ACOT in Apple Numbers: Inverse Cotangent and Angle Conversion

=DEGREES(ACOT(1)) has an expected result of 45 degrees. The inverse cotangent takes a cotangent ratio and returns a principal angle in radians, rather than interpreting the argument as an angle.

Purpose: Inverse 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:

=ACOT(num)

• num (required): cotangent ratio (unitless). Apple defines ACOT(num) for one numeric input and gives an angle between zero and π radians; it explicitly recommends DEGREES(ACOT(num)) for degrees. Apple gives ACOT(0) as π/2. No optional arguments or default values are listed.
• Arguments and defaults: there is one required numeric input; there are no optional positions and no documented defaults. =ACOT(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 ACOT is established here. No additional aliases or historical equivalences were verified; translated labels in other locales are not automatically alternative identifiers.

Official documentation: Apple ACOT 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 ACOT, start cell C2 with =ACOT(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. Put =DEGREES(C2) in D2, and fill D2 down to D9 to display the same principal angles in degrees. Do not feed D-column degrees back into ACOT as if they were cotangent ratios.

Row: 2

A: description
negative 5

B: cotangent ratio
-5

C: formula
=ACOT(B2)

C: expected radians
2.944197 rad

D: expected degrees
168.690068°

Row: 3

A: description
negative 2

B: cotangent ratio
-2

C: formula
=ACOT(B3)

C: expected radians
2.677945 rad

D: expected degrees
153.434949°

Row: 4

A: description
negative 1

B: cotangent ratio
-1

C: formula
=ACOT(B4)

C: expected radians
2.356194 rad

D: expected degrees
135.000000°

Row: 5

A: description
zero 0

B: cotangent ratio
0

C: formula
=ACOT(B5)

C: expected radians
1.570796 rad

D: expected degrees
90.000000°

Row: 6

A: description
positive 0.5

B: cotangent ratio
0.5

C: formula
=ACOT(B6)

C: expected radians
1.107149 rad

D: expected degrees
63.434949°

Row: 7

A: description
positive 1

B: cotangent ratio
1

C: formula
=ACOT(B7)

C: expected radians
0.785398 rad

D: expected degrees
45.000000°

Row: 8

A: description
positive 2

B: cotangent ratio
2

C: formula
=ACOT(B8)

C: expected radians
0.463648 rad

D: expected degrees
26.565051°

Row: 9

A: description
positive 5

B: cotangent ratio
5

C: formula
=ACOT(B9)

C: expected radians
0.197396 rad

D: expected degrees
11.309932°

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, =ACOT(MathInputs::B2)). 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.

The principal-angle convention matters

A cotangent ratio is periodic: many angles have the same cotangent, but ACOT gives only one principal answer. For finite real input the mathematical principal value lies strictly between 0 and π radians, consistent with Apple’s stated 0-to-π range. For example, =DEGREES(ACOT(-1)) is expected to be 135°, not −45°. The sign of the input decides which side of 90° the principal angle lies on.

Recovering an angle is not the same as preserving a revolution

Consider =DEGREES(ACOT(COT(RADIANS(225)))). Its independently calculated expected result is 45°, because cotangent repeats every 180°. That does not mean 225° equals 45° as a direction: an inverse trigonometric function cannot retain the original number of turns. If a project needs a full 0–360° heading, retain a quadrant or an original-angle column. Apple’s ATAN2 documents a coordinate-based alternative, but its output convention must be checked independently.

Build a sign-symmetry cross-check

Try =ACOT(-B2)+ACOT(B2) with B2=2: the mathematical expectation is π radians. This follows from the principal-angle definition and provides a useful test that negative ratios were not accidentally converted to positive. When comparing floating-point results, use a tolerance instead of a strict display-string comparison.

Comparing to ATAN, carefully

For positive x, the principal angle can be expressed mathematically as π/2 - atan(x); the expression =PI()/2-ATAN(B2) is a convenient conceptual cross-check across negative, zero and positive finite inputs. This is a mathematical identity, not evidence that Apple Numbers computes both paths with identical last bits. Don’t replace ACOT solely to make the formula look more familiar.

Data typing and error messages

Apple describes a numeric input for ACOT. 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: ACOT() is documented in the linked 15.4 help. Numbers references can include both sheet and table names, separated with double colons.
• Microsoft Excel: ACOT documentation supports the same named real-valued mathematical function. Microsoft documents Excel ACOT(number) and a principal radian result from 0 to π. This supports a same-name real-number comparison, not a guarantee of identical error messages or regional parsing.
• Google Sheets: the official Sheets function catalogue lists ACOT 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.

ACOT returns the inverse of cotangent. It is not the same as ATAN (inverse tangent), nor ACOTH (inverse hyperbolic cotangent).

• Apple ACOT 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 ACOT function — corroborates the Excel name and documents Excel-specific bounds where stated; accessed 2026-10-11 UTC.

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

For a software-specific comparison: LibreOffice Calc ACOT, Apache OpenOffice Calc ACOT, Gnumeric ACOT. 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.