Arctan Calculator
Convert a real tangent value into its principal inverse-tangent angle, or enter a principal angle to recover its tangent.
Inputs
Edit either x or y. The most recently edited value becomes the source and the other field is recalculated.
Live result
Arctan curve and current value
The curve shows the principal inverse tangent over a symmetric x-range that includes the current input. The highlighted point is the current calculation.
Reference values
Common inverse-tangent values plus the current input, calculated from the same model used by the result and export.
| x | arctan(x) radians | arctan(x) degrees | Role |
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How to use the arctan calculator
What this calculator does
This calculator evaluates the real principal inverse tangent. In the forward direction, x – tangent value is a dimensionless real number and the calculator returns the principal angle y for which tan(y) = x. In the reverse direction, you can edit y – angle and the calculator returns its tangent, provided y stays inside the principal interval. That restriction keeps the two directions one-to-one: each real x corresponds to exactly one principal angle between – 90° and 90°, excluding the endpoints. The calculator is an exact numerical transformation within floating-point precision; it does not determine a geometric quadrant from two coordinates, which would require a two-argument atan2 calculation.
When to use it
Use inverse tangent when you know a slope or rise-to-run ratio and need its angle, when you are checking a trigonometry exercise, when you need to convert a tangent value into a principal direction angle, or when you want to compare a decimal result with familiar special-angle values. The OpenStax treatment of inverse trigonometric functions explains why tangent must be restricted before it has a single-valued inverse.
How to calculate
- The calculator opens with the demonstration x = 1, which gives y = 45° = 0.785398 radians. The example XLSX is ready immediately.
- Replace x – tangent value to calculate arctan(x), or replace y – angle to calculate tan(y). The most recently edited field becomes the source.
- Choose Angle unit as Degrees or Radians. A unit change converts the current y value instead of reinterpreting the same digits.
- Read the live result, the curve, and the reference table. Select Download Excel to export the current canonical values as a validated .xlsx workbook.
- Select Reset to clear the demonstration and all calculated state. Reset leaves the data fields empty, so Download Excel is disabled until you enter a complete valid value again.
Input guide
Required when x is the source. Enter a finite real number with no unit, for example 1, – 0.5, or 1,234.5. A period is the decimal separator; conventional en-US comma grouping is accepted. Scientific notation and ambiguous decimal-comma forms are rejected. Larger positive x pushes the principal angle closer to +90°; more negative x pushes it closer to – 90°. A common mistake is treating x as an angle rather than a ratio.
Required when y is the source. Enter a finite angle in the currently selected unit, such as 30° or 0.523599 radians. To preserve the principal inverse relationship, y must be strictly between – 90° and 90°, or between – π/2 and π/2 radians. Values near either boundary create a tangent with very large magnitude. Do not enter the degree symbol in the field; the unit is supplied by the interface.
This required selector controls how y is entered and how the primary result is displayed. Degrees are convenient for geometry; radians are standard in calculus and many programming interfaces. Switching from Degrees to Radians converts 45 to about 0.785398, and switching back restores 45 within numerical precision. Changing this control does not alter x or the represented angle.
Output guide
Input x repeats the current canonical tangent ratio. Principal angle shows the current y in the chosen angle unit, while the Angle unit pill confirms that unit. Selected angle is the primary result in the active unit. Angle in degrees and Angle in radians show the same principal angle in both common systems. Tangent check evaluates tan(y) and should reproduce x within floating-point precision. Principal range states the branch used for arctan; the endpoints are excluded because tangent is not finite there.
The Arctan curve and current value visual contains two labeled series: the arctan(x) curve shows the function over a symmetric range containing the current x, and Current value marks the active x-y pair. The curve rises but flattens as |x| becomes large, illustrating the approach toward ±90°. The Reference values table lists x, arctan(x) radians, arctan(x) degrees, and Role. Its current row is generated from the same model as the result, chart point, and workbook.
Worked example
With the startup example x = 1, the calculation is y = arctan(1). Because tan(45°) = 1, the principal angle is exactly 45°, which is π/4 radians or approximately 0.785398 radians. The Tangent check returns 1. If you change y to 30° instead, the reverse direction computes tan(30°) ≈ 0.577350, and that new x becomes the current input. These identities agree with the principal-value convention documented in the NIST Digital Library of Mathematical Functions section on inverse trigonometric functions.
How the inverse tangent model works
The tangent function repeats every π radians, so it is not one-to-one across all real angles. To define arctan as a function, tangent is restricted to the interval ( – π/2, π/2), where it is strictly increasing. The inverse therefore accepts every real x and returns one y inside that interval.
The function is odd: arctan( – x) = – arctan(x). It is also monotone increasing, while its slope decreases away from zero. In calculus, its derivative is 1/(1 + x²), which is always positive for real x and tends toward zero as |x| grows. The NIST DLMF properties of inverse trigonometric functions provides the derivative and related identities.
Interpretation and common mistakes
A result near 0° means the tangent ratio is near zero. A result near +90° or – 90° means the magnitude of x is very large; the calculator never returns the endpoints for a finite real x. Do not confuse tan⁻¹(x) meaning inverse tangent with the reciprocal 1/tan(x), which is cotangent. Also distinguish one-argument arctan from atan2: arctan uses only a ratio and therefore returns a principal angle, while atan2 uses two signed coordinates to determine a full quadrant.
Display values are rounded for readability, but calculations and Excel cells retain canonical numeric precision. For special values such as x = 1 or x = √3, exact symbolic forms like π/4 or π/3 are mathematically useful, but this calculator deliberately returns numeric values so that editing, charting, and workbook export remain consistent.