SEC Function in Apple Numbers: Secant, Radians and Undefined Angles

=SEC(RADIANS(60)) is expected to return 2. RADIANS() matters: this function expects an angle in radians, not degrees.

Purpose: SEC calculates the reciprocal of the cosine of an angle. 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:

=SEC(radian-angle)

• radian-angle (required): A numeric value representing an angle in radians.
• Mathematical limitation: the angle must not be an odd multiple of π/2 radians, because its cosine is zero.
• 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 SEC 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 SEC 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. SEC 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 AngleTests with a header row: column A = Case, column B = angle, degrees, and column C = SEC 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 AngleTests, enter degrees as ordinary numeric values in column B; C uses =SEC(RADIANS(B2)) and is filled down. The values in column C are expected results, independently calculated in Python, not observed Numbers outputs.
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. RADIANS converts degree values to radians. If your source column already contains radians, use =SEC(B2) without converting again. The headers and units in this tutorial matter more than column formatting.

Row: 2

B: angle, degrees
-180

C: Numbers formula
=SEC(RADIANS(B2))

C: expected result (six decimals)
-1.000000

Row: 3

B: angle, degrees
-150

C: Numbers formula
=SEC(RADIANS(B3))

C: expected result (six decimals)
-1.154701

Row: 4

B: angle, degrees
-120

C: Numbers formula
=SEC(RADIANS(B4))

C: expected result (six decimals)
-2.000000

Row: 5

B: angle, degrees
-60

C: Numbers formula
=SEC(RADIANS(B5))

C: expected result (six decimals)
2.000000

Row: 6

B: angle, degrees
-45

C: Numbers formula
=SEC(RADIANS(B6))

C: expected result (six decimals)
1.414214

Row: 7

B: angle, degrees
-30

C: Numbers formula
=SEC(RADIANS(B7))

C: expected result (six decimals)
1.154701

Row: 8

B: angle, degrees
0

C: Numbers formula
=SEC(RADIANS(B8))

C: expected result (six decimals)
1.000000

Row: 9

B: angle, degrees
30

C: Numbers formula
=SEC(RADIANS(B9))

C: expected result (six decimals)
1.154701

Row: 10

B: angle, degrees
45

C: Numbers formula
=SEC(RADIANS(B10))

C: expected result (six decimals)
1.414214

Row: 11

B: angle, degrees
60

C: Numbers formula
=SEC(RADIANS(B11))

C: expected result (six decimals)
2.000000

Row: 12

B: angle, degrees
120

C: Numbers formula
=SEC(RADIANS(B12))

C: expected result (six decimals)
-2.000000

Row: 13

B: angle, degrees
150

C: Numbers formula
=SEC(RADIANS(B13))

C: expected result (six decimals)
-1.154701

Row: 14

B: angle, degrees
180

C: Numbers formula
=SEC(RADIANS(B14))

C: expected result (six decimals)
-1.000000

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

• =SEC(0) — expected 1 (from Apple’s documented example; rounded as shown in the help).
• =SEC(PI()/3) — expected 2 (from Apple’s documented example; rounded as shown in the help).
• =SEC(1) — expected approximately 1.85081571768093 (from Apple’s documented example; rounded as shown in the help).

These published examples are taken from the SEC 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.

The reciprocal-cosine identity

Secant is defined as sec(θ) = 1 / cos(θ) wherever cosine is nonzero. =SEC(RADIANS(60)) and =1/COS(RADIANS(60)) are independently expected to be approximately 2. Comparing these formulas can catch accidental use of degrees as radians.

Quadrant signs, evenness, and period

Cosine is even, so sec(−θ) = sec(θ). Secant has a 360° period: sec(θ + 360°) = sec(θ). Secant is positive in the first and fourth quadrants and negative in the second and third. For instance, =SEC(RADIANS(120)) is expected to equal −2, whereas =SEC(RADIANS(60)) is expected to equal 2.

The 90° singularity is a material modeling issue

At θ = 90°, cosine is mathematically zero, making secant undefined. PI()/2 and a converted RADIANS(90) may involve floating-point approximations, so an app could show a huge number instead of a specific error. Do not infer that a numeric return is physically meaningful at a documented pole. Identify and exclude near-zero cosine inputs before using the result in a scientific calculation.

A reproducible second-column comparison

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

=1/COS(RADIANS(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

For a slope-model illustration, the secant is 1/cos(θ) where θ is the angle between a segment and the horizontal, so the ratio can scale a horizontal projection into a segment length where the cosine is nonzero. This geometric interpretation is a mathematical example, not an engineering safety certification.

A deliberate input-quality check

For a user-defined safety margin, =ABS(COS(RADIANS(B2)))<0.000001 independently expresses a near-zero denominator test (expected Boolean value). The cutoff 0.000001 is an example engineering threshold, not an Apple-defined tolerance. Use it to flag a row for review before interpreting a reciprocal; never substitute an arbitrary huge number as a meaningful result.

Documented mathematical restriction: Exclude 90°, 270°, and each odd multiple of 90°; these are the poles of secant. An angle of exactly 90° is excluded for SEC.

Do not claim native error strings. Apple’s cited SEC page documents the input type and mathematical domain restriction, 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: SEC 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 SEC 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.

Should SEC(60) be 2? No. SEC(60) uses 60 radians. To compute secant of 60 degrees use SEC(RADIANS(60)).

Why does SEC blow up near 90 degrees? The cosine approaches zero at 90° (π/2 radians). Secant is the reciprocal of cosine and is undefined at that angle.

Is the result in degrees? No. The input is an angle in radians, but the output is a dimensionless reciprocal trigonometric ratio, not an angle.

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 SEC description, exact radian-angle 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 SEC 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.
• SECH — continue with a related Apple Numbers calculation.

For a software-specific comparison: Gnumeric SEC. 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.