ATAN Function in Apple Numbers: Inverse Tangent and Angles

ATAN converts a tangent ratio (or slope) into its principal angle in radians. To report an angle in degrees, enter =DEGREES(ATAN(1)). The expected answer is 45°. The standalone =ATAN(1) returns approximately 0.785398163397448 radians, not 45. Those are the examples documented by Apple’s ATAN reference.

The tangent of an angle describes opposite side ÷ adjacent side in a right triangle. ATAN reverses that relationship: given a numerical ratio such as 0.5, it returns the corresponding principal angle. Apple’s documented output interval is −π/2 through +π/2 radians (equivalently −90° through +90°). A value outside that principal interval cannot be recovered from a single tangent ratio, because several directions share the same tangent.

For example, slopes of 1 at 45° and 225° both have tangent 1, but ATAN(1) returns the principal 45° angle. When both the horizontal and vertical direction signs matter, use ATAN2(x, y) with the two coordinates rather than dividing them. ATAN alone does not resolve a full circle’s quadrant.

=ATAN(num)

Argument: num

Required
Yes

Apple-documented type
Number value

Meaning
The tangent ratio whose principal arctangent is required.

One Number argument is required; no second argument or default is documented. The input can be any ordinary finite real ratio in mathematical analysis. The output is an angle measured in radians and, per Apple’s documentation, in the principal interval −π/2 to +π/2. Apple does not specify an app error message or text-conversion rules for arbitrary nonnumeric strings in this reference. Argument-type reference; ATAN reference.

Formula punctuation, decimals, and references

Apple’s Functions overview uses an English-language formula and a comma to separate distinct arguments. If a regional setting uses the comma as the decimal separator, Apple’s guidance specifies a semicolon argument separator instead. The examples on this page use English function identifiers and a period for decimal values. A display rounded to six places is not a change to the underlying number.

In the same Numbers table, B2 refers to column B, row 2. For a reference in another table, Numbers can use Table 2::B5; for another sheet, a fully qualified reference can use Sheet 2::Table 2::B5. See Apple’s cell-reference guide. When copying down a formula, ordinary references like B2 change to B3; use the Numbers formula editor to select other tables rather than typing unverified names.

1. Create a Numbers table named Slope Angles with at least 9 rows and 3 columns. Treat row 1 as the header.
2. Put Slope ratio in B1 and Angle (degrees) in C1. Enter the numerical values in column B below, without quotation marks or degree symbols.
3. Type =DEGREES(ATAN(B2)) in C2 and fill through C9. Format C as Number with six decimal places.
4. Compare Numbers’ result with the expected calculations shown. The displayed decimal places can differ until you set Number formatting.

Row: 2

B: numeric slope
-2

C: entered formula
=DEGREES(ATAN(B2))

Expected degrees (6 d.p.)
-63.434949

Row: 3

B: numeric slope
-1

C: entered formula
=DEGREES(ATAN(B3))

Expected degrees (6 d.p.)
-45.000000

Row: 4

B: numeric slope
-0.5

C: entered formula
=DEGREES(ATAN(B4))

Expected degrees (6 d.p.)
-26.565051

Row: 5

B: numeric slope
0

C: entered formula
=DEGREES(ATAN(B5))

Expected degrees (6 d.p.)
0.000000

Row: 6

B: numeric slope
0.5

C: entered formula
=DEGREES(ATAN(B6))

Expected degrees (6 d.p.)
26.565051

Row: 7

B: numeric slope
1

C: entered formula
=DEGREES(ATAN(B7))

Expected degrees (6 d.p.)
45.000000

Row: 8

B: numeric slope
2

C: entered formula
=DEGREES(ATAN(B8))

Expected degrees (6 d.p.)
63.434949

Row: 9

B: numeric slope
10

C: entered formula
=DEGREES(ATAN(B9))

Expected degrees (6 d.p.)
84.289407

For a quick single-cell check, =ATAN(1) has the expected radian value 0.785398163397448 (Apple’s own example), while =DEGREES(ATAN(1)) is 45. All values in the table were independently computed as degrees(atan(ratio)) using Python’s standard math library. This is not an observation of how an installed version of Numbers rounds or formats cells.

1. Find a ramp’s incline from rise and horizontal run

Imagine a ramp rises 0.8 metres over a horizontal run of 10 metres. Type =DEGREES(ATAN(0.8/10)). Its expected angle is 4.573921°. The denominator must be horizontal distance, not the distance measured along the sloped surface. This distinction matters in accessibility or surveying measurements.

For a reusable table, put a measured rise into B12, a horizontal run into C12, and =DEGREES(ATAN(B12/C12)) into D12. Before calculating, check that the run is not zero and that measurements share the same unit.

2. Recover a line angle from a percent grade

A 12% grade is a slope ratio of 0.12, not a 12° angle. Use =DEGREES(ATAN(12/100)) to obtain an expected 6.842773°. If cell B15 already holds a real spreadsheet percentage formatted as 12%, use =DEGREES(ATAN(B15)) and do not divide by 100 again. Distinguish the stored numeric 0.12 from displayed text 12%.

3. Confirm a result by reversing the calculation

For the positive ratio 0.5, the expected output of =ATAN(0.5) is 0.463647609001 radians. Using =TAN(ATAN(0.5)) should yield approximately 0.5, within normal floating-point precision. This identity does not recover quadrants outside the principal interval, and decimal comparison may need a tolerance.

4. Interpret negative slopes

=DEGREES(ATAN(-1)) gives −45° as the principal angle. The negative sign tells you the slope is negative but cannot distinguish an up-left direction from a down-right direction by itself. For oriented coordinates (x, y), use =DEGREES(ATAN2(x, y)) (Numbers orders x first, y second), while excluding the origin where the direction is undefined mathematically.

• Radians mistaken for degrees: ATAN(1) returns roughly 0.7854 radians; use DEGREES for degrees. Merely formatting a cell cannot convert angle units.
• Rise/run divided in the wrong order: ATAN(run/rise) is generally a different angle. Confirm column meanings and consistent measurement units.
• Confusing angle with grade: 45° means a 100% grade, not 45%. Convert a displayed percent to its underlying ratio before calling ATAN.
• Division by zero: ATAN(B2/C2) creates a division problem if C2=0. ATAN2 avoids the division for nonorigin coordinates, but the exact app error for all edge cases was not verified here.
• Lost quadrant: A tangent ratio cannot distinguish all 360° directions. Use the separate ATAN2(x,y) function when x and y signs are known.
• Nonnumeric imported values: The official argument is a Number value. Strings such as "0.5" are not proved interchangeable with numbers in this review; clean the source cells and use the formula inspector.
• Last-decimal disagreements: Trigonometric calculations are floating-point operations. Compare the unrounded value or a sensible tolerance, not only the printed six-place display.

To diagnose, start with =DEGREES(ATAN(1)), then reference a numeric input in the same table, then a source cell in another table selected through the formula editor. That separates the mathematical operation from locale punctuation, data typing and table references.

Excel and Google Sheets both document a scalar ATAN function with the same mathematical purpose and radian outputs; their ordinary numeric examples should have mathematically equivalent expected answers. This does not establish identical error codes, coercion of text, locale punctuation, display rounding or platform version histories. For a separately oriented vector, do not simply replace ATAN with ATAN2(y,x) copied from a Python sample; Numbers’ and the spreadsheet ATAN2 syntax checked here uses x then y, while Python’s math.atan2 places y then x. Google’s published function list covers ATAN and ATAN2; Apple supplies the exact syntax. The destination application’s own documentation should be checked for unusual inputs before migrating a workbook.

• Apple ATAN — exact syntax, principal range and official numerical examples, accessed 2026-10-11.
• Apple ATAN2 — coordinate/quadrant alternative, accessed 2026-10-11.
• Apple TAN — forward tangent of radians, referenced in Apple’s ATAN See also, accessed 2026-10-11.
• Apple Functions overview — syntax conventions and locale separators, accessed 2026-10-11.
• Apple Refer to cells in formulas — Numbers table-reference format, accessed 2026-10-11.
• Apple Types of arguments and values — Number arguments, accessed 2026-10-11.
• Google Sheets function list — ATAN availability at catalogue level, accessed 2026-10-11.

• ACOS — continue with a related Apple Numbers calculation.
• ASIN — 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 ATAN, Apache OpenOffice Calc ATAN, Gnumeric ATAN. 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.