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Math calculator

Cos Calculator

Calculate cos(θ) in degrees or radians. 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 (cos, invalid-domain, xcal) · Matrix
Numerical scope
≤2 ULP vs O3 applies to the 18 published tabulated vectors (common angles 0°…180° plus 8 seeded random angles in (−50°, 50°)). It is not a guarantee over the entire real line, including huge angles or unlisted neighborhoods of zeros.
Known limitations
  • deg|rad; textbook snap
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
The real cosine y = cos(θ) for an angle in degrees or radians. Default unit is degrees.
Scope
Real cosine. The published engine is forward cos(θ). Adjacent/hypotenuse is a local triangle convenience (hypotenuse > 0, adjacent ≥ 0, adjacent ≤ hypotenuse). Use Arccos for the inverse API.
Verification
Engine tested · Source checked · v1.0.5 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.5 · CVP protocol 1.0.0-proposed · Evidence 2026-09-08.xcal
Verification revision
2026-09-08.xcal · 3/3 property · digest ece0bdda8ca9
Legacy regression
6/6 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
4/4 golden · 2/2 CVP boundary · 2/2 invalid · 3/3 property · 3/3 metamorphic · 4/4 round-trip · 18/18 O3 · 1/1 cross-interface · 8/8 cross-calculator · 4/4 URL→result→graph · 1/1 CVP contract · Manifest
Sources
Sources
Evidence
4 legacy golden · 2 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 2/2 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.5 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.

Degreescos(θ°) = cos(θ × π/180)
Radianscos(θ)
Trianglecos(θ) = adjacent / hypotenuse
iMost programming libraries take radians. This tool converts degree input to radians before evaluating. Algebraic exact values are used only when θ reduces to a listed identity (0°, 30°, 45°, …). Display may show ≈ 0 for tiny IEEE remainders; the published engine number is not rewritten.

How to use

1

Enter the angle θ

Angle mode takes θ in degrees or radians. Share URLs include theta and unit. An engine-linked Overview range slider on [−360°, 360°] (or [−2π, 2π]) writes θ through the same local engine. Domain remains all real θ.

2

Or use adjacent / hypotenuse

A local right-triangle convenience: hypotenuse > 0, adjacent ≥ 0, adjacent ≤ hypotenuse. Share URLs still use θ. Use Arccos for the published inverse.

3

Read cos(θ)

Exact labels appear only for listed identities (60°, 90°, …). The explainer reverse-checks with arccos of the principal value, even symmetry, or a 360° reduction — never arccos(y) ≈ θ when θ is outside [0°, 180°].

Example calculations

Common configurations with formula and result.

ϟ

cos(60°)

Classic half

cos(π/3)
0.5
ϟ

cos(0°)

Unit circle at (1, 0)

cos(0)
1
ϟ

cos(90°)

Unit circle at (0, 1)

cos(π/2)
0
ϟ

Adjacent / hypotenuse

cos = b/c

1/2
1/2 (θ = 60° principal)

Cosine common values

Common values at a glance.

θ (°)θ (rad)cos θ
180°π−1
150°5π/6−√3/2
135°3π/4−√2/2
120°2π/3−1/2
90°π/20
60°π/31/2
45°π/4√2/2
30°π/6√3/2
0°01
i Example: cos(60°) = 0.5. Values repeat every 360° (2π rad).

Cos calculator specification

Version 1.0.5 · Engine tested

Calculation status

Review policy · Evidence

Definition
Cosine of an angle θ is the ratio of the adjacent side to the hypotenuse in a right triangle, and the x-coordinate on the unit circle. Range is [−1, 1]. The function is even and 2π-periodic: cos(−θ) = cos(θ) and cos(θ + 360°) = cos(θ).
What it calculates
The real cosine y = cos(θ) for an angle in degrees or radians. Default unit is degrees.
Inputs
  • Angle θ (finite real)
  • Optional unit: deg (default) or rad
Outputs
  • cos(θ) in [−1, 1]
Formula
y = cos(θ_rad); θ_rad = θ° × π/180 when unit is deg
Assumptions
  • Real cosine. The published engine is forward cos(θ). Adjacent/hypotenuse is a local triangle convenience (hypotenuse > 0, adjacent ≥ 0, adjacent ≤ hypotenuse). Use Arccos for the inverse API.
  • Algebraic snap applies only when the reduced angle (mod 360°) is a listed identity such as 0°, 30°, 45°, 60°, 90°. Nearby floats, including 89.99999995°, keep IEEE-754 Math.cos.
Units
  • Dimensionless cosine value
  • Degrees or radians for the angle
Boundary conditions
  • Missing θ returns MISSING_REQUIRED_INPUT. Invalid unit returns INVALID_UNIT. Malformed numbers, NaN, and ±Infinity return INVALID_NUMBER.
  • cos(0°) = 1; cos(60°) = 1/2; cos(90°) = 0; cos(180°) = −1; cos(300°) = 1/2; cos(−θ) = cos(θ).
Numerical precision
  • IEEE-754 binary64 Math.cos after converting degrees with θ × π/180. That raw value is the published REST/SSR number unless θ reduces exactly to a listed identity.
  • Listed identities (0°, 30°, 45°, 60°, 90°, … mod 360°) return the algebraic value (so exact 90° is 0, not ~6e-17). There is no 1e-7° proximity snap.
  • Display may show ≈ 0 when |y| < 1e-12 and the input is not a listed identity. That rounding is UI-only and is not applied to the engine number.
  • The Result Card shows the degree → radian conversion next to the value (θ° = θ×π/180 rad) before cos(θ_rad). Shared URLs and REST use the same conversion.
  • The reverse check uses arccos principal value in [0°, 180°]. For negative θ it cites even symmetry; for |θ| ≥ 360° it cites 360° period; for θ in (180°, 360°) it shows the equivalent principal angle. It never writes arccos(y) ≈ θ when θ is outside [0°, 180°].
  • Domain is all finite real θ. Extremely large finite angles can lose low-order phase bits in binary64 before 2π reduction; the engine still returns Math.cos of that converted argument and does not claim a unique mathematical angle.
  • 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 y = cos(θ) using the same Overview / Focus template as Log (Overview window ±360° or ±2π; same degree/radian conversion as the engine). Angle mode has an engine-linked Overview range slider on [−360°, 360°] (or [−2π, 2π] in radians): dragging it writes θ and updates the result, reverse check, and graph through the same local engine. Adjacent/hypotenuse is a local triangle convenience: hypotenuse > 0, adjacent ≥ 0, adjacent ≤ hypotenuse.
Example
cos(60°) = 0.5; cos(π rad) = −1; cos(47.1°) ≈ 0.6807208689589178
Validation cases

22 published on this page · 6/6 tests · Production surface contract 3/3 · View evidence

  • theta=0, unit=deg → 1
  • theta=30, unit=deg → √3/2
  • theta=45, unit=deg → √2/2
  • theta=60, unit=deg → 0.5
  • theta=90, unit=deg → 0
  • theta=-47.1, unit=deg → cos(47.1°) (even); IEEE-754, no snap
  • theta=120, unit=deg → −0.5
  • theta=180, unit=deg → −1
  • theta=270, unit=deg → 0
  • theta=300, unit=deg → 0.5 (principal arccos = 60°)
  • theta=360, unit=deg → 1
  • theta=720, unit=deg → 1
  • theta=47.1, unit=deg → 0.6807208689589178 (IEEE-754)
  • theta=π, unit=rad → −1
  • theta=π/3, unit=rad → 0.5
  • theta=89.99999995, unit=deg → IEEE-754 Math.cos (not snapped to 0)
  • empty theta → error MISSING_REQUIRED_INPUT
  • theta=60, unit=foo → error INVALID_UNIT
  • theta=abc → error INVALID_NUMBER
  • theta=NaN → error INVALID_NUMBER
  • theta=Infinity → error INVALID_NUMBER
  • theta=1e16, unit=deg → IEEE-754 Math.cos after θ×π/180; low-order phase may be lost
Sources
  • NIST Digital Library of Mathematical Functions, Chapter 4 — Elementary Functions
    Supports: Cosine as a standard trigonometric function with period 2π
  • NIST DLMF §4.14 — Definitions and period of sine and cosine
    Supports: cos(θ) on the real line; even function; degree input converted via θ° × π/180
  • CalculatorX mathematical conventions — Degree default; listed-identity exact values; inverse is Arccos
    Supports: Published engine is forward cos(θ) only. Algebraic snap is identity-only; |y|<1e-12 display rounding is UI-only.
Calculation version
1.0.5

Background

Interpretation and common distinctions.

What cosine is

Cosine is one of the six basic trigonometric functions. For a right triangle:

cos(θ) = adjacent/hypotenuse

On the unit circle, cos(θ) is the x-coordinate after rotating by θ from the positive x-axis.

The result always lies in [−1, 1].

Degrees or radians

Unit Typical use Conversion
Degrees (°) Geometry, surveying θ(rad) = θ° × π/180
Radians Calculus, most code libraries θ° = θ(rad) × 180/π

This calculator evaluates cos in radians internally after converting degree input.

Periodicity and symmetry

cos(θ + 360°) = cos(θ), cos(θ + 2π) = cos(θ)

cos(−θ) = cos(θ)

(even function)

Inverse

To recover an angle from a cosine value, use Arccos (principal value in [0°, 180°]).

Graph sketch

y = cos(θ)

  • Period 360°; even: cos(−θ)=cos(θ)
  • At 0°, y=1; at 90°, y=0; at 180°, y=−1

Related: Sin, Tan, Arcsin.

CVP (Calculator Verification Protocol)

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

  • Evidence Manifest: /evidence/math.cos/1.0.5.cvp.json
  • Reproduce: /evidence/math.cos/reproduce — tabulated O3 inputs, expected values, generator, and re-run commands
  • 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 applies to the 18 published tabulated vectors (common angles 0°…180° plus 8 seeded random angles in (−50°, 50°)). It is not a guarantee over the entire real line, including huge angles or unlisted neighborhoods of zeros.declared_before_evaluation=true
  • Cross-calculator (CVP-XCAL-01): sin²θ + cos²θ = 1 vs published Sin; cos(arccos x) = x vs published Arccos for x ∈ (−1,1). Listed Tan / Arcsin do not share those identities (N/A). Huge angles are N/A after binary64 phase-bit loss.
  • 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.
  • Versions stay distinct: calculation 1.0.5 · CVP Proposed 1.0 · evidence revision 2026-09-08.xcal. Adding verification does not by itself change the cosine algorithm.
  • Expert review is optional and is not required for CVP Verified.

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

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

Key distinctions behind the calculation.

How do I find the cosine of an angle?

Enter the angle and select degrees or radians. The calculator converts to radians if needed and evaluates cos(θ).

Should I use degrees or radians?

Use whichever unit your problem states. The tool shows the equivalent in the other unit in the explainer.

Why is cos the same every 360°?

Cosine is periodic with period 360° (2π radians). Angles that differ by a full turn have the same cosine.

What is the range of cos?

Always between −1 and 1 inclusive.

What is cos(60°)?

Exactly 1/2 (0.5).

How do I go from cosine back to an angle?

Use the Arccos calculator for the principal value in [0°, 180°].