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 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 · 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 3/3
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 3/3 · 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 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.

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Frequently asked questions

Key distinctions behind the calculation.

What is the tangent function?