COTH in Apple Numbers: Hyperbolic Cotangent, Signs, and Limits

=COTH(1) has an expected result of approximately 1.313035. COTH calculates hyperbolic cotangent, not ordinary cotangent; its numeric input may be positive or negative but must not be zero.

Purpose: 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:

=COTH(num)

• num (required): nonzero real numeric input. Apple documents COTH(num), with one numeric argument that cannot equal zero. Its mathematical definition is COSH(num)/SINH(num). Apple examples: COTH(1)≈1.31303528549933; COTH(2)≈1.03731472072755; COTH(ACOTH(2))=2. No optional arguments/defaults are shown.
• Arguments and defaults: there is one required numeric input; there are no optional positions and no documented defaults. =COTH(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 COTH is established here. No additional aliases or historical equivalences were verified; translated labels in other locales are not automatically alternative identifiers.

Official documentation: Apple COTH 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 COTH, start cell C2 with =COTH(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 3

B: numeric input
-3

C: formula
=COTH(B2)

C: expected numeric result
-1.004970

Row: 3

A: description
negative 2

B: numeric input
-2

C: formula
=COTH(B3)

C: expected numeric result
-1.037315

Row: 4

A: description
negative 1

B: numeric input
-1

C: formula
=COTH(B4)

C: expected numeric result
-1.313035

Row: 5

A: description
negative 0.5

B: numeric input
-0.5

C: formula
=COTH(B5)

C: expected numeric result
-2.163953

Row: 6

A: description
positive 0.5

B: numeric input
0.5

C: formula
=COTH(B6)

C: expected numeric result
2.163953

Row: 7

A: description
positive 1

B: numeric input
1

C: formula
=COTH(B7)

C: expected numeric result
1.313035

Row: 8

A: description
positive 2

B: numeric input
2

C: formula
=COTH(B8)

C: expected numeric result
1.037315

Row: 9

A: description
positive 3

B: numeric input
3

C: formula
=COTH(B9)

C: expected numeric result
1.004970

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, =COTH(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.

Definition through hyperbolic sine and cosine

The mathematical identity coth(x) = cosh(x)/sinh(x) applies when x is nonzero. For a moderate input such as 2, enter =COSH(B2)/SINH(B2) beside =COTH(B2) and compare them. Both are expected to be approximately 1.037315. This identity checks the concept but doesn’t establish identical floating-point implementations; the direct built-in is clearer and avoids an unnecessary denominator.

Why zero is not allowed, and why small values are sensitive

SINH(0)=0, so the ratio defining COTH has a zero denominator at x=0. For a small positive input, the mathematical value behaves roughly like 1/x, and for a small negative input it is a large negative number. For example =COTH(0.1) is expected to be approximately 10.033311. Even modest measurement rounding near zero can materially change a computed result. Before using COTH on sensor readings, decide how the data pipeline will flag exact and near-zero observations.

The sign test and the large-input plateau

COTH is mathematically odd: coth(-x) = -coth(x). The predicted results for −2 and 2 are about −1.037315 and +1.037315, respectively. For large positive x, coth approaches +1 from above, while for large negative x it approaches −1 from below. This is a mathematical limit, not a claim that Apple Numbers returns exactly 1 for any particular finite input.

Inverting only within the allowed range

=COTH(ACOTH(2)) has the expected result 2 and Apple’s help includes this example. With B2=3, =ACOTH(COTH(B2)) is mathematically expected to return approximately 3. However, =ACOTH(0.5) is invalid: the real inverse is only defined for |input|>1. This is why COTH and ACOTH are related but cannot be interchanged with a simple substitution for arbitrary values.

Data typing and error messages

Apple describes a numeric input for COTH. 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: COTH() is documented in the linked 15.4 help. Numbers references can include both sheet and table names, separated with double colons.
• Microsoft Excel: COTH documentation supports the same named real-valued mathematical function. Excel documents COTH as well as an Excel-specific input magnitude restriction |number| < 2^27. Do not transfer that exact restriction or Excel error wording to Numbers without a native test.
• Google Sheets: the official Sheets function catalogue lists COTH 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.

The output for real nonzero finite x is mathematically greater than +1 for x>0 or less than −1 for x<0. It is not an angle.

• Apple COTH 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 COTH 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 COTH, Gnumeric COTH. 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.