TAN Function in Apple Numbers: Tangent, Radians and Practical Examples

The TAN function returns the tangent of an angle supplied in radians. For a 45° input, use =TAN(RADIANS(45)); the mathematically expected result is 1. Typing =TAN(45) means 45 radians, not 45 degrees.

TAN computes the tangent of an angle supplied in radians. Apple gives TAN(1) ≈ 1.5574077246549, TAN(RADIANS(45)) = 1, and TAN(3*PI()/4) = −1. The function accepts general real-number radian angles, not only those between −π and π. 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.

=TAN(radian-angle)

Argument: radian-angle

Required?
Yes

Apple-documented type
Number value

Meaning / allowable values
Apple describes any numeric radian angle as valid; the customary −π to π span is not an input restriction. No optional arguments or defaults are documented.

The TAN argument is one required Number value, representing a radian angle. There is no second argument, no mode selector, and no default angle. RADIANS turns a separate degree input into a valid TAN argument. Output range / behaviour: The output can have arbitrarily large magnitude near an odd multiple of 90°; mathematically tangent is undefined where cosine is zero.

The parameter name, argument count, range, and documented examples above come from Apple’s TAN 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. TAN 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 Degrees in B1 and Tangent (6 decimals) in C1. Enter the numeric B2:B10 inputs shown in the table; do not include a degree sign or quote characters in the actual cell values.
3. In C2 enter =TAN(RADIANS(B2)). Copy or fill C2 down through C10; 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: Degrees
-60

C: formula
=TAN(RADIANS(B2))

C: expected result (6 d.p.)
-1.732051

Row: 3

B: Degrees
-45

C: formula
=TAN(RADIANS(B3))

C: expected result (6 d.p.)
-1.000000

Row: 4

B: Degrees
-30

C: formula
=TAN(RADIANS(B4))

C: expected result (6 d.p.)
-0.577350

Row: 5

B: Degrees
0

C: formula
=TAN(RADIANS(B5))

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

Row: 6

B: Degrees
15

C: formula
=TAN(RADIANS(B6))

C: expected result (6 d.p.)
0.267949

Row: 7

B: Degrees
30

C: formula
=TAN(RADIANS(B7))

C: expected result (6 d.p.)
0.577350

Row: 8

B: Degrees
45

C: formula
=TAN(RADIANS(B8))

C: expected result (6 d.p.)
1.000000

Row: 9

B: Degrees
60

C: formula
=TAN(RADIANS(B9))

C: expected result (6 d.p.)
1.732051

Row: 10

B: Degrees
135

C: formula
=TAN(RADIANS(B10))

C: expected result (6 d.p.)
-1.000000

Degrees in column B are numerical input; column C converts them to radians before taking the tangent. The special angles 90° and 270° are intentionally excluded, because the real tangent is undefined there. Spot check: =TAN(RADIANS(45)) → 1. 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 TAN, 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. Calculate a ramp’s vertical rise from its horizontal run

Create a second small table or use ordinary empty cells in the same table: enter 1.2 in B12 as horizontal run (metres), and 10 in C12 as the ramp angle (degrees). In D12 enter:

=B12*TAN(RADIANS(C12))

The independently calculated expected rise is 0.211592 metres when shown to six decimals (1.2 × tan(10°)). This assumes the run is horizontal rather than the length measured along the slope; using the sloped length instead requires sine.

2. Convert an incline into a percentage grade

With a 30° incline in B15, enter =100*TAN(RADIANS(B15)) to calculate 57.735027% when rounded to six decimals. Percentage grade is rise divided by horizontal run times 100, which is distinct from degrees and from a cell simply formatted as Percentage. If the output cell is formatted as Percentage, use =TAN(RADIANS(B15)) without the extra 100 to avoid multiplying twice.

3. Check the inverse relationship and loss of quadrant information

=DEGREES(ATAN(5/10)) gives approximately 26.565051°. Taking TAN of that result after converting to radians yields approximately 0.5. However, tangent repeats every 180°, so the inverse alone cannot tell which quadrant an original physical direction occupied. Use extra sign/quadrant information where it matters.

4. Verify the tangent-to-sine/cosine identity

For 30°, =SIN(RADIANS(30))/COS(RADIANS(30)) has the same expected value as =TAN(RADIANS(30)). When the cosine denominator approaches zero, this ratio becomes unstable; it is a conceptual check rather than a preferred alternative near singularities.

• Degrees entered as radians: =TAN(45) is not 1; use =TAN(RADIANS(45)) for 45 degrees.
• Near 90°, 270°, etc.: tangent has no finite mathematical value at angles where cosine is zero. Floating-point approximations can instead yield a very large finite number. Do not promise that Numbers will display a particular error or infinity; its actual behaviour was not tested.
• Rounding the angle too early: even tiny angle differences close to 90° greatly change the result. Retain useful source precision.
• Wrong geometry measurement: tangent is opposite/adjacent, not opposite/hypotenuse. For sloped lengths use sine as appropriate.
• A percentage was multiplied twice: percentage-formatted output for TAN(angle) already displays the value ×100%. Do not also multiply by 100 for that format.
• Very large or nonnumeric inputs: Apple describes the argument as a numeric value, but this review did not test text coercion, overflow, or exact app error text.

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 TAN 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 TAN 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.

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

• TAN: TAN — 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 TAN 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.

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

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