Cos Calculator

What does this cos calculator compute?

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).

The published formula or method is y = cos(θ_rad); θ_rad = θ° × π/180 when unit is deg. The following references support the stated mathematics or method:

• NIST DLMF §4.14 — Trigonometric Functions: Definitions of sine, cosine, and tangent.

Which inputs and limits should you check?

Follow the selected angle unit and normalization policy. Degrees and radians are different representations of the same angle.

The worked example uses Classic half with cos(π/3). 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.cos v1.0.6 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 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

This calculator reduces degree input before conversion; radians use the declared symbolic-token or Math.cos path.

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/1.0.6.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 1.0.6 · CVP Proposed 1.0 · evidence revision 2026-10-27.full-review-v2. This revision uses signed degree reduction before conversion, strict raw inputs, and consistent ratio-mode execution.
• Expert review is optional and is not required for CVP Verified.

Numerical contract and limits

Angle mode accepts a finite binary64 angle in degrees (default) or radians. Numeric text must be decimal/scientific notation of at most 512 characters. Hexadecimal, grouped numbers, booleans and blank numbers reject. Nonzero text rounding to zero rejects NUMERIC_UNDERFLOW. Aliases theta, t, angle, thetaInput must agree; unit and unitInput normalize degree(s), °, radian(s), case and surrounding spaces. Unknown keys reject.

Degree angles are reduced by a signed remainder before quadrant reduction and conversion to radians; no proximity epsilon merges neighboring angles with a zero. Listed common degree identities use algebraic values. In radians, symbolic recognition is restricted to the exact accepted binary64 tokens 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, 7π/6, 5π/4, 4π/3, 3π/2, 5π/3, 7π/4, 11π/6, 2π, and their negatives. These tokens are floating-point approximations, not exact mathematical radians. Other radians use Math.cos.

Ratio mode uses mode=ratio, adjacent (alias adjInput) and hypotenuse (alias hypInput), with 0 ≤ adjacent ≤ hypotenuse and hypotenuse > 0. It returns adjacent/hypotenuse without an inverse/forward round trip. A nonzero ratio rounding to zero rejects RESULT_UNDERFLOW. Ratio mode rejects angle/unit fields; angle mode rejects side fields. All surfaces share this contract; a derived triangle angle is approximate.

99 independent Decimal420/620 Taylor/Machin and Fraction references cover common-angle neighbors, huge degrees/radians, subnormal inputs, aliases and ratio endpoints, with a 2 ULP fixed-vector ceiling and strict zero, sign and types. The legacy 18 tabulated O3 cases retain their own 2 ULP budget. These checks do not prove the continuous domain or bitwise equality across runtimes. Near zeros, relative conditioning is unbounded. Large source angles can lose information when first rounded to binary64. Cosine can legitimately round to ±1.

Display uses 12 significant digits, falling back to the unrounded binary64 text when rounding would change a nonzero or ±1 boundary. Graphs are illustrative samples using the same degree/radian function, not independent verification. The graph is omitted for |θ| > 1e6 and in ratio mode; the scalar result remains available. The slider only spans ±360° or ±2π. Unit selection reinterprets the current number; it does not silently convert or round the typed input.

Typing calculates locally. Sharing sends inputs in the URL; REST/MCP send inputs in request bodies, and these paths may enter server logs. Native CSV/JSON retain the full scalar and raw provided inputs. Verification and production state have separate identities; local verification does not establish that a candidate has been deployed.