ACOSH Function in Apple Numbers: Formula, Examples, and Domain Rules

=ACOSH(1) returns 0. =ACOSH(10) has the expected value about 2.993223. Find the real inverse hyperbolic cosine of an input at least 1. It is useful for recovering a nonnegative hyperbolic parameter from an observed cosh value.

Important: Real-valued domain: num ≥ 1. A value below 1 is outside the real domain; the exact Numbers error message has not been verified natively. These are expected mathematical results, not screenshots or observed outputs from an Apple Numbers installation.

Apple’s current English Formulas and Functions Help 15.4 documents the function below. This guide does not establish the first Numbers release supporting it, or certify exact feature parity on all devices and historical versions. The documentation was accessed 2026-10-11 UTC.

=ACOSH(num)

One required numeric argument, num, which must be greater than or equal to 1. There is no optional parameter or default. Apple names the parameter num.

Enter an equals sign first. Apple prints uppercase function names but allows lowercase names when typed. No second argument, separator, or optional setting is defined for this function. Decimal notation in the examples uses a period, as in 0.5. Apple’s general documentation says that for formulas with multiple arguments, a decimal-comma locale uses a semicolon separator; this particular one-argument formula has no argument separator. Change example constants to the decimal convention accepted by your configured locale, and confirm the stored B-column values are numeric. ACOSH official syntax; Apple function entry rules.

Real-valued domain: num ≥ 1. A value below 1 is outside the real domain; the exact Numbers error message has not been verified natively.

1. Open an editable Numbers sheet and add a regular table (or reuse a blank table). Name it InputData to make references unambiguous.
2. Keep the first row as a header, with Label, Input, and Result in columns A, B and C.
3. Enter the exact sample numbers from the table below into B2 downward. Format B and C as Number and show at least six decimal places in C before comparing.
4. Enter =ACOSH(B2) into C2. Confirm it references the input number (not its text label).
5. Copy the C2 formula down through the last valid test row. Use the invalid input only as a controlled domain troubleshooting exercise, not as a successful-output test.
6. Compare each cell with the expected numbers below. Last-digit differences can arise from finite precision; do not compare rounded display text as exact mathematical equality.

Row: 2

A: Label
boundary

B: Input
1

C: Formula
=ACOSH(B2)

C: Expected output (6 decimals)
0.000000

Row: 3

A: Label
just above boundary

B: Input
1.125

C: Formula
=ACOSH(B3)

C: Expected output (6 decimals)
0.494933

Row: 4

A: Label
small positive example

B: Input
2

C: Formula
=ACOSH(B4)

C: Expected output (6 decimals)
1.316958

Row: 5

A: Label
ten

B: Input
10

C: Formula
=ACOSH(B5)

C: Expected output (6 decimals)
2.993223

Row: 6

A: Label
large input

B: Input
100

C: Formula
=ACOSH(B6)

C: Expected output (6 decimals)
5.298292

Row: 7

A: Label
cosh(2) approximately

B: Input
3.76219569108

C: Formula
=ACOSH(B7)

C: Expected output (6 decimals)
2.000000

Row: 8

A: Label
invalid (domain test)

B: Input
0.5

C: Formula
Do not apply: outside real domain

C: Expected output (6 decimals)
No finite real result; actual Numbers error not observed

The figures above were calculated independently with Python’s math.acosh function for the same real inputs. They are predictions, not observations in Numbers. The body values can be entered as constants without any external connection or data import.

Reading a value from another Numbers table

If InputData is a different table on the same sheet, use =ACOSH(InputData::B2) rather than a bare B2 where ambiguity could arise. If it is on a different sheet, Apple documents the pattern SheetName::TableName::Cell, for example =ACOSH(Sheet 2::InputData::B2). Use the formula editor’s reference selection to avoid mistakes with spaces and headers. Apple reference syntax.

Recover a hyperbolic parameter from a known cosh value

If a model supplies COSH(t) = 3, and t is known to be nonnegative, use =ACOSH(3) to recover an expected t ≈ 1.762747. Unlike ordinary ACOS, this inverse is not intended for cosine values between −1 and 1.

Check the inverse relationship without assuming the input sign

For an eligible input x ≥ 1, =COSH(ACOSH(x)) is mathematically equal to x; expect only floating-point agreement. In the other order, ACOSH(COSH(t)) = ABS(t) for real t, not always t, because COSH is an even function. For example, ACOSH(COSH(-2)) is 2, not −2. This matters whenever a negative model parameter is possible.

Independent alternative for explanation—not a better production formula

Mathematically, acosh(x) = ln(x + sqrt(x²−1)) when x ≥ 1. The equivalent-style Numbers formula is =LN(B2+SQRT(B2^2-1)). It is useful to understand the calculation, but the built-in ACOSH is clearer and may be numerically more stable at extreme inputs. Do not replace ACOSH with this expanded expression as a precision claim without native tests.

Domain guard for imported measurements

A value of exactly 1 is valid. A measurement such as 0.999999 is not; do not silently clamp it unless your data policy explicitly calls for tolerance. A suitable guarded formula when an out-of-range observation should be labeled is =IF(B2<1,"Outside ACOSH domain",ACOSH(B2)). This guards the stated numerical domain only; imported text and other data types still need checking.

Numbers’ exact displayed error wording, numerical underflow/overflow thresholds, and any implicit conversion of nonnumeric cells have not been tested here. Do not infer those implementation details from Python or from another spreadsheet product. The real-domain restrictions in this guide come from Apple’s official function documentation.

ACOSH(number) is documented in Excel, and ACOSH(value) in Google Sheets; their stated real domains match Numbers (≥1). Similar mathematical expected results do not establish identical error strings, locale settings or all historical product versions.

• Numbers: ACOSH (Apple Formulas and Functions Help 15.4). One argument called num.
• Excel: ACOSH (Microsoft’s function reference). Numeric argument is called number.
• Google Sheets: ACOSH (Google Docs Editors Help). Numeric argument is called value.

As a portability check, a numeric constant and one argument will avoid table-reference differences. Numbers uses InputData::B2 to disambiguate cross-table references; Excel or Sheets worksheet/cell references use different conventions. No cross-product application executions have been performed for this article. Excel, Sheets and Numbers have their own locale and calculation settings; compare reasonable numeric tolerances, not formatting alone.

ACOS; COSH. These references explain the related mathematical operations in Apple’s documentation.

For syntax details, see Apple’s functions overview, cell-reference formats and argument value types. All linked Apple function pages above were accessible in the English 15.4 documentation during the documentation check. No native Numbers screenshots are included; the guide does not use mock application images.

Can ACOSH return a negative real number? No. Its real-valued principal result is nonnegative because the input is at least 1.

Is ACOSH the same as ACOS? No. ACOS is inverse ordinary cosine and accepts inputs from −1 to 1; ACOSH is inverse hyperbolic cosine and needs input ≥ 1.

Should I convert its output with DEGREES? Not to interpret the result as a circular geometric angle. An inverse-hyperbolic result is a real hyperbolic parameter; distinguish the model’s units first.

Official documentation: ACOSH; Apple Functions Overview; Apple Cell References; Apple Argument Types.

• 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.
• TANH — continue with a related Apple Numbers calculation.

For a software-specific comparison: LibreOffice Calc ACOSH, Apache OpenOffice Calc ACOSH, Gnumeric ACOSH. 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.