Math calculator

Arctan Calculator

Q: What is arctan?

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

Q: What is the domain of arctan?

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

Q: 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°.

Q: 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).

Q: Is atan the same as arctan?

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

Q: 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.

Calculate the principal arctangent in degrees and radians. Explore its graph, exact common angles, rounding limits, worked examples, and NIST references.

Instant result

Result

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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 · v1.0.9 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.9 · CVP protocol 1.0.0-proposed · Evidence 2026-10-16.inverse-trig-discovery-pins-v8
Verification revision: 2026-10-16.inverse-trig-discovery-pins-v8 · 8/8 property · digest 4fb2a8fe2cc5
Legacy regression: 10/10 tests · Production surface contract 6/6
Trust layers: Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
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 21/21. Error-path engine·REST·MCP 5/5 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Integration: URL → SSR result → Live graph current point (4/4). Hydration/slider/history are URL-canonical contracts, not a live browser session. Raw HTTP REST/JSON-RPC transport rejection 5/5; raw-number source and conflicting pins are transport-only checks, not engine/SSR input-source parity.
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 · 31/31 cross-interface · 4/4 cross-calculator · 4/4 URL→result→graph · 6/6 CVP contract · Manifest
Sources: NIST Digital Library of Mathematical Functions, Chapter 4 — Elementary Functions
NIST DLMF §4.23 — Inverse trigonometric functions; principal values
CalculatorX mathematical conventions — Principal branch; exact-value recognition; ratio mode is UI-only
Sources
Evidence: 7 legacy golden · 3 legacy boundary · legacy regression suite · 9/9 oracle-backed golden · 3/3 invalid · Artifact integrity PASS
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

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.

Read degrees and radians

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

Share the result

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

Example calculations

Common configurations with formula and result.

arctan(1)

Equal legs

tan(45°) = 1

45° = π/4 rad

arctan(√3)

60° angle

tan(60°) = √3

60° = π/3 rad

arctan(0)

Zero angle

tan(0°) = 0

0°

Opposite / adjacent

opp = 3, adj = 4

arctan(3/4)

≈ 36.870°

Arctan common values

Common values at a glance.

x	arctan(x) °	arctan(x) rad
−√3 ≈ −1.7320508	−60°	−π/3
−1	−45°	−π/4
−1/√3 ≈ −0.5773503	−30°	−π/6
0	0°	0
1/√3 ≈ 0.5773503	30°	π/6
1	45°	π/4
√3 ≈ 1.7320508	60°	π/3

i Example: tan(45°) = 1 ⇒ arctan(1) = 45° = π/4 rad. Limits: arctan(±∞) → ±90°.

Arctan calculator specification

Version 1.0.9 · Engine tested

Calculation status

Engine tested 10/10 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.9

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: Decimal strings are accepted and normalized to binary64 x. Nonzero decimal text that rounds to zero returns NUMERIC_UNDERFLOW; explicit zero remains valid.
REST/MCP accept only x and xInput; unknown fields return UNKNOWN_INPUT and conflicting aliases return AMBIGUOUS_INPUT.
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: Version pins reject unsupported API versions; shared pages use the current engine with an explicit mismatch notice. Historical execution and cross-runtime bitwise equality are not promised.
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. Tiny nonzero angles use scientific notation in the result card and graph labels; they are not displayed as an exact zero. 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 6/6 · 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: 1.0.9

Background

Interpretation and common distinctions.

What does this arctan calculator compute?

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).

The published formula or method is θ = arctan(x); θ° = θ × 180/π. The following references support the stated mathematics or method:

NIST DLMF §4.23 — Inverse Trigonometric Functions: Inverse sine, cosine, and tangent, including principal branches.

Which inputs and limits should you check?

Use the principal inverse-tangent branch and distinguish degree and radian outputs.

The worked example uses Equal legs with tan(45°) = 1. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.arctan v1.0.9 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

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/1.0.9.cvp.json
Independent reference: O1 (NIST DLMF model) + O3 precomputed mpmath tables (historical backend not inferred) for 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.

What happens with empty or extreme triangle lengths?

Enter both side lengths in the same unit. Empty lengths are errors, and a positive ratio that underflows to zero or overflows binary64 is rejected. The side-length convenience runs locally; shared links record the derived scalar x and reopen in Decimal mode.

How are request numbers and versions interpreted?

REST and MCP execute_calculation reject a nonzero raw JSON number such as 1e-400 with NUMERIC_UNDERFLOW before it becomes a zero calculation. Literal zero remains valid where the model permits it. Decimal strings preserve input text where declared, but do not extend binary64 precision; a value already sent as 0 cannot be recovered. exact: true identifies a recognized symbolic common-angle case for the normalized binary64 input. It does not promise lossless conversion of every source decimal or an exactly representable floating-point radian value.

Every supplied calculation-version pin must agree. Conflicting header/body pins return VERSION_PIN_CONFLICT (HTTP 409); the body does not override the header. See the request contract for accepted pin locations, input-source limits, and API contract 1.0.1. Calculation-model version and request-contract version are separate.

Other calculators in this family: Arccos Calculator, Arcsin Calculator, Cos Calculator, Pythagorean Theorem Calculator, Sin Calculator, Tan Calculator, Triangle Solver . Explore all Trigonometry & Triangles.

Frequently asked questions

Key distinctions behind the calculation.

What is arctan?