SECH Function (Gnumeric)

=SECH(0) has an independently calculated expected result of 1; =SECH(1) is approximately 0.64805427. Unlike SEC, SECH has no periodic singularities for finite real inputs.

SECH computes the hyperbolic secant, the reciprocal of the hyperbolic cosine. It can model a smooth symmetric response peaked at zero and decreasing toward zero for inputs of increasing magnitude.

SECH(x)=1/COSH(x)=2/(EXP(x)+EXP(-x)) where the operations are finite and representable. For real x, 0 < SECH(x) ≤ 1, with a peak of 1 at zero. SECH(-x)=SECH(x) (even symmetry).

=SECH(x)

SECH(x) — one required numeric argument x; no optional arguments or documented defaults. The argument is a real number (dimensionless in a mathematical model), not a degree measure.

To read a source cell rather than a literal, use =SECH(A2). Cell formulas begin with = in Gnumeric. A function with one argument has no multi-argument separator, so these examples avoid ambiguity between commas and semicolons. The tables use the English decimal-point convention, such as 0.5; Gnumeric’s local numeric format depends on locale and operating system. Check locale and spreadsheet parsing if your system expects a decimal comma. The identity statements below are mathematical deductions rather than quoted implementation promises. Formula-entry reference; Gnumeric locale reference.

The following table contains independently calculated mathematical expectations, not observed Gnumeric results. Expected numbers are rounded to eight decimal places for display; formulas may return additional digits. All examples use Gnumeric’s documented function syntax.

For a simple start, enter the first formula in any empty cell and verify that the result is plausible. Then copy a data-driven example into a column, replace constants with cell references and compare against an independent identity. SECH has no finite real singularity; a large output instead calls for checking whether the formula or function name was changed.

A symmetric weighting curve for an illustrative model

Suppose a worksheet scores a dimensionless offset from a target. Put offsets -2, -1, 0, 1, 2 in A2:A6, then enter:

=100*SECH(A2)

Copy down. Independently calculated expected scores (rounded to six decimals) are 26.580223, 64.805427, 100.000000, 64.805427, 26.580223. This is a demonstration of a smooth weight, not a statistically calibrated probability or physical law. The input unit and scaling need to be defined by your application.

Check the hyperbolic identity

For an input in A2, compare:

=SECH(A2)
=1/COSH(A2)

At A2=2, both are expected to be approximately 0.26580223. Differences beyond expected floating-point rounding warrant checking formula references and the installed function support.

Inverse-profile interpretation

If you are working with a catenary of the form y=a*COSH(x/a) with a>0, then the dimensionless ratio a/y equals SECH(x/a) mathematically. For an illustrative value a=2, x=2, compare =SECH(2/2) with =2/(2*COSH(2/2)); the expected ratio is 0.64805427. Confirm units and model assumptions before applying to real measurements.

• Confusing it with SEC: SECH is the reciprocal of hyperbolic COSH, not the reciprocal of ordinary COS. It is not periodic and has no finite real singularities.
• Misreading degrees: Gnumeric labels its argument simply as a number. Do not use RADIANS unless your own model first requires converting an angular input to a dimensionless parameter; there is no ordinary trigonometric degree conversion here.
• Huge magnitudes: For large |x|, the mathematical result is positive but may round or underflow to zero in finite precision; direct equivalents using EXP or COSH may overflow earlier. Gnumeric’s exact overflow and underflow presentation was not natively checked.
• Text/error inputs: Confirm the source cell is numeric. The official short reference does not document the precise error token for each invalid input.
• Identity check: In a very large-magnitude regime, =1/COSH(x) and =SECH(x) can have different finite-precision behaviour even when algebraically equivalent.

How to diagnose a surprising value: inspect the underlying numeric cell, check units (radians versus dimensionless hyperbolic inputs), test a simple known case, compare with the identity only in a numerically safe range, and confirm the workbook’s locale. Record the application version and the exact error shown before adding specific product-error claims. The official short documentation does not fully specify behaviour for blanks, text values, machine overflow, or all singularities.

Gnumeric’s official 1.12 Mathematics manual labels SECH not Excel compatible, and describes OpenFormula export as 1/COSH(x). However, Microsoft’s up-to-date support documentation separately lists SECH in modern Excel (including Excel 2016, 2019, 2021, 2024 and Microsoft 365). This historical Gnumeric classification must not be interpreted as “Excel has no SECH.” Microsoft lists the Excel-specific restriction |number| < 2^27; Gnumeric’s short synopsis does not establish that same bound. Use 1/COSH(x) as an algebraic alternative when the destination application lacks SECH, subject to numerical-range limitations.

Compatibility may also depend on worksheet file format and the exact importer/exporter. Merely documenting a function in two programs does not prove bit-for-bit identical precision, error codes or conversion behaviour. See Gnumeric’s manual and Microsoft’s current SECH documentation, accessed 2026-10-12.

Is SECH equal to SEC? No. It is 1/COSH(x) rather than 1/COS(x).

Can SECH be negative? Not for finite real inputs; it is strictly positive mathematically.

Why do values look like zero? SECH decays rapidly as input magnitude increases. Cell formatting and finite-precision underflow can obscure a small positive mathematical value.

Should I convert degrees to radians? Not as a routine step: unlike ordinary trigonometric functions, SECH takes a numeric hyperbolic argument.

COSH, SINH, TANH, SEC, CSCH.

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.

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.