Math calculator

Tan Calculator

Q: 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.

Q: When is tan undefined?

Mathematically when cos(θ) = 0, i.e. θ = 90° + 180°k. This engine classifies an input as UNDEFINED when its distance to that pole is under 1e-12°. That is a numerical classification policy, not a claim that those floats are mathematically exact. Outside that window, nearby angles stay finite with a NEAR_SINGULARITY warning.

Q: What is the period of tan?

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

Q: 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.

Q: How do I find the angle from a tangent value?

Use inverse mode here (principal value in (−90°, 90°)) or the Arctan calculator for more detail.

Calculate tan(θ) = sin(θ)/cos(θ) in degrees or radians. Fast local calculation with shareable, machine-readable results.

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 (tan, invalid-domain, xcal) · Matrix
Numerical scope: ≤2 ULP vs O3 applies to the 16 published tabulated vectors (common angles away from poles plus 8 seeded random angles in (−50°, 50°)). It is not a guarantee over the entire real line, including poles, huge angles, or unlisted neighborhoods of zeros.
Known limitations: deg|rad; pole classification / snap / |y| cleanup are documented numerical policies
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: Real tangent only. Inverse is Arctan, not this published engine.
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 · Evidence 2026-09-08.policy
Verification revision: 2026-09-08.policy · 5/5 property · digest 494caf2c4f61
Legacy regression: 25/25 tests · Production surface contract 5/5
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 · 10/10 CVP boundary · 8/8 invalid · 5/5 property · 4/4 metamorphic · 4/4 round-trip · 16/16 O3 · 1/1 cross-interface · 11/11 cross-calculator · 4/4 URL→result→graph · 1/1 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 · 4/4 oracle-backed golden · 8/8 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)

iLibraries evaluate tan in radians. This tool converts degree input first. UNDEFINED is a documented 1e-12° pole classification around 90° + 180°k, not a claim of mathematical exactness. Nearby classified-finite angles return a large value with a NEAR_SINGULARITY warning. Inverse mode returns the principal angle in (−90°, 90°).

How to use

Choose tan(θ) or angle from tan

The published engine computes tan(θ). Inverse mode is a local convenience; use Arctan for the published inverse.

Pick degrees or radians

Default unit is degrees. Avoid odd multiples of 90° (undefined). Share URLs include theta and unit.

Read the result

Exact labels appear for listed common-angle snaps. Classified poles (distance < 1e-12° from 90° + 180°k) return UNDEFINED; nearby angles stay finite with a Near singularity warning.

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 1.0.4 · Engine tested

Calculation status

Engine tested 25/25 tests · Production surface contract 5/5
Named expert review Not performed
Calculation version 1.0.4

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: Real tangent only. Inverse is Arctan, not this published engine.
Documented pole classification: UNDEFINED when distance to 90° + 180°k is < 1e-12°. That threshold is a numerical classification policy, not a claim that those inputs are mathematically exact odd multiples of 90°. |cos(θ)| is not used as a pole test.
If the input is not classified as a pole and distance-to-pole < 0.01°, the engine returns a finite tan(θ) with warnings: [{ code: NEAR_SINGULARITY }]. status remains ok.
Documented common-angle snap: after reducing mod 360°, an input within 1e-12° of a listed common angle (0, ±30, ±45, ±60, …) is replaced by the algebraic value (0, ±1/√3, ±1, ±√3). This is a snap policy, not mathematical exactness. Nearby inputs such as 60.00000005° keep IEEE-754 Math.tan.
Documented cleanup: if no snap applies and |y| < 1e-12, the engine rewrites y to 0. That changes the IEEE-754 result for some non-zero inputs; it is a cleanup policy, not a mathematical identity.
Units: Dimensionless tangent value
Degrees or radians for the angle
Boundary conditions: Missing θ returns MISSING_REQUIRED_INPUT. Invalid unit returns INVALID_UNIT. Malformed numbers return INVALID_NUMBER.
tan(0°) = 0; tan(45°) = 1; tan(30°) = 1/√3; tan(60°) = √3; tan(−45°) = −1; tan(180°) = 0; tan(225°) = 1; tan(360°) = 0.
tan(90°), tan(270°), tan(−90°), tan(π/2 rad) return UNDEFINED.
tan(89°) and tan(91°) are large finite values, not UNDEFINED.
Periodicity: tan(θ + 180°) = tan(θ), e.g. tan(405°) = tan(45°) = 1.
Numerical precision: IEEE-754 binary64 Math.tan after converting degrees with θ × π/180, unless a documented pole classification, common-angle snap, or |y| cleanup applies.
Common-angle snap uses a 1e-12° classification window on the reduced angle. It is not bitwise identity with the listed degree, and it is not a claim that the input is mathematically exact.
Documented cleanup: remaining |y| < 1e-12 becomes 0 in the published REST/SSR number. The page may show an exact label only for listed snaps.
tan(θ + 180°) = tan(θ) is exact at listed snaps. Off-snap, degree → radian conversion can leave a few to tens of ULP; that identity is not a ≤2 ULP binary64 claim.
Interactive evaluation runs in the browser. Shared URLs and REST use the same engine server-side. CVP interface evidence is UI (SSR) plus URL → result → graph; hydration, slider, and history are not a live browser session.
The Result Card includes an engine-linked Live graph of y = tan(θ) using the same Overview / Focus template as Log. Vertical asymptotes at odd multiples of 90° are drawn as faint dashed lines; the curve is gapped across poles, not connected.
Example: tan(45°) = 1 (listed snap); 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 5/5 · 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. Pole classification, common-angle snap, and |y| cleanup are documented numerical policies, not mathematical exactness. Near-pole values stay finite with NEAR_SINGULARITY.
Calculation version: 1.0.4

Background

Interpretation and common distinctions.

What does this tan calculator compute?

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).

The published formula or method is y = tan(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?

Check the angle unit and the calculator’s policy near tangent singularities; finite-looking values do not remove a singularity.

The worked example uses Unit slope with tan(π/4). 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.tan v1.0.4 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 tangent is

Tangent is one of the six basic trigonometric functions. For an acute angle in a right triangle:

tan(θ) = opposite/adjacent

Equivalently:

tan(θ) = (sin(θ))/(cos(θ))

The name comes from a line segment tangent to the unit circle. The function is odd and has period 180° (π):

tan(−θ) = −tan(θ), tan(θ + 180°) = tan(θ)

When tan is undefined

tan(θ) is undefined whenever cos(θ) = 0:

θ = 90° + 180° k (k ∈ ℤ)

Examples: ± 90°, ± 270°, π/2, 3π/2.

Degrees or radians

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

Example: tan(π/4) = tan(45°) = 1.

Graph sketch

y = tan(θ)

Period 180° (π); odd function
Vertical asymptotes at odd multiples of 90° are drawn as faint dashed lines; the curve is gapped, not connected across the pole
Classified poles return UNDEFINED, not Infinity; nearby angles stay finite with a Near singularity warning

Use Arctan for the published inverse.

CVP (Calculator Verification Protocol)

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

Evidence Manifest: /evidence/math.tan/1.0.4.cvp.json
Reproduce: /evidence/math.tan/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 16 published tabulated vectors (common angles away from poles plus 8 seeded random angles in (−50°, 50°)). It is not a guarantee over the entire real line, including poles, huge angles, or unlisted neighborhoods of zeros.declared_before_evaluation=true
Cross-calculator (CVP-XCAL-01): arctan(tan θ) ≈ θ vs published Arctan on the principal range; 1 + tan²θ = sec²θ vs Cos; 1 + cot²θ = csc²θ vs Sin off zero. Listed Arcsin does not share those identities (N/A). Huge angles and poles 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 and is not a claim of live UI verified.
Versions stay distinct: calculation 1.0.4 · CVP Proposed 1.0 · evidence revision 2026-09-08.policy. Adding verification does not by itself change the tangent algorithm.
Public claim: CalculatorX CVP Verified · Core · Proposed 1.0 · Evidence available · Independent expert review not performed. That is CalculatorX's own verification, not a third-party audit.
Expert review is optional and is not required for CVP Verified.

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?