inputs and independent check results. This record is not a native spreadsheet test log.