ACOS Function in Apple Numbers: Inverse Cosine and Angles

ACOS turns a cosine ratio in the closed interval [−1, 1] into the principal angle in radians, between 0 and π. =DEGREES(ACOS(0.5)) has the expected result 60°.

ACOS computes the principal inverse cosine of a numeric cosine ratio. Apple specifies ACOS(num) with num between −1 and 1, and outputs a principal result between 0 and π radians. Its official examples include DEGREES(ACOS(0.5)) = 60 and ACOS(SQRT(2)/2) ≈ 0.785398163397448. The distinction between a ratio, an angle in radians, and an angle in degrees matters: spreadsheet formulas do not automatically supply a degree symbol to a numerical result. The examples below make the conversion explicit so they are reproducible when pasted into a clean Numbers table.

A single formula is rarely enough to validate a large model. Start with the small examples here, compare your real column data types and measurement units, then extend the formula only after checking actual cell values. This avoids turning plausible-looking trigonometry into an incorrect report.

=ACOS(num)

Argument: num

Required?
Yes

Apple-documented type
Number value

Meaning / allowable values
The numeric input must be in [−1, +1]. There are no optional arguments or default values in the documented syntax.

The function accepts one required Number value, representing a dimensionless ratio. No second argument or default input is documented. Its result is in radians, so DEGREES is an optional outer conversion and not another argument of the function. Output range / behaviour: The result is in radians from 0 through π inclusive (0° to 180°).

The parameter name, argument count, range, and documented examples above come from Apple’s ACOS function reference. Apple’s Types of arguments and values defines a number argument as a numeric literal, numeric expression or reference to a cell containing numeric content. This review did not establish how arbitrary text strings, arrays, overflow values, or formatted unit strings are coerced in an actual Numbers runtime.

Locale, decimal separators and Numbers tables

The working formulas use English-language function names and a period for decimal values such as 0.5. Apple’s Functions overview specifies commas between arguments in its examples, but a semicolon when the user’s decimal separator is a comma. ACOS itself takes just one argument, so its direct call has no separator; a nested function with two or more arguments still needs the appropriate locale delimiter. Function name letter case is not significant in Apple’s documented convention.

In one Numbers table, B2 means that table’s B2 cell when typed within it. If referencing a different table, Apple’s reference value guidance shows the Table Name::B2 double-colon pattern (and includes the sheet name when needed). It is safer to select a cell in another table through the formula editor instead of hand-typing a guessed table name. These reference rules apply to formulas below just as they do to other Numbers calculations.

1. Insert or reuse an otherwise empty Numbers table. Give it the name Angle Lab and use column B for input values and column C for formula output.
2. Put Cosine ratio in B1 and Principal angle, degrees (6 decimals) in C1. Enter the numeric B2:B8 inputs shown in the table; do not include a degree sign or quote characters in the actual cell values.
3. In C2 enter =DEGREES(ACOS(B2)). Copy or fill C2 down through C8; this intentionally uses same-table references and avoids guessing cross-sheet names.
4. Format the C column to show six decimal places. The table entries below are expected mathematical results; your actual Numbers display may differ according to precision and the app’s number formatting. Compare the numerical results, not merely the visual text.

Row: 2

B: Cosine ratio
-1

C: formula
=DEGREES(ACOS(B2))

C: expected result (6 d.p.)
180.000000

Row: 3

B: Cosine ratio
-0.866025403784439

C: formula
=DEGREES(ACOS(B3))

C: expected result (6 d.p.)
150.000000

Row: 4

B: Cosine ratio
-0.5

C: formula
=DEGREES(ACOS(B4))

C: expected result (6 d.p.)
120.000000

Row: 5

B: Cosine ratio
0

C: formula
=DEGREES(ACOS(B5))

C: expected result (6 d.p.)
90.000000

Row: 6

B: Cosine ratio
0.5

C: formula
=DEGREES(ACOS(B6))

C: expected result (6 d.p.)
60.000000

Row: 7

B: Cosine ratio
0.707106781186548

C: formula
=DEGREES(ACOS(B7))

C: expected result (6 d.p.)
45.000000

Row: 8

B: Cosine ratio
1

C: formula
=DEGREES(ACOS(B8))

C: expected result (6 d.p.)
0.000000

B contains cosine ratios (numeric values), and C returns the principal angle in degrees. The negative inputs demonstrate obtuse results, while 1 maps to 0° and −1 maps to 180°. Spot check: =DEGREES(ACOS(0.5)) → 60°. If the spot check disagrees, investigate the source type, angle units, formula punctuation and entered values before using more complicated models.

Why these sample inputs matter

The table includes positive and negative inputs as appropriate for ACOS, a zero boundary where meaningful, and characteristic angle values that can be reasoned about independently of a spreadsheet. It is deliberately not a random dataset: you can hand-check several rows using special-angle trigonometric identities and compare with the independent high-precision numerical outputs stored in the supporting calculation records. The expected display rounds the mathematical answer to six decimal places; it is not a promise that a particular Numbers version internally rounds at this stage.

1. Find the angle of a right triangle from adjacent and hypotenuse

Enter 4 in B12 as the adjacent side and 5 in C12 as the hypotenuse (same units). In D12 enter =DEGREES(ACOS(B12/C12)). The independent mathematical result is approximately 36.869898°. This method applies to the angle between the adjacent side and hypotenuse. If you have opposite/hypotenuse instead, consider ASIN.

2. Calculate an included angle from a normalized dot product

For two vectors, if the dot product (B15) is 5 and the product of their nonzero magnitudes (C15) is 10, =DEGREES(ACOS(B15/C15)) yields 60°. In a full vector workbook, calculate the dot product and magnitudes separately, inspect zero-length vectors, and ensure the ratio lies in [−1,1]. ACOS alone cannot compute those precursor quantities automatically.

3. Know what the principal output leaves out

=COS(RADIANS(300)) equals 0.5 mathematically, but =DEGREES(ACOS(0.5)) returns 60°, not 300°. ACOS returns the standard 0°–180° angle and does not determine clockwise versus counterclockwise direction; signed bearings require more information.

4. Verify using COS and relate to ASIN

For x=0.5, =COS(ACOS(x)) should be approximately 0.5. For any x in [−1,1], ASIN and ACOS principal outputs add to π/2 radians (90°). With x=0.5, that is 30° + 60°; this is a mathematical identity, not an independently observed Numbers result.

• Input outside ±1: ACOS(1.01) is not real-valued; Apple restricts inputs to [−1,1]. No specific Numbers error string has been natively confirmed.
• Radian output mistaken for degrees: ACOS(0.5) is approximately 1.047198 radians; wrap with DEGREES to show 60°.
• Ambiguous angle: a cosine ratio loses rotation direction; ACOS returns a principal angle, not a 360° bearing.
• Zero-length vectors: the denominator of a normalized dot product cannot be zero. Verify the vector magnitudes before division.
• Measurement or floating-point spillover: a calculated ratio can drift slightly above 1 or below −1 due to precision. Investigate the source and acceptable tolerance; do not silently clamp unreviewed invalid data.
• Wrong triangle side: for a right-triangle angle, adjacent/hypotenuse belongs to cosine; opposite/hypotenuse belongs to sine.
• Typed text or incompatible imports: Apple documents a number value. App-specific coercion, error messages, or release parity were not tested.

Practical verification procedure: First type a known numeric literal into a spare cell and evaluate the one-line spot-check formula. Next reference a cell in the same table. Finally, reference a different table with the name inserted by Numbers. This sequence separates function support from formula syntax, locale differences, and wrong-table references. If the result is close but not identical in its last decimals, display more precision and test an appropriate tolerance rather than asserting an exact match.

• Microsoft Excel: Microsoft’s current ACOS documentation confirms the same named scalar function, its central radian/ratio behaviour, and the normal conversion to degrees where applicable. For the elementary examples here, the mathematical operation is equivalent. That is not evidence of identical error strings, import coercion, floating-point behaviour, regional separators, or version history.
• Google Sheets: Google’s official function list lists ACOS and its core trigonometric purpose. This supports functional overlap for ordinary mathematical inputs, but this batch did not run any formulas in Google Sheets or compare edge cases. Locale-specific entry syntax should be checked in the destination spreadsheet.
• Apple Numbers editions and platforms: Apple’s reviewed public help is 15.4. It does not prove first release availability or parity across historical Mac, iPhone, iPad, or browser versions. Test in the target installation before converting an important workbook.

Avoid copying data validation or display formatting assumptions from one spreadsheet program into another without examining them. For example, a cell containing the string "0.5" is not the same documented input type as a cell containing the number 0.5, even if its displayed characters match.

• COS — official Apple reference (read 2026-10-11)
• DEGREES — official Apple reference (read 2026-10-11)
• SIN — official Apple reference (read 2026-10-11)

• ACOS: ACOS — official help, accessed 2026-10-11.
• Syntax conventions: Apple Functions overview — official help, accessed 2026-10-11.
• Number types and table references: Apple Types of arguments and values — official help, accessed 2026-10-11.
• Excel cross-check: Microsoft’s ACOS function — accessed 2026-10-11; basic scalar and angle units only.
• Google Sheets cross-check: Google Sheets function list — accessed 2026-10-11; feature listing, not native formula test.

• ASIN — continue with a related Apple Numbers calculation.
• ATAN — continue with a related Apple Numbers calculation.
• ATAN2 — continue with a related Apple Numbers calculation.
• TAN — continue with a related Apple Numbers calculation.
• SIN — continue with a related Apple Numbers calculation.

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