SECH Function in Apple Numbers: Hyperbolic Secant and Examples

=SECH(0) has an expected result of 1. SECH is a hyperbolic function; its numeric argument is not a degree-based circular angle.

Purpose: SECH calculates the reciprocal of hyperbolic cosine. The examples in this guide are documentation-backed or independent mathematical calculations. No formula has been executed in Apple Numbers; all numeric output tables are expected, not observed.

Apple’s English Formulas and Functions Help 15.4 specifies:

=SECH(num)

• num (required): A numeric value, supplied directly or through a numeric cell reference.
• Mathematical limitation: Apple lists a numeric argument without additional mathematical domain exclusion for real inputs.
• Argument count/order: one required argument, no optional arguments or documented defaults, no multi-argument order ambiguity.
• Result: a calculated real number for an in-domain real argument, subject to rounding and the app’s numeric precision.
• Aliases: the dated official English catalogue identifies SECH as its own canonical function, with no verified additional aliases. Regional display names and historical edition differences are not assumed to be distinct functions.
• Version/platform scope: the cited Apple documentation is help edition 15.4, consulted 2026-10-12 UTC. This edition number is not proof of the first Numbers release supporting the function, nor proof of identical runtime behavior across Mac, iPhone, iPad or the web. No native platform was tested.

Official evidence: Apple SECH function reference.

What to type in different regions

English formula examples on this page use the period for decimals, for example 0.5. Apple’s Functions overview says if your document/settings use a comma as the decimal separator, type semicolons as separators in multi-argument formulas entered by hand. SECH has only one argument, so it contains no argument separator. Do not insert a comma or semicolon into its one-argument form. Names of functions are case-insensitive in Apple’s documented formula entry convention.

1. Create a Numbers table named HyperbolicTests with a header row: column A = Case, column B = numeric input, and column C = SECH result.
2. Enter the numeric column-B values shown below, leaving them as numbers (not quoted text). The Case labels are optional for calculation but help with auditing.
3. In HyperbolicTests, put the numeric observations (not text strings) in column B. In C2 enter =SECH(B2) and fill down. These outputs are independently calculated expectations, not native Numbers observations.
4. Format the C results with at least six decimal places. Compare values to the table using a small tolerance; formatting and floating-point differences can change the least significant digits.
5. Neither CSCH nor SECH requires RADIANS or DEGREES. If your underlying scientific problem has units, transform the variables in the scientific model before treating them as dimensionless hyperbolic inputs.

Row: 2

B: numeric input
-5

C: Numbers formula
=SECH(B2)

C: expected result (six decimals)
0.013475

Row: 3

B: numeric input
-2

C: Numbers formula
=SECH(B3)

C: expected result (six decimals)
0.265802

Row: 4

B: numeric input
-1

C: Numbers formula
=SECH(B4)

C: expected result (six decimals)
0.648054

Row: 5

B: numeric input
0

C: Numbers formula
=SECH(B5)

C: expected result (six decimals)
1.000000

Row: 6

B: numeric input
1

C: Numbers formula
=SECH(B6)

C: expected result (six decimals)
0.648054

Row: 7

B: numeric input
2

C: Numbers formula
=SECH(B7)

C: expected result (six decimals)
0.265802

Row: 8

B: numeric input
5

C: Numbers formula
=SECH(B8)

C: expected result (six decimals)
0.013475

Dataset verification: each value in the last column was independently computed using Python 3 standard-library math (sin/cos with degree conversion for CSC/SEC; sinh/cosh for CSCH/SECH). No Numbers execution was observed. Entering these into the app yourself is still a required native validation step.

Apple’s own published formula examples

• =SECH(0) — expected 1 (from Apple’s documented example; rounded as shown in the help).
• =SECH(1) — expected approximately 0.648054273663885 (from Apple’s documented example; rounded as shown in the help).
• =SECH(5) — expected approximately 0.0134752822213046 (from Apple’s documented example; rounded as shown in the help).

These published examples are taken from the SECH reference. Our independent calculations of the mathematical values agree within 1×10⁻¹¹ (except trivial exact integer comparisons). Apple documentation corroborates the claimed values; a local Numbers screenshot or screen output was not obtained.

Within the named table, B2 is sufficient for an ordinary relative reference. To reference the same input from a different table, use a qualified pattern such as AngleTests::B2 or HyperbolicTests::B2, with the exact table name of your document. Other-sheet references also require the sheet name in the editor’s qualified reference. Apple’s official cell-reference documentation explains the token and cross-sheet reference behavior. When table or sheet names contain spaces, selecting the actual source cell within the formula editor is more reliable than guessing a typed path. Copying C2 down uses relative B-column references unless you deliberately anchor components; do not assume B2 remains fixed after fill-down.

Why zero is a valid input

SECH(x) = 1 / COSH(x). Since COSH(0)=1, =SECH(0) is exactly 1. This differs from CSCH(0), which is undefined. When importing a model with two similar-looking functions, this distinction helps identify a mistaken function name.

Symmetry and bounded output

Hyperbolic cosine is even; therefore SECH(−x) = SECH(x). For real x, COSH(x) is at least 1, so the real-valued hyperbolic secant lies in (0, 1]. For example, =SECH(-2) and =SECH(2) are both expected to be approximately 0.265802. This property makes SECH useful for mathematical envelopes and smooth symmetric weighting curves, with inputs and units selected by the reader’s model.

Comparing distant and central observations

In the example table, the center at x=0 peaks at 1 and the weights for ±5 are about 0.013475. Large abs(x) mathematically drives the value toward zero without ever reaching it for a finite real x. The exact precision, underflow behavior, or overflow policy in Apple Numbers for extreme inputs has not been checked; test the production range in the native app before relying on boundary values.

A reproducible second-column comparison

A mathematical comparison uses the reciprocal of COSH. Its predicted value at any table row matches the corresponding SECH table value in ideal real-number arithmetic wherever the function is defined. Important: =1/COSH(... here is shorthand for =1/COSH(Ellipsis); type the actual formula shown below:

=1/COSH(B2)

Use column D for the reciprocal calculation and compare columns C and D. This catches misidentified units, a mistyped function name, and out-of-domain values before a calculation is reused in a workbook. Separate evaluations may differ by floating-point rounding, and a near-zero denominator is inherently unstable.

Practical worked scenario

A centered input signal can be mapped through SECH to a smooth bell-shaped nonnegative weight, equal to 1 at zero and decreasing symmetrically. It is a mathematical model component—not a probability distribution by itself and not a substitute for validating model assumptions.

A deliberate input-quality check

For CSCH, =ABS(B2)<0.000001 is an example screening rule for values near the invalid zero input; use a separate quality column to flag them. For SECH, =ABS(B2)>5 flags readings in the diminishing tails and is not a built-in domain restriction. These are user-chosen quality controls, not Apple’s error or precision thresholds.

Documented mathematical restriction: All finite real inputs are mathematically valid; implementation limits for extremely large values are not established by the cited Numbers help. Zero is valid for SECH: its documented value is 1.

Do not claim native error strings. Apple’s cited SECH page documents the input type and definition, but it does not establish every cell-specific error message, coercion rule for booleans and dates, overflow/underflow threshold, or double-precision behavior. An implementation may return errors or finite-looking artifacts near a pole. For published advice, test in your actual edition before certifying edge behavior.

• Apple Numbers: SECH is named and documented in Apple’s 15.4 reference linked above. Formulas can use Numbers-style table-qualified and sheet-qualified cell references.
• Microsoft Excel: Microsoft’s published SECH function documentation identifies an Excel function with the same name and single required numeric argument. Microsoft’s help specifies its own magnitude restriction |number| < 2²⁷. That limit and its error strings are Excel-specific; the Apple reference cited here does not establish the same numeric threshold.
• Google Sheets: compatibility was not independently confirmed for this batch, so do not assume availability in every Sheets edition, identical locale behavior, or precision/error parity. Consult the official Sheets function directory when migrating a real workbook.
• Migration warning: replace Numbers table-qualified references with the target spreadsheet’s syntax, explicitly check angle units and source types, and recalculate ordinary/negative/boundary examples in the destination app. Formula-name resemblance is not a native compatibility test.

Is SECH(0) invalid like CSCH(0)? No. SECH(0)=1 because COSH(0)=1. CSCH is the distinct function that excludes zero.

Do I enter degrees into SECH? No. Its numeric input is a hyperbolic-function argument, not a circular angle measured in degrees.

Is the result in degrees? No. Hyperbolic inputs and outputs here are numeric; the result is not a measurement in degrees.

How should I handle tiny differences from the six-decimal values? Format numeric results with the intended precision and compare within a tolerance appropriate to the model. Do not equate display rounding with exact mathematical equality.

• Apple function evidence: official SECH description, exact num argument, mathematical limits, identity and examples. Formulas and Functions Help 15.4; accessed 2026-10-12 UTC.
• Apple formula conventions: Functions overview, Help 15.4; accessed 2026-10-12 UTC. Supports case-insensitive function names, required arguments, and semicolon delimiters under decimal-comma settings.
• Apple references: Refer to cells in formulas, Help 15.4; accessed 2026-10-12 UTC. Supports same-table and cross-table reference guidance.
• Microsoft-specific comparison: Excel SECH documentation, accessed 2026-10-12 UTC; only the named function and Excel’s documented constraints, not runtime equivalence.
• Official catalogue baseline: Apple English documentation 15.4, accessed 2026-10-12 UTC, together with the saved research/apple-numbers/catalogue-2026-10-11/official-catalogue.json. Prior inventory is 355 named entries, with alias/edition completeness not yet independently certified.

• ACOT — continue with a related Apple Numbers calculation.
• COT — continue with a related Apple Numbers calculation.
• CSC — continue with a related Apple Numbers calculation.
• CSCH — continue with a related Apple Numbers calculation.
• SEC — continue with a related Apple Numbers calculation.

For a software-specific comparison: Gnumeric SECH. 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.