Math calculator

Tan Calculator

Calculate tangent in degrees or radians with stable angle reduction, clear pole errors, near-singularity warnings, full-precision exports and shareable results.

Instant result

Result

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Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · angle deg/rad, exact declared poles and finite near-singularity warning envelope

Input interpretation: Enter values to calculate.
Result: —
Verified scope: angle deg/rad, exact declared poles and finite near-singularity warning envelope
Assurance: Core
Declared partition coverage: PASS · 3/3 declared partitions (degree, radian, invalid) · Matrix
Numerical scope: 121fixedvectors
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: The real tangent y = tan(θ) = sin(θ)/cos(θ) when defined. Default unit is degrees.
Scope: Finite binary64 model. Common degree angles and17 listed normalized binary64 radian magnitudes within2π andtheir signed counterparts use symbolic recognition; generic radian values use Math.tan. No epsilon pole window or tiny-result cleanup.
Verification: Engine tested · Source checked · v1.0.5 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · angle deg/rad, exact declared poles and finite near-singularity warning envelope· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.5 · CVP protocol 1.0.0-proposed · Evidence 2026-10-28.meta-metrics-v2
Verification revision: 2026-10-28.meta-metrics-v2 · 2/2 property · digest 9a7f190fd7c1
Legacy regression: 25/25 tests · Production surface contract 6/6
Trust layers: Verification VERIFIED · Production STALE · Overall CVP state: VERIFIED_STALE under CVP 1.0.0-proposed
CVP status: STALE · attestation snapshot core 64820c88e5ca ≠ live cvp_core_sha256 9a7f190fd7c1 (2026-10-28.meta-metrics-v2)
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 28/28 (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.
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 121/121 golden · 10/10 CVP boundary · 36/36 invalid · 2/2 property · 2/2 metamorphic · 149/149 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Sources: NIST Digital Library of Mathematical Functions, Chapter 4 — Elementary Functions
NIST DLMF §4.14 — Trigonometric functions
CalculatorX mathematical conventions — Poles are UNDEFINED, not Infinity; inverse is Arctan
Sources
Evidence: 15 legacy golden · 10 legacy boundary · legacy regression suite · 121/121 oracle-backed golden · 36/36 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.

Ratiotan(θ) = opposite / adjacent

Sin / costan(θ) = sin(θ) / cos(θ)

Degreestan(θ°) = tan(θ × π/180)

Undefined whencos(θ) = 0 (θ = 90° + 180°k)

iDegrees reduce before conversion using stable tan/cotangent offsets. Only exact odd90-degree poles and listed radian pole tokens reject UNDEFINED; neighboring floats remain finite. Use Arctan for the inverse.

How to use

Enter an angle

Use finite decimal/scientific theta, at most512characters. Select degrees or radians. Aliases must agree.

Read the scalar and warning

Only exact declared pole tokens reject. A near-singularity warning is a finite result with heightened input sensitivity.

Share or export the complete result

Exports preserve full scalar, warnings and original input; Angle from tan opens the published Arctan calculator.

Example calculations

Common configurations with formula and result.

tan(45°)

Unit slope

tan(π/4)

tan(30°)

Classic

tan(π/6)

1/√3 ≈ 0.57735

tan(60°)

√3

tan(π/3)

√3 ≈ 1.73205

Undefined

tan(90°)

cos = 0

undefined

From tan

tan⁻¹(1)

principal

45° or π/4

Tangent common values

Common values at a glance.

θ (°)	θ (rad)	tan θ
−60°	−π/3	−√3
−45°	−π/4	−1
−30°	−π/6	−1/√3
0°	0	0
30°	π/6	1/√3
45°	π/4	1
60°	π/3	√3
90°	π/2	undefined

i Example: tan(45°) = 1; tan(π/4) = 1. Values repeat every 180° (π rad).

Tan calculator specification

Version 1.0.5 · Engine tested

Calculation status

Engine tested 25/25 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.5

Review policy · Evidence

Definition: Tangent of an angle θ is the ratio of opposite to adjacent in a right triangle, equal to sin(θ)/cos(θ), and the slope of the ray on the unit circle. It is undefined when cos(θ) = 0 (odd multiples of 90°). Period is 180° (π); the function is odd: tan(−θ) = −tan(θ).
What it calculates: The real tangent y = tan(θ) = sin(θ)/cos(θ) when defined. Default unit is degrees.
Inputs: Angle θ (finite real, not an odd multiple of 90°)
Optional unit: deg (default) or rad
Outputs: tan(θ), or error UNDEFINED at classified poles
Optional warning NEAR_SINGULARITY when θ is extremely close to a pole
Formula: y = tan(θ_rad); θ_rad = θ° × π/180 when unit is deg
Assumptions: Finite binary64 model. Common degree angles and17 listed normalized binary64 radian magnitudes within2π andtheir signed counterparts use symbolic recognition; generic radian values use Math.tan. No epsilon pole window or tiny-result cleanup.
Inverse tangent is the separate published Arctan capability. Typing calculates locally; shareURLs andREST/MCP send parameters andmayenterlogs.
Units: Dimensionless tangent value
Degrees or radians for the angle
Boundary conditions: Missing/null/blank numeric=>MISSING_REQUIRED_INPUT; badunit=>INVALID_UNIT; badnumber=>INVALID_NUMBER;conflictingaliases=>CONFLICTING_INPUT;unknownkeys=>UNKNOWN_INPUT.
Exact theta=90+180k degrees =>UNDEFINED. Listed radian ±π/2/±3π/2 tokens =>UNDEFINED. Adjacent finite floats remain finite with warnings if magnitude exceeds5729.57789313059.
Finite results are scalar numbers, or {value,warnings:[{code,severity,detail,suggestion}]} near singularity. Scalar limits are never silently replaced byzero.
Numerical precision: 121 independent Decimal420/620 Taylor/Machin reference vectors, 2ULP fixed-vector ceiling; zero/sign/types/warnings strict, actual ULP measured.16 legacy O3 values retained separately. No global/cross-runtime guarantee.
Signed remainder before degree conversion; reciprocal tangent near poles avoids overflow and pole-distance cancellation. Nonzero result rounding tozero rejects RESULT_UNDERFLOW. Nonzero raw decimal rounding tozero rejects NUMERIC_UNDERFLOW.
Display12 significant digits falls back tofullbinary64 if rounding erasesnonzero orchanges±1. Exports preservefullscalar,warnings,rawinput.
Nearpoles input sensitivity isunbounded; large binary64 spacing loses phase before computation. Graph is engine-linked illustration,not independent evidence; omitted for |theta|>1e6 or |tan(theta)|>8.
Example: tan(45°) = 1 (listed common-angle identity); tan(90°) is classified UNDEFINED; tan(89.99999995°) is a large finite value with NEAR_SINGULARITY
Validation cases: 17 published on this page · 25/25 tests · Production surface contract 6/6 · View evidence
theta=45, unit=deg → 1
theta=30, unit=deg → 1/√3
theta=60, unit=deg → √3
theta=0, unit=deg → 0
theta=-45, unit=deg → -1
theta=180, unit=deg → 0
theta=225, unit=deg → 1
theta=360, unit=deg → 0
theta=405, unit=deg → 1 (periodicity tan(θ+180°)=tan(θ))
theta=89, unit=deg → ≈57.29 (finite, not UNDEFINED)
theta=90, unit=deg → error UNDEFINED
theta=270, unit=deg → error UNDEFINED
theta=-90, unit=deg → error UNDEFINED
theta=π/4, unit=rad → 1
theta=π/2, unit=rad → error UNDEFINED
theta=89.99999995, unit=deg → finite value + warning NEAR_SINGULARITY
empty theta → error MISSING_REQUIRED_INPUT
Sources: NIST Digital Library of Mathematical Functions, Chapter 4 — Elementary Functions
Supports: Tangent as sin/cos with period π and poles where cos = 0
NIST DLMF §4.14 — Trigonometric functions
Supports: tan(θ) undefined at odd multiples of π/2; degree input converted via θ° × π/180
CalculatorX mathematical conventions — Poles are UNDEFINED, not Infinity; inverse is Arctan
Supports: Published engine is forward tan(θ) only. Only exact degree poles and listed radian pole tokens reject, without proximity epsilon or tiny-result cleanup. Nearby floats stay finite and may carry NEAR_SINGULARITY.
Calculation version: 1.0.5

Background

Interpretation and common distinctions.

Tangent is sin(theta)/cos(theta), with period180° orπ. Enter finite decimal/scientific text (maximum512characters); hexadecimal, booleans, grouped numbers and conflicting aliases reject. Null or empty numeric fields reject. A nonzero text token such as1e-400 rejects NUMERIC_UNDERFLOW; nonzero result underflow rejects RESULT_UNDERFLOW.

Degrees reduce before conversion. Exact odd90-degree poles reject; the listed binary64 radian tokens ±π/2 and±3π/2 also reject by symbolic policy. Nearby floats remain finite. A magnitude above5729.57789313059 yields a full finite value and NEAR_SINGULARITY warning, not an undefined result.

The primary result uses12 significant display digits, with a full-binary64 fallback at nonzero and±1 boundaries. Native CSV/JSON preserve the full scalar, complete warning envelope and original raw input. Angle summaries preserve the full accepted binary64 value.

The graph is omitted when |theta|>1e6 or |tan(theta)|>8. Large angle spacing can lose phase information before calculation; near poles small input changes are strongly amplified. Graphs and inverse calculations are not independent verification. For an angle from a tangent, use the published Arctan calculator, with its own validation, exports and sharing.

Model sources: NIST DLMF4.14, DLMF4.19 sine/cosine series. The numerical scope is 121 independent CPython Decimal420/620 Taylor/Machin fixed references, strict zero/sign/types/warnings and2ULP ceiling;16 old O3 numeric values remain separately checked. This is not a full-domain or cross-runtime error proof or external certification.

Typing stays local. Shared URLs send parameters to the server and may enter logs; REST/MCP also send inputs. Capability, Evidence1.0.5, CVP Manifest. CalculatorX maintains this page; verification and production states must be read separately.

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

Frequently asked questions

Key distinctions behind the calculation.

What is the tangent function?

In a right triangle, tan(θ) = opposite / adjacent. Equivalently tan(θ) = sin(θ)/cos(θ). On the unit circle it relates to the slope of the ray at angle θ.

When is tan undefined?

Exact odd multiples of90 degrees and the listed binary64 radian tokens ±π/2,±3π/2 reject UNDEFINED. Adjacent floats are not snapped. A finite magnitude above5729.57789313059 carries NEAR_SINGULARITY.

What is the period of tan?

180° or π radians: tan(θ + 180°) = tan(θ).

Is tangent odd?

Yes. tan(−θ) = −tan(θ).

What is tan(45°)?

Exactly 1.

Should I use degrees or radians?

Use the unit in your problem. Most programming and statistics libraries take radians; convert with θ_rad = θ° × π/180.

How do I find the angle from a tangent value?

Use the linked Arctan calculator for the principal angle, with its own validated inputs and sharing.