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Math calculator

Arctan Calculator

Calculate arctan(x) = tan⁻¹(x) for any real x. Fast local calculation with shareable, machine-readable results.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance
Input interpretation
Enter values to calculate.
Result
—
Assurance
Core
Declared partition coverage
PASS · 3/3 declared partitions (arctan, invalid-domain, xcal) · Matrix
Known limitations
  • any finite real x; mathematical principal range (−90°, 90°); binary64 may round to ±90°
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
The principal real arctangent θ = arctan(x) in degrees and radians. Default display is degrees.
Scope
Real inverse tangent with principal branch (−π/2, π/2) (NIST DLMF §4.23(ii)). Infinity is not accepted as an input.
Verification
Engine tested · Source checked · v1.0.7 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.7 · CVP protocol 1.0.0-proposed
CVP identity
8/8 property · digest cf317536e8dd
Legacy regression
10/10 tests · Production surface contract 3/3
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP / URL→result→graph — Success 1/1. Integration: URL → SSR result → Live graph current point (4/4). Hydration/slider/history are URL-canonical contracts, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
9/9 golden · 3/3 CVP boundary · 3/3 invalid · 8/8 property · 3/3 metamorphic · 4/4 round-trip · 21/21 O3 · 1/1 cross-interface · 4/4 cross-calculator · 4/4 URL→result→graph · 1/1 CVP contract · Manifest
Sources
Sources
Evidence
7 legacy golden · 3 legacy boundary · legacy regression suite · 9/9 oracle-backed golden · 3/3 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.7 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Definitionθ = arctan(x) = tan⁻¹(x)
Degreesθ° = arctan(x) × 180/π
Right triangleθ = arctan(opposite / adjacent)
iDomain is all real numbers. Principal range is (−π/2, π/2). General solutions differ by kπ. As x → ±∞, arctan(x) → ±π/2.

How to use

1

Enter the tangent value x

The published engine computes arctan(x) for any finite real x. Opposite/adjacent is a local right-triangle convenience (positive side lengths); share URLs use x.

2

Read degrees and radians

The main result is in degrees; the explainer also shows radians (and exact π forms for common values).

3

Share the result

Share URLs include x, for example /calc/math/arctan?x=1.

Example calculations

Common configurations with formula and result.

ϟ

arctan(1)

Equal legs

tan(45°) = 1
45° = π/4 rad
ϟ

arctan(√3)

60° angle

tan(60°) = √3
60° = π/3 rad
ϟ

arctan(0)

Zero angle

tan(0°) = 0
0°
ϟ

Opposite / adjacent

opp = 3, adj = 4

arctan(3/4)
≈ 36.870°

Arctan common values

Common values at a glance.

xarctan(x) °arctan(x) rad
−√3 ≈ −1.7320508−60°−π/3
−1−45°−π/4
−1/√3 ≈ −0.5773503−30°−π/6
00°0
1/√3 ≈ 0.577350330°π/6
145°π/4
√3 ≈ 1.732050860°π/3
i Example: tan(45°) = 1 ⇒ arctan(1) = 45° = π/4 rad. Limits: arctan(±∞) → ±90°.

Arctan calculator specification

Version 1.0.7 · Engine tested

Calculation status

Review policy · Evidence

Definition
Arctangent is the inverse of tangent. For any real x, arctan(x) is the unique angle θ in (−π/2, π/2) (−90° to 90°) such that tan(θ) = x. Notation: arctan(x), atan(x), or tan⁻¹(x).
What it calculates
The principal real arctangent θ = arctan(x) in degrees and radians. Default display is degrees.
Inputs
  • Any finite real tangent value x
Outputs
  • deg: principal angle in degrees. Mathematical range (−90, 90); binary64 may equal ±90 for huge finite x
  • rad: the same angle in radians. Mathematical range (−π/2, π/2); binary64 may equal ±π/2 for huge finite x
Formula
θ = arctan(x); θ° = θ × 180/π
Assumptions
  • Real inverse tangent with principal branch (−π/2, π/2) (NIST DLMF §4.23(ii)). Infinity is not accepted as an input.
  • Opposite/adjacent in the UI is a right-triangle length convenience (opposite ≥ 0, adjacent > 0, same unit). It is not atan2. The published engine remains single-argument arctan(x). Use Decimal for a signed ratio, and Tan for the forward map.
  • When x is exactly a listed common value (0, ±1/√3, ±1, ±√3), the engine returns the exact algebraic angle such as 45° = π/4. Nearby floats are not snapped.
Units
  • Degrees (°)
  • Radians (rad)
Boundary conditions
  • Missing x returns MISSING_REQUIRED_INPUT.
  • Non-numeric, object, array, boolean, NaN, or Infinity inputs return INVALID_NUMBER.
  • arctan(0) = 0°; arctan(1) = 45°; arctan(√3) = 60°; arctan(−1) = −45°.
  • Large finite |x| approaches ±90°. The mathematical principal range is the open interval (−90°, 90°). IEEE-754 binary64 results may round to ±90° for extremely large finite inputs (for example 1e20 and Number.MAX_VALUE). When the result rounds to ±90°, tan(θ) cannot recover x and verification.passed may be false.
Numerical precision
  • Computation uses IEEE-754 binary64 Math.atan. Canonical deg/rad are that numeric result unless x is bitwise-equal to a listed common value.
  • Exact algebraic angles are returned only when x === a listed common value.
  • A value within 1e-9 of a common x may include a display hint but does not replace deg/rad.
  • Canonical deg/rad are not display-flushed. UI may show 0° for |θ°| below the display threshold (compact labels use 1e-12°; the result card uses 1e-15°). REST and SSR return the IEEE-754 values in { deg, rad, verification }.
  • Interactive evaluation runs in the browser. Shared URLs and REST use the same engine server-side.
  • The Result Card includes an engine-linked Live graph of θ = arctan(x). Unbounded domains use Overview / Focus (not Full range): Overview window −6 ≤ x ≤ 6, with faint horizontal +90° / −90° limit lines. Vertical axis labelled θ (°). Decimal mode has an engine-linked Quick input slider covering the Overview window [−6, 6] — that window is a control range, not the domain. Dragging it writes x and updates the result, reverse check, and graph through the same local engine. Opposite/adjacent is a local right-triangle convenience: adjacent > 0, opposite ≥ 0, same unit; not atan2.
Example
arctan(1) = 45° = π/4
Validation cases

19 published on this page · 10/10 tests · Production surface contract 3/3 · View evidence

  • x=1 → 45°
  • x=√3 → 60°
  • x=0 → 0°
  • x=-1 → −45°
  • x=1/√3 → 30°
  • x=-√3 → −60°
  • x=-1/√3 → −30°
  • x=1e-6 → ≈ 5.7296e-5° (not snapped to 0°)
  • x=1e10 → ≈ 89.99999999427042° (finite, not 90°)
  • x=-1e10 → ≈ −89.99999999427042° (finite, not −90°)
  • x=1e20 → 90° (binary64 rounding of the open-interval limit)
  • x=-1e20 → −90° (binary64 rounding of the open-interval limit)
  • x=Number.MAX_VALUE → 90° (binary64 rounding)
  • x=1e-20 → ≈ 5.729577951308232e-19° (canonical; not flushed to 0)
  • x=Infinity → error INVALID_NUMBER
  • x=NaN → error INVALID_NUMBER
  • x=not-a-number → error INVALID_NUMBER
  • empty x → error MISSING_REQUIRED_INPUT
  • ratio adjacent=0 (UI) → error (adjacent must be positive; published engine remains arctan(x))
Sources
  • NIST Digital Library of Mathematical Functions, Chapter 4 — Elementary Functions
    Supports: Inverse trigonometric functions as elementary inverses of sine, cosine, and tangent
  • NIST DLMF §4.23 — Inverse trigonometric functions; principal values
    Supports: Principal arctan on the real line with range (−π/2, π/2); degree output is θ × 180/π
  • CalculatorX mathematical conventions — Principal branch; exact-value recognition; ratio mode is UI-only
    Supports: Published engine is arctan(x) of one real argument (not atan2).
Calculation version
1.0.7

Background

Interpretation and common distinctions.

What arctan is

Arctangent is the inverse of tangent:

arctan(x) = tan⁻¹(x)

If tan(45°) = 1, then:

arctan(1) = 45° = (π)/4 rad

It answers: “which principal angle has this tangent?”

Tangent is periodic, so many angles share the same tangent. This calculator returns the principal value in (−π/2,π/2). Other solutions differ by kπ.

Domain and range

Real arctan
Domain all real x
Range (−π/2, π/2) rad ≈ (−90°, 90°)

As x → +∞, arctan(x) → 90°; as x → −∞, arctan(x) → −90°. In IEEE-754 binary64, extremely large finite |x| may round to ± 90°.

Right-triangle use

Opposite / adjacent mode is a right-triangle length convenience, not atan2.

  • opposite ≥ 0
  • adjacent > 0
  • both sides in the same unit

The published engine remains the single-argument map θ = arctan(x). Negative coordinates of a 2D vector need atan2, which this calculator does not compute.

θ = arctan(opposite/adjacent)

Example: opposite = 3, adjacent = 4 → arctan(0.75) ≈ 36.87° (an acute angle in (0°, 90°)).

Identities

arctan(−x) = −arctan(x)

tan(arctan(x)) = x

sin(arctan(x)) = x/(√(1+x²)), cos(arctan(x)) = 1/(√(1+x²))

For x > 0:

arctan(x) + arctan(1/x) = (π)/2

(and −π/2 when x < 0).

Optional calculus forms:

  • Derivative: d/dxarctan(x) = 1/(1+x²)
  • Integral: intarctan(x) dx = xarctan(x) − 1/2ln(1+x²) + C

Notation tip

tan⁻¹(x) means inverse tangent, not 1/tan(x). The reciprocal of tangent is cot(x). Prefer arctan when you want to avoid that ambiguity.

Use Tan to go angle → tangent, and Arcsin / Arccos for the other inverse functions.

CVP (Calculator Verification Protocol)

Math pilot under CVP 1.0 Proposed. Profile: core (pure inverse-tangent transform, not an engineering model).

  • Evidence Manifest: /evidence/math.arctan/1.0.7.cvp.json
  • Independent reference: O1 (NIST DLMF model) + O3 mpmath/MPFR expected values and numerical behavior + O2 live identities
  • Numerical policy: IEEE-754 binary64; ≤2 ULP vs O3; declared_before_evaluation=true
  • Cross-calculator (CVP-XCAL-01): tan(arctan(x)) ≈ x vs published Tan. Listed Arcsin / Arccos / Sin do not share that identity (N/A). Saturating binary64 endpoints (±90°) are also N/A for this identity.
  • Interfaces: UI (SSR), REST, and MCP must agree. A separate integration check covers URL → result → Live graph. ui-ssr is not a live browser session.
  • Expert review is optional and is not required for CVP Verified.

Other calculators in this family: Tan, Arcsin, Arccos, Sin .

tan Tan › asin Arcsin › acos Arccos › sin Sin ›
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Frequently asked questions

Key distinctions behind the calculation.

What is arctan?

Arctan (arctangent) is the inverse of tangent. It returns the principal angle whose tangent equals the input x.

What is the domain of arctan?

All real numbers. There is no [−1, 1] restriction like arcsin or arccos.

What range does arctan return?

Mathematically the principal value is in (−π/2, π/2) radians, or (−90°, 90°), approached as |x| → ∞. In IEEE-754 binary64, extremely large finite |x| may round to ±90°.

Is tan⁻¹ the same as 1/tan?

No. tan⁻¹(x) means inverse tangent (arctan), not the reciprocal (cotangent). Cotangent is 1/tan(x) or cot(x).

What is arctan(1)?

45°, or π/4 radians.

What is arctan(√3)?

60°, or π/3 radians.

Is atan the same as arctan?

Yes. atan is common in programming; arctan and tan⁻¹ are the usual math notations.

Is this atan2?

No. The published engine is arctan(x) of one real argument, with principal range (−90°, 90°). Opposite/adjacent only accepts positive right-triangle side lengths. For a 2D direction angle from signed coordinates, use atan2.