SECH in Calligra Sheets: Syntax, Examples and Numerical Limits

Use SECH to find the hyperbolic secant, the reciprocal of hyperbolic cosine. For example, =SECH(0) is documented by KDE as 1, while =SECH(1) has an independently calculated expected result of approximately 0.6480542737. The pinned implementation divides 1 by COSH internally; this is source evidence, not a measured spreadsheet output. [E1, E2, E3]

The defining operation is 1/cosh(x). The hyperbolic sine/cosine function is not the same as ordinary circular sine/cosine. The numerical expectations below were independently calculated with Python’s mathematical standard library; no formula has been run in a native Calligra Sheets installation. The function’s presence in the KDE handbook and 26.08.2 source proves documentation/registration, not installed-package availability. [E1, E2]

=SECH(number)

KDE’s shipped function-description XML gives the literal syntax SECH(Float), with one <Parameter> of type Float and a floating-point return type. The sole required argument is a real numeric value or cell reference containing a number. No second argument, mode switch or default is declared. The source wrapper reads args[0] and forwards it into the corresponding hyperbolic calculation (COSH). The definition is 1/cosh(x). [E1, E2]

Mathematical domain: For every finite real x, cosh(x)≥1, hence the true mathematical result satisfies 0<SECH(x)≤1. This does not establish numeric overflow/underflow behavior for arbitrary large computer values. [E2, E3]

Radian and locale notation: KDE’s XML calls the argument an “Angle (radians).” The trigonometric-family nomenclature should not be mistaken for ordinary periodic sine and cosine: SINH/COSH define hyperbolic functions. For any workflow that explicitly starts from degrees, convert the intended angle with RADIANS first; in many exponential/hyperbolic models the input is simply a dimensionless number x. The one-argument formulas do not require an argument separator; multi-argument examples here use semicolons as in KDE’s English handbook (=ROUND(SECH(1);6)). A localized installation may require different decimal/list separators; check the formula editor in that specific installation. [E1, E5]

Version and platform: This guide is scoped to Calligra 26.08.2 (KDE Gear release 2026-10-08), source commit 72123cc643f6625351139e7df294544255a1056f. Official docs and source checked 2026-10-13 UTC. KDE distributes Calligra across packages that may differ by operating system or release; no Linux Flatpak, distribution-package, or Windows execution was performed. The catalogued function is not independently runtime-certified by this manuscript. [E1, E2, E4]

1. In a blank worksheet, enter Input x in A1 and SECH(x) in B1. Optionally enter Reciprocal reference in C1.
2. Enter the nine numeric values shown in column A, one per cell A2:A10. Enter them as numbers without quotes and use the locale’s decimal syntax.
3. Enter =SECH(A2) in B2 and fill/copy down to B10. In C2 enter =1/COSH(A2) and fill down if you want a second spreadsheet expression to compare. It relies on the same underlying family of calculations; this is not an independent native cross-platform test.
4. Format B (and C, if used) to roughly ten decimal places. Compare with the expected results below, which are independent mathematical calculations. A blank cell, text, error or unusual locale may behave differently; those cases were not native-tested.

Calculation method: for SECH, Python 3 calculates 1/math.cosh(x) using the exact x-values above, before formatting the decimal output to ten places. For example, the pinned KDE XML example says SECH(0) equals 1; independently cosh(0)=1, so 1/cosh(0)=1. For x=±2, the expected result is approximately 0.2658022288. [E1, E2, E3]

A calculated value is not a statement that a native Calligra sheet was executed. Slightly different trailing digits can occur from floating-point rounding. The source proof and calculated expectations are kept separate in this guide. [E2, E3]

• Even symmetry: mathematically, SECH(-x)=SECH(x) for every finite real x. Its maximum is 1 at x=0. [E3]
• Do not substitute an ordinary function: SECH is not the same as SEC. A hyperbolic function is based on exponential identities, unlike the ordinary reciprocal cosine function. [E2, E3]
• KDE handbook cross-check: The pinned KDE XML example says SECH(0) equals 1; independently cosh(0)=1, so 1/cosh(0)=1. For x=±2, the expected result is approximately 0.2658022288. [E1, E3]
• Important edge: =SECH(0) is the well-defined value 1, explicitly documented by KDE. For very large |x|, the mathematical result remains positive but may be rounded to zero or produce an error in finite precision; native Calligra details are unverified. [E2, E3]

A researcher can use SECH as a mathematical weighting function centered at zero, without claiming it is a KDE-specific worksheet template. Place Baseline in D1 and Input x in E1. Put 10 in D2 and 1 in E2. In F2 enter:

=$D$2*SECH(E2)

The independently calculated expected value is 6.480542737. At x=0, the weight is 1, so the scaled output is 10. At x=-1, it is also about 6.480542737 because SECH is even. For a reusable sensitivity profile, fill E2:E10 with -4 through 4 and copy F2 down. This is a derived calculation use case; it has not been executed inside Calligra Sheets. [E2, E3]

At H2=1, put =SECH(H2) into I2 and =TANH(H2) into J2. The mathematical identity sech²(x)+tanh²(x)=1 means =ROUND(I2^2+J2^2;9) in K2 has the independently calculated expected result 1.000000000. Copy these three formulas down after entering x-values 0, 0.5, 1 and 2 in H2:H5.

This is a mathematical cross-check, not proof of native Calligra correctness: each target function still requires a native runtime evaluation. For a simpler source-level cross-check at x=1 use =SECH(1) and =1/COSH(1); both are independently expected to be 0.6480542737. Avoid very large inputs in the reciprocal formula because the intermediate hyperbolic cosine can exceed representable finite precision. [E2, E3]

The following table distinguishes mathematical limitations, source-inferred details, and untested native behavior. The exact displayed errors in a real 26.08.2 build are not confirmed by source registration alone. [E2, E3]

Numerical precautions

1. Prefer moderate finite inputs for cross-checking. The reciprocal calculation calls a hyperbolic function first; the result may be very small or involve a very large intermediate for extreme inputs. The exact precision, overflow and display behavior varies by platform and was not tested in Calligra. [E2, E3]
2. Check any source data for numbers stored as text, unexpected locale decimal punctuation and formulas that already contain errors. The XML’s Float type does not, on its own, establish all coercion rules. [E1, E5]
3. When comparing two systems, check meaningful sample values and edge cases. A mathematically equal positive number does not prove equivalent error handling, rounding or formatting. [E2, E6]

Microsoft’s official Excel SECH reference documents the same named mathematical operation with one required numeric argument in radians. For ordinary finite numeric inputs, the source-described Calligra expression and Microsoft’s documented Excel function have the same mathematical definition; this is not a claim that importing a workbook has been tested in either program. [E2, E6]

Excel’s documentation imposes an absolute argument bound below 2^27 and specifies Excel #NUM! outside its constraints and Excel #VALUE! for nonnumeric input. Those are Excel-specific documented rules; the pinned Calligra wrapper has no identical explicit bound at its own call site, and the exact Calligra error display, numeric range, import handling and Excel parity remain unverified. Do not copy the Excel limits into a Calligra reference as if they had been natively established. [E2, E6]

Portability check: confirm the Calligra/Excel version, formula name, data types, decimal/list separators, zero and very large inputs, and calculated values. Other spreadsheet programs need their own official, version-specific evidence; this guide does not assert their runtime behavior.

Can SECH produce a negative value? For any finite real mathematical input, no: cosh(x) is at least 1 so its reciprocal is positive. A displayed negative would require checking the formula, data type and app behavior.

What is SECH(0)? KDE documents a result of 1, consistent with the independently calculated reciprocal of cosh(0).

Is SECH(2) the same as SECH(-2)? Yes mathematically. Both are expected to be approximately 0.2658022288.

• COSH in Calligra Sheets.
• TANH in Calligra Sheets.
• RADIANS in Calligra Sheets.
• PI in Calligra Sheets.

For first-party instructions, see the KDE Calligra Sheets function handbook, pinned 26.08.2 XML, pinned C++ registration/implementation and formula editor help. Source links were checked 2026-10-13 UTC. [E1, E2, E5]

The numbered references support the claims above at the indicated scope. Access/verification date: 2026-10-13 UTC.

• [E1] KDE Sheets official handbook — SECH and release-pinned trig.xml — registered user-facing syntax SECH(Float), one Float parameter, floating return, handbook example. The HTML handbook is not a native release execution.
• [E2] Pinned trig.cpp registration and C++ wrapper — source-level registration and expression 1/cosh(x). The released wrapper calls calc->div(1,calc->cosh(args[0])); native display/error behavior was not observed.
• [E3] The displayed inputs and outputs were independently checked with Python standard-library hyperbolic functions and arithmetic; these are mathematical expectations, not Calligra executions.
• [E4] KDE Gear 26.08.2 release announcement, dated 2026-10-08, release-scope only; pinned source revision 72123cc643f6625351139e7df294544255a1056f.
• [E5] KDE Sheets formula editor — formula entry and locale considerations; not a native test of this page.
• [E6] Microsoft Excel official SECH documentation — Excel one-argument syntax, 2^27 limit and Excel-specific errors, not Calligra proof.

Validation: Official documentation and pinned source reviewed; examples independently calculated. Native Calligra execution and screenshots were not performed.