COSH Function in Apple Numbers: Formula, Examples, and Catenary Curves

=COSH(0) has an expected result of 1; =COSH(1) has an expected result of about 1.543081. COSH returns the hyperbolic cosine of a numeric input. Unlike ordinary COS, its real-valued output is always at least 1 mathematically.

COSH is useful when a model uses hyperbolic cosine, including the idealized hanging-chain or catenary curve. This is a mathematical modelling function, not a more precise substitute for COS or a display-formatting command. The figures in this guide are expected results independently calculated from mathematics and cross-checked with Apple’s examples. They are not observations from Numbers running on a device.

Apple’s English Formulas and Functions Help 15.4, checked 2026-10-11 UTC, documents the following exact expression:

=COSH(num)

• num — required: a number value, as Apple labels the argument. It is the input to the hyperbolic cosine function. No optional second argument, fallback, or default is documented.
• Argument order: the sole argument goes inside the parentheses; for a cell use =COSH(B2). Enter an initial equals sign in a Numbers formula cell. Formula names are not case-sensitive according to Apple’s overview.
• Regional notation: these examples assume an English-style decimal point (0.5, -0.5). In decimal-comma regions enter numbers using the locale-compatible numeric syntax. Apple’s semicolon rule applies when separating multiple arguments, not to this one-argument function. In another function such as IF, commas in English may become semicolons when the locale uses decimal commas.
• Platform/version qualification: 15.4 is the documentation edition, not a verified first-supported Numbers release. This single-function reference does not establish parity across every Mac, iPhone, iPad, iCloud, or earlier Numbers edition; none of those applications was run during preparation.

Official claims: Apple COSH — syntax, numeric parameter, and examples; Apple functions overview — formula entry and locale separators.

1. Create a blank Numbers document and insert a regular table named InputData (or rename a blank table to InputData). Keep row 1 as headers.
2. Enter Label in A1, Input in B1, and Result in C1. Type the numeric B-column values below; do not copy the descriptions into B.
3. Format column B as Number and column C as Number with at least six decimal places. Cell formatting changes the display, not the underlying stored precision.
4. Type =COSH(B2) in C2. Fill the formula down to C10. Check that copied formulas use B3, B4, and so on, rather than B2 repeatedly.
5. Compare the expected six-decimal figures with your cells. The table below was calculated independently using Python math.cosh and is not a record of Numbers execution. If a displayed value disagrees, confirm whether it is a real numeric input and that the correct function was selected.

Row: 2

A: Label
negative five

B: Numeric input
-5

C: Formula
=COSH(B2)

C: Expected six-decimal output
74.209949

Row: 3

A: Label
negative two

B: Numeric input
-2

C: Formula
=COSH(B3)

C: Expected six-decimal output
3.762196

Row: 4

A: Label
negative one

B: Numeric input
-1

C: Formula
=COSH(B4)

C: Expected six-decimal output
1.543081

Row: 5

A: Label
negative half

B: Numeric input
-0.5

C: Formula
=COSH(B5)

C: Expected six-decimal output
1.127626

Row: 6

A: Label
zero

B: Numeric input
0

C: Formula
=COSH(B6)

C: Expected six-decimal output
1.000000

Row: 7

A: Label
positive half

B: Numeric input
0.5

C: Formula
=COSH(B7)

C: Expected six-decimal output
1.127626

Row: 8

A: Label
one

B: Numeric input
1

C: Formula
=COSH(B8)

C: Expected six-decimal output
1.543081

Row: 9

A: Label
two

B: Numeric input
2

C: Formula
=COSH(B9)

C: Expected six-decimal output
3.762196

Row: 10

A: Label
five

B: Numeric input
5

C: Formula
=COSH(B10)

C: Expected six-decimal output
74.209949

Refer to another table or sheet

If C2 belongs to a different Numbers table from the inputs, an explicit table reference can help: =COSH(InputData::B2). Apple’s documentation also describes sheet-qualified references in the form Sheet Name::Table Name::B2 when needed. Use the formula editor and click the intended cell instead of manually guessing tokenization for names with spaces. A same-table B2 reference generally needs no table qualifier. Apple’s cell-reference guide provides the syntax and copy/paste rules.

Even symmetry: negative inputs do not make negative outputs

Hyperbolic cosine is even: cosh(-x)=cosh(x). Thus =COSH(-2) and =COSH(2) each have an independently computed expected value of approximately 3.762196. A comparison like =COSH(B2)-COSH(-B2) should be mathematically 0 at ordinary inputs; a tiny floating-point discrepancy would not imply a different function.

A catenary (hanging-chain) modelling example

For an idealized catenary, a common mathematical model is y = a cosh(x/a) with a > 0, chosen by the model rather than discovered by COSH. Set horizontal x in B2 and use =2*COSH(B2/2) when a=2. This gives an expected 2.000000 at x=0, and about 3.086161 at x=2. The curve is symmetric around the origin. This is a mathematical example, not an engineering-design method for real loads, cables, or installations.

Exponential identity and numerical caveat

The mathematical identity is cosh(x) = (e^x + e^(−x))/2. For an ordinary finite worksheet input, =(EXP(B2)+EXP(-B2))/2 can illustrate the result. In production prefer =COSH(B2): separately evaluated exponentials can overflow for large magnitude inputs. We have not benchmarked the relative accuracy in Apple Numbers.

Check a hyperbolic identity

At x=1, =COSH(1)^2-SINH(1)^2 is mathematically 1. Floating-point arithmetic may make the observed cell differ slightly in its least significant digits. The built-in ACOSH is a real inverse of COSH on the nonnegative branch: ACOSH(COSH(-2)) is mathematically 2, not −2, because COSH discards the sign of the original argument.

Decide whether the input is a number before diagnosing a math error

The Apple function reference specifies a numeric argument. Imported CSV values may look numeric while being stored as text; a custom-format display can also obscure precision. Test with a freshly typed number such as 1 first, then inspect the imported input. This guide does not claim any undocumented implicit conversion of text, empty cells, logical values, or dates into a number.

Numerical and version limitations: Mathematically, cosh(x) ≥ 1 for every real x, but the finite machine representation and user-visible error behavior at extremely large magnitudes are not established in Apple Numbers. A complex-number extension is not claimed by this guide. Specific Apple Numbers error text, implicit coercion, exact large-value limits, behavior of non-finite inputs, and platform-specific differences have not been tested. Small last-digit differences are possible with floating-point calculation and display rounding; do not require exact string equality for approximate examples.

Excel and Google Sheets also document a function called COSH. In ordinary real-number mathematics the function definition is the same, so the example figures are suitable comparison expectations. This does not establish identical application behavior for imported text, rounding, errors, extreme floating-point values, cross-table syntax, or release support. Check these when transferring a spreadsheet.

• Apple Numbers: uses tables as formula-reference units, so another table can be referenced as InputData::B2.
• Excel / Google Sheets: frequently use sheet-centric references such as Sheet1!B2; those are not a Numbers table-qualified reference.
• Google Sheets: consult Google’s official function list for its function signature. For Excel, Microsoft’s official COSH reference documents a one-number argument, but confirm the intended Excel edition separately.
• Regional settings: test the file’s actual decimal and argument separators in the target spreadsheet. The one-argument COSH formula itself has no separator to translate.

• Apple: COSH function — official syntax, argument type and Apple examples. Formulas and Functions Help 15.4, consulted 2026-10-11 UTC.
• Apple: Functions overview — case conventions, formulas, and locale argument separators; consulted 2026-10-11 UTC.
• Apple: Refer to cells in formulas — same-table and cross-table reference forms; consulted 2026-10-11 UTC.
• Apple: Functions Help welcome — general function browser/platform context, not a first-release guarantee; consulted 2026-10-11 UTC.
• Google Sheets function list — cross-application signature existence; consulted 2026-10-11 UTC.
• Microsoft Excel COSH documentation — documented Excel function name and signature, consulted 2026-10-11 UTC. Exact release/locale parity was not tested.

• ACOSH — continue with a related Apple Numbers calculation.
• ASINH — continue with a related Apple Numbers calculation.
• ATANH — 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 COSH, Apache OpenOffice Calc COSH, Gnumeric COSH. 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.