Tangent Calculator

Welcome to the Tangent Calculator — a high-precision tool that calculates tan(θ) for any angle from 1 to 1000 decimal places. Enter an angle in degrees or radians, see a step-by-step MathJax-rendered solution, explore the interactive unit circle diagram, and get clear warnings at vertical asymptotes (90°, 270°, …).

What is the Tangent Function?

The tangent function (tan) is one of the six fundamental trigonometric functions. It is defined as the ratio of the sine to the cosine:

Geometrically, on the unit circle, tan(θ) represents the y-coordinate where the terminal side of angle θ intersects the vertical tangent line at x = 1. It also equals the slope of the radius line from the origin at angle θ.

Key Properties of Tangent

• Period: π radians (180°) — tan(θ + 180°) = tan(θ)
• Domain: All real numbers except θ = 90° + k·180° (where cos θ = 0)
• Range: All real numbers (−∞, +∞)
• Odd function: tan(−θ) = −tan(θ)
• Asymptotes: Vertical asymptotes at odd multiples of 90° (π/2 rad)

Tangent Values at Special Angles

Tangent Sign by Quadrant

The tangent function is positive where sine and cosine share the same sign, and negative where they differ:

Mnemonic: "All Students Take Calculus" (ASTC) — All trig functions positive in Q I; only Sin in Q II; only Tan in Q III; only Cos in Q IV.

Key Tangent Identities

How to Use This Calculator

1. Enter your angle: Type any numeric value in the angle field. International number formats are supported (e.g., 1.234 or 1,234).
2. Select angle unit: Choose Degrees or Radians.
3. Set precision: Enter the number of decimal places (1–1000). Standard is 10; scientific work may require 50–100+.
4. Click Calculate: View the tangent value, interactive unit circle diagram, step-by-step solution, and complete trig values.

Why High-Precision Tangent?

• Arbitrary precision: Go far beyond the typical 15–16 digit limit of standard calculators, up to 1000 decimal places.
• Research-grade accuracy: Powered by mpmath for reliable arbitrary-precision arithmetic.
• Asymptote awareness: Detects undefined points at 90° + k·180° (π/2 + k·π) and warns clearly.
• Educational output: MathJax-rendered formulas, unit circle visualization, and special angle detection.

Frequently Asked Questions

What is the tangent function in trigonometry?

The tangent function (tan) is defined as the ratio of the sine to the cosine: tan(θ) = sin(θ)/cos(θ). It represents the slope of the line from the origin at angle θ, and geometrically, it is the y-value where this line intersects the vertical tangent line x = 1 on the unit circle.

Where is the tangent function undefined?

Tangent is undefined at odd multiples of 90° (or π/2 radians): 90°, 270°, −90°, etc. At these angles, cos(θ) = 0, making the ratio sin(θ)/cos(θ) undefined. The tangent graph has vertical asymptotes at these points.

What are the tangent values at special angles?

The exact tangent values at special angles are: tan(0°) = 0, tan(30°) = √3/3 ≈ 0.577, tan(45°) = 1, tan(60°) = √3 ≈ 1.732, tan(90°) = undefined, tan(120°) = −√3, tan(135°) = −1, tan(150°) = −√3/3, and tan(180°) = 0.

Why does tan(45°) equal 1?

At 45°, both sine and cosine have the same value (√2/2). Since tan(θ) = sin(θ)/cos(θ), we get tan(45°) = (√2/2)/(√2/2) = 1. Geometrically, the radius line at 45° has a slope of exactly 1.

What is the period of the tangent function?

The tangent function has a period of π radians (180°), meaning tan(θ + 180°) = tan(θ). This is shorter than sine and cosine (which have period 2π/360°) because both sin and cos change sign after 180°, and their ratio stays the same.

In which quadrants is tangent positive or negative?

Tangent is positive in Quadrant I (0°–90°) and Quadrant III (180°–270°), where sine and cosine share the same sign. Tangent is negative in Quadrant II (90°–180°) and Quadrant IV (270°–360°), where sine and cosine have opposite signs.

Additional Resources

• Tangent – Wikipedia
• Trig function graphs – Khan Academy
• Tangent – Wolfram MathWorld