COTH Function (Gnumeric)

COTH returns the hyperbolic cotangent of a number. Enter =COTH(2) for the independent mathematical expectation of approximately 1.0373147207. The operation is COTH(x)=COSH(x)/SINH(x)=1/TANH(x) wherever the denominator is nonzero. It is not the same as COT, which is the ordinary trigonometric cotangent. The official Gnumeric manual documents one required number argument. Gnumeric COTH.

=COTH(x)

The documented syntax contains no optional arguments or defaults. Mathematically COTH(0) is undefined because SINH(0)=0, whereas COTH(-x)=-COTH(x) for nonzero real x. The official manual does not enumerate a precise #...! token, text coercion or numeric overflow behavior; their exact behaviour has not been verified in an installed Gnumeric application. Gnumeric COTH.

Enter the following numbers in A2:A7. Type =COTH(A2) in B2 and fill down, or enter the shown formulas individually. Display at least nine decimals to distinguish values close to one.

The expected values were computed independently using both 1/tanh(x) and cosh(x)/sinh(x) in Python, with agreement within floating-point tolerance. They are not observed Gnumeric outputs.

What the numbers show: COTH is an odd function. Positive outputs exceed 1, and negative outputs are less than −1; near zero its magnitude becomes large. Unlike TAN, hyperbolic tangent is bounded between −1 and 1 for finite real inputs, so reciprocal values have magnitude above 1.

For a mathematical model, the Langevin function is L(x)=COTH(x)-1/x, for x≠0. This is a useful example of combining Gnumeric functions without pretending a physical simulation has been validated.

Expected model results: B10 ≈ 0.3130352855, B11 ≈ 0.5373147207. To verify the identity independently, compute =COSH(A10)/SINH(A10)-1/A10 and compare within numerical tolerance. Both expressions are illustrative formulas, not observed Gnumeric outputs.

An illustrative guarded formula is:

=IF(A10=0,"Undefined at zero",COTH(A10)-1/A10)

This is a illustrative formula, not an executed safeguard; handle nonnumeric inputs and locale separators explicitly for real spreadsheets. Do not rely on subtractive cancellation very close to zero: the two large terms approach one another, and floating-point precision may be lost.

• Input zero: the underlying mathematical function has a singularity. Do not assume a specific Gnumeric error text without a native test.
• Tiny nonzero magnitude: COTH(x) behaves approximately like 1/x, so large magnitudes and precision loss are expected. Avoid exact equality assertions.
• Large positive or negative input: the mathematical value approaches +1 or −1, respectively; display rounding may hide the difference.
• Units: x is a numeric argument to a hyperbolic function. It is not a degree angle, and RADIANS(x) is not intrinsically required.
• Imported text/blanks: the manual labels x as number but does not fully describe coercion; validate cell content and record native behavior before claiming a specific error.

Gnumeric’s public catalogue records COTH: Complete / No Testsuite. This is a presence/implementation label, not a statement that the worked formulas ran here. Catalogue. Microsoft’s current Excel help documents COTH(number) and a separate Excel-specific numeric magnitude limit. Do not transfer Excel’s limit or error behavior automatically to Gnumeric. Microsoft COTH.

ACOTH, COSH, SINH, TANH, COT.

The official Gnumeric reference establishes the function’s purpose and argument structure. The arguments and worked examples were editorially reviewed and the stated expected results were independently calculated. The tagged Gnumeric 1.12.62 numerical implementation was additionally checked for the hyperbolic-cotangent and inverse-hyperbolic-cotangent identities.

Review date: 2026-10-12. No native application run: the worked outputs are documentation-based or independently calculated expectations, not measurements from installed Gnumeric. Ordinary numeric cases and the stated text-construction examples are checked; blank/text coercion, machine extremes and workbook round trips are not certified. The formula editor in your installed version is the authority for localized separators.

Example-check record · Editorial standards.