Calculate arctan(x) or atan2(y,x) with full quadrant support. Get results in degrees, radians, gradians, and turns with quadrant identification, common values table, and atan-vs-atan2 comparison.
| x | arctan(x) Degrees | Exact Radians |
|---|---|---|
| −∞ (limit) | −90° | −π/2 |
| −√3 ≈ −1.7321 | −60° | −π/3 |
| −1 | −45° | −π/4 |
| −1/√3 ≈ −0.5774 | −30° | −π/6 |
| 0 | 0° | 0 |
| 1/√3 ≈ 0.5774 | 30° | π/6 |
| 1 | 45° | π/4 |
| √3 ≈ 1.7321 | 60° | π/3 |
| +∞ (limit) | 90° | π/2 |
| Feature | arctan(x) | atan2(y, x) |
|---|---|---|
| Arguments | 1 (ratio y/x) | 2 (y and x separately) |
| Range | (−90°, 90°) | (−180°, 180°] |
| Quadrant awareness | No — only Q I and Q IV | Yes — all four quadrants |
| Handles x = 0 | No (division by zero) | Yes (returns ±90°) |
| Use case | Basic angle from ratio | Direction, heading, phase |
| JS function | Math.atan(x) | Math.atan2(y, x) |
| Property | arctan(x) |
|---|---|
| Domain | (−∞, +∞) — all real numbers |
| Range | (−π/2, π/2) or (−90°, 90°) |
| Odd function | arctan(−x) = −arctan(x) |
| Limits | lim x→+∞ arctan(x) = π/2; lim x→−∞ = −π/2 |
| Derivative | d/dx arctan(x) = 1/(1 + x²) |
| Integral | ∫ 1/(1+x²) dx = arctan(x) + C |
| Identity | arctan(x) + arccot(x) = π/2 for x > 0 |
The **Inverse Tangent (arctan) Calculator** computes the angle whose tangent equals a given value, with two modes: standard arctan(x) for a single ratio and atan2(y, x) for full quadrant awareness. Switch between modes to compare how the single-argument and two-argument forms differ in range and quadrant handling.
The arctangent function accepts any real number, making it uniquely useful among the three primary inverse trig functions — unlike arcsin and arccos, which are limited to [−1, 1]. It appears in every engineering discipline: analog filter design (phase response), robotics (steering angle from displacement), game development (aiming direction), and calculus (integration of 1/(1+x²)).
In arctan mode, the calculator returns the angle in (−90°, 90°) along with verification that tan(θ) = x. In atan2 mode, it takes separate y and x inputs and returns the full (−180°, 180°] angle, identifies the quadrant, provides the vector magnitude, and compares the atan2 result to plain atan(y/x). Visual bars track the angle in both the standard and the 0°–360° range.
Eight presets per mode cover classic values: 0°, 30°, 45°, 60°, −45°, and large values approaching ±90° in arctan mode; all four quadrants and axis directions in atan2 mode. A common values table lists the standard angles, and a feature-by-feature comparison table shows exactly when to use arctan versus atan2.
Inverse Tangent (arctan) Calculator — arctan & atan2 Modes helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like Degrees, Radians, Gradians in one pass.
arctan(x): θ = tan⁻¹(x), domain (−∞, ∞), range (−π/2, π/2). atan2(y,x): θ = atan2(y,x), range (−π, π]. Conversion: degrees = radians × 180/π. Identity: arctan(x) + arccot(x) = π/2 (x > 0). Derivative: d/dx arctan(x) = 1/(1 + x²).Result: 45°
Using value=1, unit=degrees, the calculator returns 45°. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
This calculator is tailored to inverse tangent (arctan) calculator — arctan & atan2 modes workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
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arctan(x) takes a single ratio and returns an angle in (−90°, 90°), losing quadrant info. atan2(y, x) takes y and x separately and returns the full (−180°, 180°] angle, correctly identifying all four quadrants.
The domain is all real numbers (−∞, +∞). The range is (−π/2, π/2) or (−90°, 90°). The limits at ±∞ are ±π/2.
Because tangent grows without bound as the angle approaches 90°. The limit is 90° but it is never attained for any finite input.
Multiply by 180/π. For example, arctan(1) = π/4 radians × 180/π = 45°.
Whenever you have two components (y and x) rather than a pre-computed ratio. atan2 avoids division by zero when x = 0, and correctly places the angle in the right quadrant.
Yes. arctan(−x) = −arctan(x) for all real x, meaning the graph is symmetric about the origin.
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