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Math calculator

Arccos Calculator

Calculate arccos(x) = cos⁻¹(x) for x in [−1, 1]. 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 · 2/2 declared partitions (arccos, invalid-domain) · Matrix
Known limitations
  • x ∈ [−1, 1]; principal range [0, π]
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
The principal real arccosine θ = arccos(x) in degrees and radians. Default display is degrees.
Scope
Real inverse cosine with principal branch [0, π] (NIST DLMF §4.23(ii)). Complex arccos is not returned.
Verification
Engine tested · Source checked · v1.0.4 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.4 · CVP protocol 1.0.0-proposed
CVP identity
8/8 property · digest 2dd501560e45
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 · 5/5 CVP boundary · 5/5 invalid · 8/8 property · 4/4 metamorphic · 5/5 round-trip · 30/30 O3 · 1/1 cross-interface · 4/4 URL→result→graph · 1/1 CVP contract · Manifest
Sources
Sources
Evidence
5 legacy golden · 5 legacy boundary · legacy regression suite · 9/9 oracle-backed golden · 5/5 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.4 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 = arccos(x) = cos⁻¹(x)
Degreesθ° = arccos(x) × 180/π
Radiansθ = arccos(x) (SI / calculus default)
iThe real arccos is defined only for −1 ≤ x ≤ 1. The principal value lies in [0, π].

How to use

1

Enter the cosine value x

x must satisfy −1 ≤ x ≤ 1. Share URLs include x, for example /calc/math/arccos?x=0.5.

2

Read degrees and radians

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

3

Check with cosine

Verify with cos(θ) ≈ x using the Cos calculator if needed.

Example calculations

Common configurations with formula and result.

ϟ

arccos(0.5)

Classic acute angle

cos(60°) = 0.5
60° = π/3 rad
ϟ

arccos(0)

Right angle

cos(90°) = 0
90° = π/2 rad
ϟ

arccos(1)

Zero angle

cos(0°) = 1
0° = 0 rad
ϟ

arccos(−1)

Straight angle

cos(180°) = −1
180° = π rad

Arccos common values

Common values at a glance.

xarccos(x) °arccos(x) rad
−1180°π
−√3/2 ≈ −0.8660254150°5π/6
−√2/2 ≈ −0.7071068135°3π/4
−0.5120°2π/3
090°π/2
0.560°π/3
√2/2 ≈ 0.707106845°π/4
√3/2 ≈ 0.866025430°π/6
10°0
i Example: cos(60°) = 0.5 ⇒ arccos(0.5) = 60° = π/3 rad.

Arccos calculator specification

Version 1.0.4 · Engine tested

Calculation status

Review policy · Evidence

Definition
Arccosine is the inverse of cosine. For x in [−1, 1], arccos(x) is the unique angle θ in [0, π] (0° to 180°) such that cos(θ) = x. Notation: arccos(x) or cos⁻¹(x).
What it calculates
The principal real arccosine θ = arccos(x) in degrees and radians. Default display is degrees.
Inputs
  • Real cosine value x ∈ [−1, 1]
Outputs
  • deg: principal angle in degrees, range [0, 180]
  • rad: the same angle in radians, range [0, π]
Formula
θ = arccos(x); θ° = θ × 180/π
Assumptions
  • Real inverse cosine with principal branch [0, π] (NIST DLMF §4.23(ii)). Complex arccos is not returned.
  • When x is exactly a listed common value (0, ±1/2, ±√2/2, ±√3/2, ±1), the engine returns the exact algebraic angle such as 60° = π/3. 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.
  • x < −1 or x > 1 returns NOT_REAL. Direct user input is not clamped onto the domain, even if |x| − 1 is only a few 1e-13.
  • arccos(1) = 0°; arccos(0.5) = 60°; arccos(0) = 90°; arccos(−1) = 180°.
Numerical precision
  • Computation uses IEEE-754 binary64 Math.acos. Canonical deg/rad are that numeric result unless x is bitwise-equal to a listed common value.
  • Exact algebraic angles (integer degrees and nπ/m radians) are returned only when x === a listed common value (for example x = 0.5 → 60° = π/3).
  • Direct inputs are not repaired onto [−1, 1]. x = 1.0000000000005 is NOT_REAL. Derived IEEE rounding (future a/b internals) may use ULP repair; this published engine does not.
  • A value within 1e-9 of a common x may include a display hint (≈ π/3) but does not replace deg/rad.
  • Otherwise, |θ°| < 1e-12 is cleaned to 0. REST and SSR return { deg, rad, verification }; exact inputs also set exact: true. The page shows Check: cos(θ) = x.
  • 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 = arccos(x) using the same Full range / Focus template as bounded inverse trig (domain −1 ≤ x ≤ 1; vertical axis labelled θ (°)).
Example
arccos(0.5) = 60° = π/3
Validation cases

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

  • x=0.5 → 60°
  • x=0 → 90°
  • x=1 → 0°
  • x=-1 → 180°
  • empty x → error MISSING_REQUIRED_INPUT
  • x=2 → error NOT_REAL
  • x=0.5000000009 → ≈59.99999994° (not snapped to 60°)
  • x=1.0000000000005 → error NOT_REAL
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 arccos on [−1, 1] with range [0, π]; degree output is θ × 180/π
  • CalculatorX mathematical conventions — Principal branch; exact algebraic values only for exact inputs
    Supports: Published engine is arccos(x). Direct inputs outside [−1, 1] are NOT_REAL with no domain clamp.
Calculation version
1.0.4

Background

Interpretation and common distinctions.

What arccos is

The arccosine function is the inverse of cosine:

arccos(x) = cos⁻¹(x)

If cos(60°) = 0.5, then:

arccos(0.5) = 60° = (π)/3 rad

It answers: “which angle (in the principal range) has this cosine?”

Domain and range

Real arccos
Domain x ∈ [−1, 1]
Range [0, π] rad = [0°, 180°]

Many programming libraries return radians by default; this tool shows degrees as the main result and radians alongside.

Relation to other inverse trig functions

For x ∈ [−1, 1]:

arccos(x) + arcsin(x) = (π)/2

Also, asec(x) = arccos(1/x) when |x| ≥ 1.

Use Cos to go the other direction (angle → cosine), and Arcsin / Arctan for the other inverse functions.

Notation tip

cos⁻¹(x) means inverse cosine, not 1/cos(x). The reciprocal of cosine is sec(x).

CVP (Calculator Verification Protocol)

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

  • Evidence Manifest: /evidence/math.arccos/1.0.4.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
  • 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: Cos, Arcsin, Arctan, Sin .

cos Cos › asin Arcsin › atan Arctan › sin Sin ›
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Frequently asked questions

Key distinctions behind the calculation.

What is arccos?

Arccos (arccosine) is the inverse of cosine. It returns the angle whose cosine equals the input x.

What is the domain of arccos?

For real numbers, x must be in [−1, 1]. Outside that interval the real arccos is undefined.

What range does arccos return?

The principal value is in [0, π] radians, or 0° to 180°.

Is cos⁻¹ the same as 1/cos?

No. cos⁻¹(x) means inverse cosine (arccos), not the reciprocal (secant). The reciprocal is written 1/cos(x) or sec(x).

Does the calculator use degrees or radians?

Both: the primary display is degrees; radians (and exact π labels for common inputs) appear in the explanation.

How is arccos related to arcsin?

For x in [−1, 1], arccos(x) + arcsin(x) = π/2 (90°).