COT in Apple Numbers: Degrees, Radians, and Undefined Angles

=COT(RADIANS(45)) has an expected result of 1. COT returns the cotangent of an angle in radians, so a measurement recorded in degrees must first be converted with RADIANS().

Purpose: Cotangent supports mathematical, scientific, engineering, or geometry calculations where the formula’s input and permitted domain match this documented definition. The worked numbers on this page are independently calculated expectations and have not been observed running in Apple Numbers.

Apple’s English Formulas and Functions Help 15.4 lists the exact signature:

=COT(radian-angle)

• radian-angle (required): a numeric angle in radians. Convert degrees with RADIANS before COT. Apple specifies COT(radian-angle): one required numeric angle expressed in radians. Its documented mathematical domain excludes multiples of π because the cotangent is undefined there. Apple examples include COT(1)≈0.642092615934331, COT(PI()/4)=1, and COT(RADIANS(30))≈1.73205080756888. No optional arguments/defaults are listed.
• Arguments and defaults: there is one required numeric input; there are no optional positions and no documented defaults. =COT(B2) refers to a cell in the same table. For one argument there is no argument-order ambiguity.
• Regional input: the guide uses English function identifiers and period decimal separators, such as 0.5. If the document uses decimal commas, type numeric literals in its local format; Apple’s function overview says multi-argument formulas typed by hand use semicolons when the decimal separator is a comma. That separator rule does not change this function’s one-argument signature.
• Platforms/releases: the 15.4 number identifies the edition of Apple’s Functions Help linked here. It is not proof of the first Numbers release supporting this function or identical behaviour on every Mac, iPhone, iPad and web version. No native edition was executed for this guide.
• Aliases: only the English identifier COT is established here. No additional aliases or historical equivalences were verified; translated labels in other locales are not automatically alternative identifiers.

Official documentation: Apple COT function; Apple Functions overview. Consulted 2026-10-11 UTC.

1. Create a table named MathInputs in Numbers. Keep row 1 as headers, with the inputs in column B and calculated results starting in column C.
2. Type the B-column values from the table below as real numbers, not quoted strings or cells formatted as Text. Enter the row labels in column A; they are for the reader, not arguments.
3. For COT, start cell C2 with =COT(RADIANS(B2)) and fill down to the last row. Use at least six decimal places in the result column so displayed rounding is easy to inspect.
4. Compare the formulas and values in C2 through C9 against the independently predicted figures. Formula fills should refer to B3, B4, etc.

Row: 2

A: description
negative 120

B: numeric input
-120

C: formula
=COT(RADIANS(B2))

C: expected numeric result
0.577350

Row: 3

A: description
negative 60

B: numeric input
-60

C: formula
=COT(RADIANS(B3))

C: expected numeric result
-0.577350

Row: 4

A: description
negative 45

B: numeric input
-45

C: formula
=COT(RADIANS(B4))

C: expected numeric result
-1.000000

Row: 5

A: description
negative 30

B: numeric input
-30

C: formula
=COT(RADIANS(B5))

C: expected numeric result
-1.732051

Row: 6

A: description
positive 30

B: numeric input
30

C: formula
=COT(RADIANS(B6))

C: expected numeric result
1.732051

Row: 7

A: description
positive 45

B: numeric input
45

C: formula
=COT(RADIANS(B7))

C: expected numeric result
1.000000

Row: 8

A: description
positive 60

B: numeric input
60

C: formula
=COT(RADIANS(B8))

C: expected numeric result
0.577350

Row: 9

A: description
positive 120

B: numeric input
120

C: formula
=COT(RADIANS(B9))

C: expected numeric result
-0.577350

Validation labels: The values above are expected, computed by Python standard-library mathematics, rounded to six decimals. They are not native Numbers test observations. Numbers may display fewer/more digits depending on cell format; small floating-point differences should be evaluated with a tolerance.

Use Numbers table references when inputs are elsewhere

Within MathInputs, B2 is sufficient. If the formula is in a different Numbers table, Apple’s documented qualified-reference pattern allows MathInputs::B2 (for example, =COT(MathInputs::B2) with the input in radians). For a different sheet, the documented pattern is Sheet Name::MathInputs::B2 if disambiguation is required. Click the cell in the formula editor to obtain the correct token if your real sheet/table names contain spaces. Copying relative references normally changes the referenced row; lock $ components deliberately if you need a fixed source cell. Apple cell references.

Cotangent is a ratio, not an angle

For mathematical angle θ, cot(θ) = cos(θ)/sin(θ) wherever sin(θ) ≠ 0. For example, with B2=30 degrees, =COS(RADIANS(B2))/SIN(RADIANS(B2)) is an independent algebraic cross-check of =COT(RADIANS(B2)); both are expected to be around 1.732051. Prefer the built-in function for readability; a ratio of separately computed values can behave differently at floating-point extremes.

Why 0°, 180°, and 360° are invalid

Cotangent is undefined where the sine denominator is zero, meaning every integer multiple of π radians. Accordingly, 0°, 180° and 360° are mathematically excluded. A finite-precision evaluation of PI() may not create an exactly zero sine internally: it could display a huge number instead of an error, depending on the implementation. Apple’s documentation defines the mathematical restriction, not a tested exact Numbers error string for every floating-point representation.

Repeat pattern versus change of sign

Cotangent repeats every 180°: cot(θ + 180°) = cot(θ). Thus 45° and 225° are both expected to produce 1. But cot(−45°) = −1, showing odd symmetry. These relationships make useful validations for survey geometry and slope models when you know whether your input values represent degrees or radians.

Transform a slope into a complementary angle

If a right-triangle slope or rise/run ratio is known, COT can describe the reciprocal of tangent for a nonzero angle. For an angle of 30°, tangent is approximately 0.577350, while cotangent is approximately 1.732051. Interpret those as geometric ratios, not as degrees. When a dataset contains the ratio but you need the principal angle, ACOT is the inverse operation.

Data typing and error messages

Apple describes a numeric input for COT. The official page establishes the documented mathematical domain; it does not enumerate every error code, rounding threshold, overflow behaviour, or conversion rule for text, blanks, Boolean values and dates. Therefore this guide avoids asserting a specific Numbers error message for a nonnumeric cell or out-of-domain value. First test the direct numeric input recommended in Apple’s examples, then inspect any imported text or regional number formatting. For extreme magnitudes, don’t claim exact machine precision or Apple-specific limits without a native test.

• Apple Numbers: COT() is documented in the linked 15.4 help. Numbers references can include both sheet and table names, separated with double colons.
• Microsoft Excel: COT documentation supports the same named real-valued mathematical function. Excel documents COT(number) with radians and an Excel-specific magnitude restriction |number| < 2^27. Apple’s cited COT help does not establish the same numerical limit or error-code wording.
• Google Sheets: the official Sheets function catalogue lists COT under its Math functions, with one argument. That establishes existence and documented syntax in Sheets, not identical error coercion, floating-point precision, or release support.
• On migration: recalculate several positive, negative, and boundary-adjacent examples using the target application; convert Numbers table-qualified references into the destination’s own worksheet reference syntax. Distinguish a numeric expected value from a screen result seen in an app. Locale and sheet-name handling need target-file verification.

COT and COTH are different. COT is periodic and requires an angle in radians; COTH is a hyperbolic ratio for a nonzero real input.

• Apple COT reference — exact signature, numeric argument, domain/meaning, and Apple’s own listed examples; Help 15.4, accessed 2026-10-11 UTC.
• Apple Functions overview — function naming, required arguments, and decimal-comma separator rule; Help 15.4, accessed 2026-10-11 UTC.
• Apple cell-reference formats — same-table and qualified references, and relative/absolute copying behaviour; Help 15.4, accessed 2026-10-11 UTC.
• Apple help welcome — high-level application/platform context only; accessed 2026-10-11 UTC.
• Google Sheets function list — corroborates the separate Sheets function name; accessed 2026-10-11 UTC.
• Microsoft Excel COT function — corroborates the Excel name and documents Excel-specific bounds where stated; accessed 2026-10-11 UTC.

• ACOT — 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.
• SECH — continue with a related Apple Numbers calculation.

For a software-specific comparison: LibreOffice Calc COT, Gnumeric COT. 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.